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INDEX
Aabbreviations, 1627Abel equation, 443, 525active functions, 1628adding integration constants, 1665additive separable solution, 1374, 1459additively separable, 1651, 1710adiabatic equation, 1262adiabatic exponent, 1262adiabatic nonisentropic gas flow, 1204adsorption coefficients, 1231Airy equation, 861Airy functions, 1193algebraic equations, 1441algebraic equations with even powers, 1441algebraic systems of equations, 1441alternative definitions of functions, 1628analysis of some ordinary differential
equations, 1447analytical derivation of numerical methods,
1680, 1729analyticalnumerical solutions, 1680, 1728analytical solutions
of nonlinear systems, 1659, 1719visualizations, 1629, 1691
anonymous functions, 1741ansatz methods for constructing traveling
wave solutions, 1641, 1700antikink, 1636, 1695antisoliton, 1636, 1695application program interface, 1736applied shear stress at boundary, 1252arbitrary functions (included in PDEs)
dependent ondifferent arguments, 1164linear combination of unknowns, 1133,
1157, 1173, 1178, 1185product of powers of unknowns, 1145,
1162ratio of unknowns, 1137, 1159, 1175,
1181, 1187sum or difference of squares of unknowns,
1146, 1163unknowns in complex way, 1148
arithmetic operators, 1627, 1738array, 1629assignment/unassignment operators, 1627associativity equations, 972autoBacklund transformation, 681, 1413,
1635, 1695axial flows, 1303axisymmetric fluid flows, 1274axisymmetric jet, 910
boundary layer approximation, 1335
axisymmetric steady laminar hydrodynamicboundary layer equations, 909
axisymmetric unsteady laminar boundarylayer equations, 929
BBacklund transformations, 1413, 1635, 1695
autoBacklund transformations, 681, 1413,1635, 1695
canonic for evolution equations, 1420constructed via RF pairs, 1415for secondorder equations, 1413
basic arithmetic operators, 1689basic relations used in symmetry analysis of
systems of equations, 1527basic steps of Painleve test for nonlinear
equations, 1570basic transformation rules, 1689BBM equation, 958Bellman type equations and related equa
tions, 805Beltarmi surface, 253Benney moment equations, 764Bernoulli equation, 394, 421Bessel functions, 1160, 1255, 1259bilinear functional equations, 1775Blasius problem, 904, 1333block, 1691blowup regimes, 216boolean, 1739boolean expression, 1628, 1689Born–Infeld equation, 766, 1401boundary conditions, 1752boundary layer
approximation for axisymmetric jet, 1335axisymmetric steady laminar hydrody
namic, 909axisymmetric unsteady laminar, 929equations, 903for nonNewtonian fluid, 915nonlinear diffusion equations, 346, 402nonlinear thermal equations, 346on flat plate, 1333on Vshaped body, 1334steadystate equations, 903steadystate equations for Newtonian
fluid, 903steadystate equations for nonNewtonian
fluids, 911steadystate laminar, 903symmetries of steadystate equations, 1528unsteady, 917unsteady equations for Newtonian
fluid, 917
Page: 1841
1841
1842 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
boundary layer (continued)unsteady equations for nonNewtonian
fluids, 930with pressure gradient, 907
boundary layer equation for nonNewtonianfluid of general form, 917
boundary layer equation for powerlaw fluidwith pressure gradient, 913
boundary layer equations (Prandtl equations),1333
boundary value problem, 1679, 1727Boussinesq equation, 368, 1504
in canonical form, 987modifications, 987
Boussinesqtype equation, 1028Boussinesq transformation, 1289breaking soliton equation, 1030breather, 492brief introduction to Maple, 1625brief introduction to Mathematica, 1687brief introduction to MATLAB, 1735builtin functions, 1740Burgers equation, 186, 1409, 1414, 1572,
1637, 1673, 1677, 1697, 1723, 1725,1743, 1761
Burgers–Huxley equation, 189Burgers–Korteweg–de Vries equation, 884Burgers vortex, 1290
CCalogero equation, 745, 746Calogero typeequation, 1400Camassa–Holm equation, 968Camassa–Holm multipeakon solutions, 965canonic Backlund transformations of
evolution equations, 1420canonical form
Boussinesq equation, 987for systems of equations, 1611Kadomtsev–Petviashvili equation, 1000Korteweg–de Vries equation, 857modified Korteweg–de Vries equation, 867of elliptic equations, 1392of gas dynamics systems, 1611of hyperbolic equations, 1390–1392of parabolic equations, 1390
Cauchy–Lagrange integral, 1201Cauchy problem, 1357, 1379, 1611
for Hamilton–Jacobi equation, 1380qualitative features of solutions, 1611
centered rarefaction wave, 1359Chaplygin equations, 1202Chaplygin gas equation, 765characteristic, 1393characteristic curves, 1647characteristic equation, 1389, 1393, 1647,
1706characteristic system, 1355characteristic velocity, 1615Charpit equations, 1647, 1706
checking whether nonlinear systems ofequations of mathematical physics passPainleve test, 1576
chromatography equations, 1231Clairaut equation, 118, 152, 172
in implicit form, 122partial differential, 1375
Clarkson–Kruskal direct method, 1503classic worksheet, Maple, 1626classical Cauchy problem, 1647, 1707, 1709,
1756classical Couette–Poiseuille solution, 1263classical hodograph transformation, 1400classical method of separation of variables,
1651, 1711classical method of symmetry reductions,
1513classical selfsimilar solutions, 1435classical travelingwave solutions, 1431classification of secondorder nonlinear
equations, 1389classification results for nonlinear first and
secondorder equations, 1565cnoidal waves, 858coefficient of reflection, 1588coefficients of equations contain
exponential functions, 11hyperbolic functions, 14logarithmic functions, 16powerlaw functions, 3trigonometric functions, 17
Cole–Hopf transformation, 1573for Burgers equation, 1637, 1696
combined KdVmKdV equation, 871commutator of operators, 1579compacton, 889comparing exact solution generation
methods, 1662comparison of asymptotic and numerical
solutions, 1684, 1733compatibility condition, 1581complete elliptic integral, 858complete integral, 1373, 1647, 1650, 1706,
1709complete Lin–Reissner–Tsien equation, 757complete solution, 1647, 1706complex separation of variables in nonlinear
partial differential equations, 1462composite data types, 1741condition
compatibility, 1581condition, for integrability of Pfaffian
equation by single relation, 1601Courant–Friedrichs–Levy, 1763invariance, 1516, 1527, 1595invariant, 1527invariant surface, 1533jump, 1362, 1363, 1369Lax, 1616, 1617noslip, 1253Rankine–Hugoniot, 1126, 1128, 1759Rankine–Hugoniot jump, 1363, 1369, 1615
Page: 1842
INDEX 1843
condition (continued)stability, 1369stability, for generalized solution, 1369strip, 1651, 1710transversality, 1380
conditionsboundary, 1752evolutionary, 1616
conical vortex flow, 1290connection between method of differential
constraints and other methods, 1561conservation form, 1678, 1725, 1762, 1763conservation law at discontinuity, 1363conservation laws, 1365, 1593
for some nonlinear equations of mathematical physics, 1593
constructing analytical solutions in terms ofpredefined functions, 1629
constructing complete integral using two firstintegrals, 1377
constructing exact solutions using symboliccomputation, 1662
constructing finitedifference approximations,1677, 1725, 1760
constructing generalized separable solutions,1661, 1721
constructing numerical solutions, 1683, 1731in terms of predefined functions, 1672,
1722constructing selfsimilar solutions, 1645,
1705constructing separable solutions, 1651, 1710constructing solutions along characteristics,
1647, 1706constructing solutions of nonlinear equations
that fail Painleve test, using truncatedexpansions, 1577
constructing solutions using predefinedfunctions, 1692
constructing solutions via transformations,1635, 1694
constructing travelingwave solutions, 1637,1659, 1697, 1719
constructive solid geometry, 1754contact transformations, 1403
for ordinary differential equations, 1403continuity equation, 1197, 1247, 1610continuous point group, 1526control structures, 1741convective fluid motions, 1329coordinates of first and second prolongations,
1515Couette–Poiseuille plane flow, 1251Couette–Poiseuille type solution, 1283Courant–Friedrichs–Levy condition, 1763Crocco transformation, 906, 909, 1422Crocco variables, 748, 1107cylindric Burgers equation, 191cylindrical Korteweg–de Vries equation, 866cylindrical vortex flow, 1290
Ddata structures, 1741data types, 1690decomposed geometry, 1752Degasperis–Procesi equation, 968description of simplified scheme for
constructing solutions based onpresetting one system of coordinatefunctions, 1465
desktop, 1737determining equation, 1555development environment, 1736different types of cells, 1688differential constraints, 1539, 1545
of arbitrary order, 1542differential equation
adiabatic, 1262Airy, 861BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian
fluid, 917boundary layer, for powerlaw fluid with
pressure gradient, 913Boussinesq, 368, 987, 1504Boussinesq, in canonical form, 987Boussinesqtype, 1028breaking soliton, 1030Burgers, 186, 1409, 1414, 1572, 1637,
1673, 1677, 1697, 1723, 1725, 1743,1761
Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400Camassa–Holm, 968Chaplygin, 765characteristic, 1389, 1393, 1647, 1706Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757continuity, 1197, 1247, 1610cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968determining, 1555double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392Emden–Fowler, 248, 442, 444, 447, 462,
464, 647, 648, 655Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,
346fast diffusion, 211firstorder quasilinear, 1355first Painleve, 861, 989
Page: 1843
1844 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
differential equation (continued)Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouville type, 1030fourth Painleve, 988Fujita–Storm, 212Gardner, 871Gauss, 1772general fifthorder KdV, 1034generalized Burgers, 288generalized Burgers–Korteweg–de
Vries, 1045, 1503generalized Calogero, 748generalized Emden–Fowler, 327, 477, 511,
512, 513, 516, 527, 528, 697, 698generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878Helmholtz, 380, 397, 399, 420, 423, 558,
570, 601, 938, 1289homogeneous Monge–Ampere, 775, 1430Hopf, 5Hunter–Saxton, 745hydrodynamic boundary layer, with
pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392hypergeometric, 1012integrating factor of Pfaffian, 1601inviscid Burgers, 1674, 1763inviscid Burgers (explicit method), 1678,
1725inviscid Burgers (initial value problem),
1762inviscid Burgers (shock waves and
rarefaction waves), 1757inviscid Burgers (shock waves and weak
solutions), 1759Ito, 1037Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical
form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749KK, 1036Klein–Gordon, 1641, 1656, 1701, 1716
k(n,n), 891Kolmogorov–Petrovskii–Piskunov
(KPP), 279, 1633, 1694Korteweg–de Vries, 857, 1574, 1580,
1583, 1638, 1698Korteweg–de Vries–Burgers, 884Korteweg–de Vries, in canonical form, 857Korteweg–de Vries type, with exponential
nonlinearity, 874Korteweg–de Vries type, with logarithmic
nonlinearity, 874Korteweg–de Vries type, with powerlaw
nonlinearity, 872KPPtype, 1633, 1694Kuramoto–Sivashinsky, 982Laplace, 394, 409, 421, 556, 558, 579,
646, 938, 1392Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759linear determining, 1556, 1557linear heat, 807, 1252linear Schrodinger, 865, 1186linear wave, 1392Liouville, 144, 165, 470, 540, 1117, 1414Liouville type, nthorder, 1109mKdV, 867model, for dynamics of nonlinear
strings, 534model, of gas dynamics, 5, 25, 1360model, of nonlinear waves with damping,
6modification of coupled KdV, 1172modified Burgers, 191modified Burgers–Korteweg–de Vries, 886modified Harry Dym, 891modified k(n,n), 892modified Korteweg–de Vries, 862, 866,
869, 1583modified Korteweg–de Vries–Burgers, 886modified Korteweg–de Vries, in canonical
form, 867Monge–Ampere, 774, 1393ndimensional modified Schrodinger, with
cubic nonlinearity, 431ndimensional Schrodinger, with cubic
nonlinearity, 430Newell–Whitehead, 1534ninthorder KdVtype, 1041nonlinear diffusion, 1645, 1705nonlinear diffusion, with cubic source,
1577nonlinear firstorder partial differential,
1373nonlinear Klein–Gordon, 499, 1494nonlinear Kolmogorov type, 377, 398nonlinear Poisson, 1679, 1727, 1750nonlinear Poisson–Boltzmann, 1755nonlinear Schrodinger, 348, 425, 1582nonlinear wave, 1400, 1519, 1536, 1673,
1678, 1723, 1727, 1764
Page: 1844
INDEX 1845
differential equation (continued)nthorder Liouville type, 1109of axisymmetric steady hydrodynamic
boundary layer, 1508of Burgers hierarchy, 1067of complete Korteweg–de Vries hierarchy,
1069of helical surface, 8of Korteweg–de Vries hierarchy, 1068of light rays, 63of minimal surfaces, 768, 1597of modified Korteweg–de Vries hierarchy,
1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756onesoliton solution, mKdV, 867onesoliton solution, potential KdV, 876Ostrovsky, 968palindromic, 1442Pfaffian, 1600Pfaffian, condition for integrability by
single relation, 1601Plebanski, first heavenly, 802Plebanski, second heavenly, 803Poisson, 379, 397, 423, 580, 588, 938,
1008porous medium, 210potential, of onedimensional flow of
compressible gas, 768potential Korteweg–de Vries (KdV), 876potential modified Korteweg–de Vries
(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262Riccati, 140, 142, 162, 164, 423, 425, 453,
482, 520, 1012, 1277Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, of general form, 358, 360,
363, 364Schrodinger, with cubic nonlinearity, 348,
350, 351, 352, 355Schrodinger, with powerlaw nonlinearity,
351, 353Schwarzian Korteweg–de Vries, 870secondorder nonlinear partial differential,
1392secondorder quasilinear partial differential,
1393second Painleve, 861, 867, 988, 989seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,
1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649telegraph, 494, 538, 583, 585thin film, 985
thirdorder Liouville type, 972Thomas, 544threedimensional Khokhlov–Zabolotskaya,
752threedimensional nonlinear Schrodinger,
of general form, 429threedimensional Schrodinger, with cubic
nonlinearity, 428tractrix, 253Tricomi, 659twodimensional Khokhlov–Zabolotskaya,
749twodimensional linear heat, 1256twodimensional nonlinear Schrodinger, of
general form, 427twodimensional Schrodinger, with cubic
nonlinearity, 425twodimensional Schrodinger, with
powerlaw nonlinearity, 427Tzitzeica, 541unnormalized Boussinesq, 990unnormalized Burgers, 187unnormalized Kadomtsev–Petviashvili,
1002unnormalized Korteweg–de Vries, 865unnormalized modified Korteweg–de
Vries, 870variant of modified Burgers–Korteweg–de
Vries, 886vector Burgers, 417Weber, 1193Yermakov, 309, 518
differential equationsadmitting variational form, 1595associativity, 972axisymmetric steady laminar hydrodynamic
boundary layer, 909axisymmetric unsteady laminar boundary
layer, 929basic steps of Painleve test, 1570Bellman type, 805Benney moment, 764boundary layer (Prandtl equations), 1333elliptic, canonical form, 1392evolution, canonic Backlund transforma
tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231coefficients contain exponential functions,
11coefficients contain hyperbolic functions,
14coefficients contain logarithmic functions,
16coefficients contain powerlaw functions, 3coefficients contain trigonometric
functions, 17cubic in derivatives, 134Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201
Page: 1845
1846 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
differential equations (continued)Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197evolution, nonlinear in second derivative,
346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder with two independent variables
quadratic in derivatives, 43firstorder with three or more independent
variables, 125firstorder with two independent variables
of general form, 99firstorder quasilinear, 3for convective fluid motions, 1329fourthorder, 977heat and mass transfer, 697heat and mass transfer, in quiescent
or moving media with chemicalreactions, 390
heat and mass transfer, with complicatingfactors, 734
heat, with powerlaw or exponentialtemperaturedependent thermaldiffusivity, 409
hydrodynamic boundary layer, 903hydrodynamic type, order reduction, 1422hyperbolic, canonical form, 1390invariance under scaling transformations,
1431invariance under translations, 1429involving arbitrary differential operators,
1101involving arbitrary functions, 77, 279, 390,
499, 546, 682involving arbitrary functions of derivatives,
113involving arbitrary functions of four
variables, 123involving arbitrary functions of indepen
dent variables, 107involving arbitrary functions of one
variable, 19, 113involving arbitrary functions of three
variables, 120involving arbitrary functions of two
variables, 30, 116, 168involving arbitrary parameters, 43, 540involving arbitrary powers of derivatives,
102, 136involving cosine, 276involving cotangent, 278involving derivatives in radicands, 101involving exponential nonlinearities, 587involving first derivative in t, 811, 857,
977involving first derivative in t and linear in
highest derivative, 1031involving fourth powers of derivatives, 99involving hyperbolic cosine, 268
involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,
279involving mixed derivatives, 1000, 1103involving one arbitrary power of derivative,
107involving powerlaw functions of
derivatives, 156involving powerlaw nonlinearities, 583involving products of derivatives, 133involving products of derivatives with
respect to different variables, 131involving roots and moduli of derivatives,
136involving second derivative in t, 896, 987,
1088involving secondorder mixed derivatives,
949involving sine, 277involving squares of derivatives, 133involving squares of one or two deriva
tives, 125involving squares of three derivatives, 130involving tangent, 278involving thirdorder mixed derivatives,
958involving two or more second derivatives,
839involving two or three arbitrary powers of
derivatives, 111linear in mixed derivative, 745, 847mathematical physics, Painleve test, 1565mathematical physics, transformations,
1395Navier–Stokes, 1247Navier–Stokes, general properties, 1249Navier–Stokes, in Boussinesq approxima
tion, 1329Navier–Stokes, in various coordinate
systems, 1247nonlinear, involving arbitrary parameters,
99nonlinear diffusion boundary layer, 402nonlinear elliptic, geometrical models,
1754nonlinear elliptic, in two space dimensions,
1748nonlinear, in highest derivatives, 404nonlinear, in two independent variables,
1392nonlinear, involving arbitrary linear
differential operators, 1065nonlinear, of general form, 150, 169, 1392nonlinear, of thermal (diffusion) boundary
layer, 346nonlinear, parabolic, 1677, 1725nonlinear, secondorder, 1516nonlinear, simple separation of variables,
1459
Page: 1846
INDEX 1847
differential equations (continued)nonlinear telegraph, with two space
variables, 583nonlinear, with arbitrary number of
variables involving arbitrary functions,162
nonlinear, with arbitrary number ofvariables involving arbitrary parameters,158
nonlinear, with four independent variables,154
nonlinear, with three variables involvingarbitrary functions, 139
nonlinear, with three variables quadratic inderivatives, 125
nonstationary, 942nonstationary hydrodynamic (Navier–
Stokes equations), 1013nonstationary, of motion of viscous
incompressible fluid, 1013of atmospheric circulation in equatorial
region, 1225of breeze and monsoons, 1223of Burgers and Korteweg–de Vries
hierarchies, 1067of dynamic convection in the sea, 1227of flows in the baroclinic layer of the
ocean, 1229of general form, involving arbitrary
functions of single argument, 358of general form, involving arbitrary
functions of two arguments, 362of general form, involving first derivative
in t, 1070of heat and mass transfer in anisotropic
media, 412of higher orders, 1031of ideal incompressible fluid, 942of ideal plasticity, 1238of mass transfer in quiescent or moving
media with chemical reactions, 406of motion of ideal fluid (Euler equations),
938, 1197of nonlinear vibrations stratified medium,
1131of sixth to ninthorder, 1039of stationary transonic planeparallel gas
flow, 1122of viscous incompressible fluid, 1003ordinary differential, 1445ordinary differential, Painleve test, 1566Painleve, 1566parabolic, canonical form, 1390passing Painleve test, 1572Pfaffian, 1600Pfaffian, not satisfying integrability
condition, 1602Plebanski heavenly, 802quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772quasilinear, 1393, 1647, 1649, 1707, 1708
quasilinear, discontinuous solutions, 1360quasilinear, in conservative form, 1368quasilinear, of general form, 1368quasilinear, qualitative features, 1360quasilinear, with n independent variables,
1356reducible to Korteweg–de Vries equation,
875secondorder, classification, 1389secondorder, elliptic, with three or more
space variables, 713secondorder, elliptic, with two space
variables, 641secondorder, evolution, 1547secondorder, hyperbolic, 1551secondorder, hyperbolic, with one space
variable, 433secondorder, hyperbolic, with two or more
space variables, 553secondorder, involving mixed derivatives
and some other, 745secondorder, involving real functions of
real variables, 1165secondorder, nonlinear, for laser systems,
1170secondorder, of general form, 811, 1553secondorder, parabolic, with one space
variable, 175secondorder, parabolic, with two or more
space variables, 367semilinear, in two independent variables,
1389shallow water, 1124stationary, 938stationary hydrodynamic, (Navier–Stokes
equations), 1003steady boundary layer, for nonNewtonian
fluids, 911steady hydrodynamic boundary layer, for
Newtonian fluid, 903steady hydrodynamic boundary layer,
symmetries, 1528steadystate hydrodynamic boundary
layer, 903thirdorder, 857threeargument functional, 1497threedimensional, 1240threedimensional, involving arbitrary
functions, 730threedimensional, of ideal plasticity, 1240three and ndimensional, 428with arbitrary dependence on derivatives,
148with arbitrary number of independent
variables, 39with cubic nonlinearities involving
arbitrary functions, 355with exponential nonlinearities, 254, 469,
662with hyperbolic nonlinearities, 268, 485,
675
Page: 1847
1848 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
differential equations (continued)with logarithmic nonlinearities, 271, 388,
486, 677with n independent variables, 739, 853with n space variables, 417with powerlaw nonlinearities, 175, 433,
641, 808with powerlaw nonlinearity in derivatives,
161, 167with power nonlinearity in derivatives, 146with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables
involving arbitrary functions, 624with three space variables involving
arbitrary parameters, 604with three space variables involving
exponential nonlinearities, 722with three space variables involving
powerlaw nonlinearities, 713with trigonometric nonlinearities, 276, 389,
680with two independent variables, 1355with two independent variables involving
arbitrary functions, 19with two independent variables involving
arbitrary parameters, 3with two independent variables, nonlinear
in two or more highest derivatives, 849with two space variables involving
arbitrary functions, 589with two space variables involving
exponential nonlinearities, 384, 574with two space variables involving
powerlaw nonlinearities, 367, 553twodimensional, 425, 1238twodimensional Euler, for incompressible
ideal fluid (plane flows), 1198unsteady boundary layer, for Newtonian
fluid, 917unsteady boundary layer, for non
Newtonian fluids, 930von Karman (fourthorder elliptic
equations), 1192differential substitutions, 1409differentiation method, 1492direct and inverse scattering problems, 1587direct method of symmetry reductions and
differential constraints, 1562direct method of symmetry reductions of
nonlinear equations, 1503doubleprecision floating point, 1740double sineGordon equation, 492double sinhGordon equation, 485Dym equation, 878dynamic output, 1691dynamics of shallow water, 1610
Eeditor, MATLAB, 1737
Eigen–Schuster model, 1137Ekman flow, 1282elementary symmetric polynomial, 1444elementary theory of using invariants for
solving equations, 1439ellipsoidal vortex, 1278elliptic equation, 1392Emden–Fowler equation, 248, 442, 444, 447,
462, 464, 647, 648, 655Enneper equation, 491equation
adiabatic, 1262Airy, 861BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian
fluid, 917boundary layer, for powerlaw fluid with
pressure gradient, 913Boussinesq, 368, 987, 1504Boussinesq, in canonical form, 987Boussinesqtype, 1028breaking soliton, 1030Burgers, 186, 1409, 1414, 1572, 1637,
1673, 1677, 1697, 1723, 1725, 1743,1761
Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400Camassa–Holm, 968Chaplygin, 765characteristic, 1389, 1393, 1647, 1706Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757continuity, 1197, 1247, 1610cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968determining, 1555double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392Emden–Fowler, 248, 442, 444, 447, 462,
464, 647, 648, 655Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,
346fast diffusion, 211firstorder quasilinear, 1355first Painleve, 861, 989Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouvilletype, 1030
Page: 1848
INDEX 1849
equation (continued)fourth Painleve, 988Fujita–Storm, 212functional, 1498, 1500, 1767, 1769functional, describing travelingwave
solutions, 1431Gardner, 871Gauss, 1772Gelfand–Levitan–Marchenko, integral, 863,
868, 990, 1002Gelfand–Levitan–Marchenko type,
integral, 1584general fifthorder KdV, 1034generalized Burgers, 288generalized Burgers–Korteweg–de
Vries, 1045, 1503generalized Calogero, 748generalized Emden–Fowler, 327, 477, 511,
512, 513, 516, 527, 528, 697, 698generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108generalized reciprocal polynomial, 1443Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878Helmholtz, 380, 397, 399, 420, 423, 558,
570, 601, 938, 1289homogeneous Monge–Ampere, 775, 1430Hopf, 5Hunter–Saxton, 745hydrodynamic boundary layer, with
pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392hypergeometric, 1012inviscid Burgers, 1674, 1763inviscid Burgers, explicit method, 1678,
1725inviscid Burgers, initial value problem,
1762inviscid Burgers, shock waves and
rarefaction waves, 1757inviscid Burgers, shock waves and weak
solutions, 1759Ito, 1037Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical
form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749
KK, 1036Klein–Gordon, 1641, 1656, 1701, 1716k(n,n), 891Kolmogorov–Petrovskii–Piskunov
(KPP), 279, 1633, 1694Korteweg–de Vries, 857, 1574, 1580,
1583, 1638, 1698Korteweg–de Vries, in canonical form, 857Korteweg–de Vries–Burgers, 884Korteweg–de Vries type, with exponential
nonlinearity, 874Korteweg–de Vries type, with logarithmic
nonlinearity, 874Korteweg–de Vries type, with powerlaw
nonlinearity, 872KPPtype, 1633, 1694Kuramoto–Sivashinsky, 982Laplace, 394, 409, 421, 556, 558, 579,
646, 938, 1392Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759linear determining, 1556, 1557linear heat, 807, 1252linear Schrodinger, 865, 1186linear wave, 1392Liouville, 144, 165, 470, 540, 1117, 1414mKdV, 867model, for dynamics of nonlinear
strings, 534model, of gas dynamics, 5, 25, 1360model, of nonlinear waves with damping,
6modification of coupled KdV, 1172modified k(n,n), 892modified Burgers, 191modified Burgers–Korteweg–de Vries, 886modified Harry Dym, 891modified Korteweg–de Vries, 862, 866,
869, 1583modified Korteweg–de Vries, in canonical
form, 867modified Korteweg–de Vries–Burgers, 886Monge–Ampere, 774, 1393ndimensional modified Schrodinger, with
cubic nonlinearity, 431ndimensional Schrodinger, with cubic
nonlinearity, 430Newell–Whitehead, 1534ninthorder KdVtype, 1041nonlinear diffusion, 1645, 1705nonlinear diffusion, with cubic source,
1577nonlinear firstorder partial differential,
1373nonlinear Klein–Gordon, 499, 1494nonlinear Kolmogorov type, 377, 398nonlinear Poisson, 1679, 1727, 1750nonlinear Poisson–Boltzmann, 1755nonlinear Schrodinger, 348, 425, 1582
Page: 1849
1850 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
equation (continued)nonlinear wave, 1400, 1519, 1536, 1673,
1678, 1723, 1727, 1764nthorder Liouville type, 1109of axisymmetric steady hydrodynamic
boundary layer, 1508of Burgers hierarchy, 1067of complete Korteweg–de Vries hierarchy,
1069of helical surface, 8of Korteweg–de Vries hierarchy, 1068of light rays, 63of minimal surfaces, 768, 1597of modified Korteweg–de Vries hierarchy,
1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756onesoliton solution, mKdV, 867onesoliton solution, potential KdV, 876Ostrovsky, 968palindromic, 1442Pexider, 1771Pfaffian, 1600Pfaffian, condition for integrability by
single relation, 1601Pfaffian, integrating factor, 1601Plebanski first heavenly, 802Plebanski second heavenly, 803Poisson, 379, 397, 423, 580, 588, 938,
1008porous medium, 210potential, of onedimensional flow of
compressible gas, 768potential Korteweg–de Vries (KdV), 876potential modified Korteweg–de Vries
(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262reciprocal, 1442Riccati, 140, 142, 162, 164, 423, 425, 453,
482, 520, 1012, 1277Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, of general form, 358, 360,
363, 364Schrodinger, with cubic nonlinearity, 348,
350, 351, 352, 355Schrodinger, with powerlaw nonlinearity,
351, 353Schwarzian Korteweg–de Vries, 870secondorder nonlinear partial differential,
1392secondorder quasilinear partial differential,
1393second Painleve, 861, 867, 988, 989seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,
1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035
spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649telegraph, 494, 538, 583, 585thin film, 985thirdorder Liouville type, 972Thomas, 544threedimensional, Khokhlov–Zabolotskaya,
752threedimensional, Schrodinger, nonlinear,
of general form, 429threedimensional, Schrodinger, with cubic
nonlinearity, 428tractrix, 253Tricomi, 659twodimensional, heat, linear, 1256twodimensional, Khokhlov–Zabolotskaya,
749twodimensional, Schrodinger, nonlinear, of
general form, 427twodimensional, Schrodinger, with cubic
nonlinearity, 425twodimensional, Schrodinger, with
powerlaw nonlinearity, 427Tzitzeica, 541unnormalized Boussinesq, 990unnormalized Burgers, 187unnormalized Kadomtsev–Petviashvili,
1002unnormalized Korteweg–de Vries, 865unnormalized modified Korteweg–de
Vries, 870variant of modified Burgers–Korteweg–de
Vries, 886vector Burgers, 417Weber, 1193Yermakov, 309, 518
equationsadmitting variational form, 1595algebraic, 1441algebraic, with even powers, 1441associativity, 972axisymmetric steady laminar hydrodynamic
boundary layer, 909axisymmetric unsteady laminar boundary
layer, 929basic steps of Painleve test, 1570Bellman type, 805Benney moment, 764bilinear functional, 1775boundary layer (Prandtl equations), 1333elliptic, canonical form, 1392evolution, canonic Backlund transforma
tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231cubic in derivatives, 134Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197
Page: 1850
INDEX 1851
equations (continued)evolution, nonlinear in second derivative,
346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder, quasilinear, 3firstorder, with three or more independent
variables, 125firstorder, with two independent variables
of general form, 99firstorder, with two independent variables
quadratic in derivatives, 43for convective fluid motions, 1329fourthorder, 977functionaldifferential, reducible to bilinear
equation, 1776Gelfand–Levitan–Marchenko type, 1584heat, with powerlaw or exponential
temperaturedependent thermaldiffusivity, 409
heat and mass transfer, 697heat and mass transfer, in quiescent
or moving media with chemicalreactions, 390
heat and mass transfer, with complicatingfactors, 734
hydrodynamic boundary layer, 903hydrodynamic type, order reduction, 1422hyperbolic, canonical form, 1390invariance under scaling transformations,
1431invariance under translations, 1429involving arbitrary differential operators,
1101involving arbitrary functions, 77, 279, 390,
499, 546, 682involving arbitrary functions of derivatives,
113involving arbitrary functions of four
variables, 123involving arbitrary functions of indepen
dent variables, 107involving arbitrary functions of one
variable, 19, 113involving arbitrary functions of three
variables, 120involving arbitrary functions of two
variables, 30, 116, 168involving arbitrary parameters, 43, 540involving arbitrary powers of derivatives,
102, 136involving cosine, 276involving cotangent, 278involving derivatives in radicands, 101involving exponential nonlinearities, 587involving first derivative in t, 811, 857,
977involving first derivative in t and linear in
highest derivative, 1031involving fourth powers of derivatives, 99
involving hyperbolic cosine, 268involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,
279involving mixed derivatives, 1000, 1103involving one arbitrary power of derivative,
107involving powerlaw functions of
derivatives, 156involving powerlaw nonlinearities, 583involving products of derivatives, 133involving products of derivatives with
respect to different variables, 131involving roots and moduli of derivatives,
136involving second derivative in t, 896, 987,
1088involving secondorder mixed derivatives,
949involving sine, 277involving squares of derivatives, 133involving squares of one or two deriva
tives, 125involving squares of three derivatives, 130involving tangent, 278involving thirdorder mixed derivatives,
958involving two or more second derivatives,
839involving two or three arbitrary powers of
derivatives, 111linear in mixed derivative, 745, 847mathematical physics, Painleve test, 1565mathematical physics, transformations,
1395Navier–Stokes, 1247Navier–Stokes, general properties, 1249Navier–Stokes, in Boussinesq approxima
tion, 1329Navier–Stokes, in various coordinate
systems, 1247nonlinear, diffusion boundary layer, 402nonlinear, elliptic, geometrical models,
1754nonlinear, elliptic, in two space dimen
sions, 1748nonlinear, functional, containing complex
argument, 1776nonlinear, functional, reducible to
bilinear, 1775nonlinear, functional, solutions, 1496nonlinear, in highest derivatives, 404nonlinear, in two independent variables,
1392nonlinear, involving arbitrary linear
differential operators, 1065nonlinear, involving arbitrary parameters,
99nonlinear, of general form, 150, 169, 1392
Page: 1851
1852 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
equations (continued)nonlinear, of thermal (diffusion) boundary
layer, 346nonlinear, parabolic, 1677, 1725nonlinear, secondorder, 1516nonlinear, simple separation of variables,
1459nonlinear, telegraph, with two space
variables, 583nonlinear, with arbitrary number of
variables containing arbitrary functions,162
nonlinear, with arbitrary number ofvariables containing arbitrary parameters,158
nonlinear, with four independent variables,154
nonlinear, with three variables containingarbitrary functions, 139
nonlinear, with three variables quadratic inderivatives, 125
nonstationary, 942nonstationary, hydrodynamic (Navier–
Stokes equations), 1013nonstationary, of motion of viscous
incompressible fluid, 1013of atmospheric circulation in equatorial
region, 1225of breeze and monsoons, 1223of Burgers and Korteweg–de Vries
hierarchies, 1067of dynamic convection in the sea, 1227of flows in the baroclinic layer of the
ocean, 1229of general form, involving arbitrary
functions of single argument, 358of general form, involving arbitrary
functions of two arguments, 362of general form, involving first derivative
in t, 1070of heat and mass transfer in anisotropic
media, 412of higher orders, 1031of ideal incompressible fluid, 942of ideal plasticity, 1238of mass transfer in quiescent or moving
media with chemical reactions, 406of motion of ideal fluid (Euler equations),
938, 1197of nonlinear vibrations, stratified medium,
1131of sixth to ninthorder, 1039of stationary transonic planeparallel gas
flow, 1122of viscous incompressible fluid, 1003ordinary differential, 1445ordinary differential, Painleve test, 1566Painleve, 1566parabolic, canonical form, 1390passing Painleve test, 1572Pfaffian, 1600
Pfaffian, not satisfying integrabilitycondition, 1602
Plebanski heavenly, 802pseudodifferential, 1483, 1484quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772quasilinear, 1393, 1647, 1649, 1707, 1708quasilinear, discontinuous solutions, 1360quasilinear, in conservative form, 1368quasilinear, of general form, 1368quasilinear, qualitative features, 1360quasilinear, with n independent variables,
1356reducible to Korteweg–de Vries equation,
875secondorder, classification, 1389secondorder, elliptic, with three or more
space variables, 713secondorder, elliptic, with two space
variables, 641secondorder, evolution, 1547secondorder, hyperbolic, 1551secondorder, hyperbolic, with one space
variable, 433secondorder, hyperbolic, with two or more
space variables, 553secondorder, involving mixed derivatives
and some other, 745secondorder, involving real functions of
real variables, 1165secondorder, nonlinear, for laser systems,
1170secondorder, of general form, 811, 1553secondorder, parabolic, with one space
variable, 175secondorder, parabolic, with two or more
space variables, 367semilinear, in two independent variables,
1389shallow water, 1124stationary, 938stationary, hydrodynamic (Navier–Stokes
equations), 1003steady boundary layer, for nonNewtonian
fluids, 911steady hydrodynamic boundary layer, 903steady hydrodynamic boundary layer, for
Newtonian fluid, 903steady hydrodynamic boundary layer,
symmetries, 1528thirdorder, 857threeargument, functional, 1497threedimensional, 1240threedimensional, involving arbitrary
functions, 730threedimensional, of ideal plasticity, 1240three and ndimensional, 428twodimensional, 425, 1238twodimensional, Euler, for incompressible
ideal fluid (plane flows), 1198
Page: 1852
INDEX 1853
equations (continued)unsteady boundary layer, for Newtonian
fluid, 917unsteady boundary layer, for non
Newtonian fluids, 930von Karman (fourthorder elliptic
equations), 1192with arbitrary dependence on derivatives,
148with arbitrary number of independent
variables, 39with coefficients involving exponential
functions, 11with coefficients involving hyperbolic
functions, 14with coefficients involving logarithmic
functions, 16with coefficients involving powerlaw
functions, 3with coefficients involving trigonometric
functions, 17with cubic nonlinearities involving
arbitrary functions, 355with exponential nonlinearities, 254, 469,
662with hyperbolic nonlinearities, 268, 485,
675with logarithmic nonlinearities, 271, 388,
486, 677with n independent variables, 739, 853with n space variables, 417with powerlaw nonlinearities, 175, 433,
641, 808with powerlaw nonlinearity in derivatives,
161, 167with power nonlinearity in derivatives, 146with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables,
involving arbitrary functions, 624with three space variables, involving
arbitrary parameters, 604with three space variables, involving
exponential nonlinearities, 722with three space variables, involving
powerlaw nonlinearities, 713with trigonometric nonlinearities, 276, 389,
680with two independent variables, 1355with two independent variables, involving
arbitrary functions, 19with two independent variables, involving
arbitrary parameters, 3with two independent variables, nonlinear
in two or more highest derivatives, 849with two space variables, involving
arbitrary functions, 589with two space variables, involving
exponential nonlinearities, 384, 574with two space variables, involving
powerlaw nonlinearities, 367, 553error function, 187errors in constructing solutions by symbolic
computations, 1668Euler equations, 938, 1197, 1681, 1729
for barotropic gas flow, 1201in Gromeka–Lamb form, 1201in various coordinate systems, 1197
Euler–Lagrange equation, 1595Euler transformation, 122, 346, 769, 957,
1403, 1407Eulerian coordinates, 1680, 1728evaluation of function, 1628, 1740evolution equations nonlinear in second
derivative, 346evolutionary conditions, 1616exact methods for nonlinear partial differen
tial equations, 1353exact nonlinear problem, 1681, 1729exact solutions of nonlinear partial differen
tial equations, 1exact solutions of various types of nonlinear
PDEs, 1634exact solutions with simple separation of
variables, 1459examples of constructing conservation laws
using Noetherian symmetries, 1596examples of constructing exact generalized
separable solutions, 1467examples of constructing exact solutions,
1503, 1534examples of constructing functional separable
solutions, 1492examples of constructing invariant solutions
to nonlinear equations, 1522examples of constructing invariant solutions
to nonlinear partial differentialequations, 1451
examples of finding exact solutions ofsecond and thirdorder equations, 1465
examples of finding symmetries of nonlinearequations, 1516
examples of Lax pairs for nonlinearequations of mathematical physics,1580
examples of selfsimilar solutions tomathematical physics equations, 1432
examples of solutions having movablesingularities, 1565
examples of solutions of some specificfunctional equations, 1770, 1771
examples of solutions of specific functionalequations, 1773
examples of solving Cauchy problem, 1381examples of viscosity (nonsmooth) solutions,
1385existence and uniqueness theorem, 1357,
1359, 1379, 1380expfunction method, 1643, 1703explicit central difference method, 1678,
1727, 1764
Page: 1853
1854 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
explicit finite difference scheme, 1761, 1763explicit method, 1677, 1678, 1725, 1727exponential isotherm, 1235export facilities, 1751external flow around cylinder, 1253
FFalkner–Skan problem, 907, 1334false, 1739fast diffusion, 251fast diffusion equation, 211fifthorder equations, 1031file name, 1740finding linear subspaces invariant under given
nonlinear operator, 1480finite element method, 1748first hodograph transformation, 1399firstorder chemical reaction, 1135, 1137firstorder differential constraints, 1539
for PDEs, 1547firstorder equations
classification results, 1565describing convective mass transfer with
volume reaction, 1602general solution, 1355hydrodynamic type, involving three or
more equations, 1197in one unknown, overdetermined system,
1599quasilinear, 3, 1355nonlinear, 1373, 1659, 1719, 1720nonlinear, methods for solving, 1373nonlinear, with three or more independent
variables, 125nonlinear, with two independent variables
of general form, 99quasilinear, hyperbolic, 1607quasilinear, methods for solving, 1355solution, general, 1355with three or more independent variables,
125with two independent variables of general
form, 99with two independent variables quadratic
in derivatives, 43firstorder hydrodynamic and other systems
involving three or more equations, 1197firstorder hyperbolic systems of quasilinear
equations, 1607firstorder nonlinear equations with three or
more independent variables, 125firstorder nonlinear equations with two
independent variables of generalform, 99
firstorder quasilinear equations, 3firstorder quasilinear partial differential
equation, 1355first Painleve equation, 861, 989first Stokes problem, 1251Fisher equation, 176, 1639, 1699, 1744
FitzHugh–Nagumo equation, 179, 1746fixed singularities of solutions, 1565fKdV equation, 1034flow
adiabatic nonisentropic gas, 1204around cylinder, external, 1253barotropic gas, Euler equations, 1201Couette–Poiseuille plane, 1251Ekman, 1282Hagen–Poiseuille, 1253Homann, 1302in confuser, 1261in diffuser, 1261in rotating layer, 1283irrotational gas, 1201Jeffery–Hammel, 1260Karman, 1290onedimensional, compressible gas, 768onedimensional, polytropic ideal gas,
1125potential, 1334potential gas, 1201quasiplane, 1285stationary transonic planeparallel
gas, 1122steady transonic gas, 657von Karman, 1290
flowsaxial, 1303axisymmetric, 1274conical vortex, 1290cylindrical vortex, 1290generalized Couette–Poiseuille plane, 1329in baroclinic layer of ocean, 1229onedimensional barotropic, ideal
compressible gas, 1127onedimensional rotation fluid, 1257plane, twodimensional Euler equations for
incompressible ideal fluid, 1198plane, twodimensional solutions in
cylindrical coordinates, 1270plane, twodimensional solutions in
rectangular Cartesian coordinates, 1263plane, unidirectional, 1251quasiplane, 1285threedimensional stagnationpoint
type, 1302unidirectional, in tubes of various cross
sections, 1253unidirectional, plane, 1251unsteady transonic gas, 756von Karmantype rotationally symmetric,
1316with twononzero velocity components,
1282fluid streamlines, 1253, 1257Fokas–Yortsos equation, 237forward difference method, 1677, 1725,
1761forward time backward space method, 1676fourthorder equations, 977fourthorder Liouville type equation, 1030
Page: 1854
INDEX 1855
fourth Painleve equation, 988front end, 1688Fuchs index, 1570Fuchs indices, 1567, 1568, 1570Fujita–Storm equation, 212function
error, 187handle, 1741Jacobi elliptic, 990mathematical library, 1736mfiles, 1737modified Bessel, 1260name of, 1740
functionaldifferential equations reducible tobilinear equation, 1776
functional equation, 1498, 1500, 1767, 1769describing travelingwave solutions, 1431Pexider, 1771
functional equationsbilinear, 1775nonlinear, containing complex argument,
1776nonlinear, reducible to bilinear, 1775nonlinear, solutions, 1496
functional separable solutions, 1487, 1652,1711
functional separation of variables, 1655,1715
functional separation of variables bydifferentiation, 1656, 1716
functionsactive functions, 1628anonymous, 1741Bessel, 1160, 1255, 1259builtin, 1740inert, 1628Kelvin, 1256nested, 1741predefined, 1628, 1740pure, 1690special, 1740written in mfiles, 1740
GGalileo transformation, 1514Gardner equation, 871gas dynamic type systems linearizable with
hodograph transformation, 1122Gauss equation, 1772Gelfand–Levitan–Marchenko integral
equation, 863, 868, 990, 1002Gelfand–Levitan–Marchenko type integral
equations, 1584general consistency method for two
equations, 1542general description of method of differential
constraints, 1546general fifthorder KdV equation, 1034general form
equations involving first derivativein t, 1070
for symmetry reduction, 1506of conservation laws, 1593of contact transformations for partial
differential equations, 1405of functional differential equations, 1464of point transformations, 1395of selfsimilar solutions, 1431of solutions, 1464of travelingwave solutions, 1429
general integral, 1373, 1374general properties of Navier–Stokes
equations, 1249general scheme
for analysis of nonlinear partial differentialequations, 1570
for constructing generalized travelingwavesolutions, 1489
of using invariants for solving mathematical equations, 1439, 1440
general solutionof characteristic system, 1355of firstorder PDE, 1355
generalizations to pseudodifferentialequations, 1483
generalized Blasius problem, 912generalized Burgers equation, 288generalized Burgers–Korteweg–de Vries
equation, 1045, 1503generalized Calogero equation, 748generalized Cauchy problem, 1649, 1650,
1708, 1709generalized classical method of characteris
tics, 1385generalized Couette–Poiseuille plane
flows, 1329generalized Emden–Fowler equation, 327,
477, 511, 512, 513, 516, 527, 528, 697,698
generalized Falkner–Skan problem, 914generalized Kadomtsev–Petviashvili equation,
1029generalized Khokhlov–Zabolotskaya
equation, 751generalized Korteweg–de Vries equation,
871generalized Kuramoto–Sivashinsky equation,
983generalized Landau–Ginzburg equation, 360generalized Langmuir isotherm, 1233generalized Liouville equation, 1108generalized method of characteristics, 1647,
1650, 1709generalized porous medium equation with
nonlinear source, 340generalized reciprocal polynomial equation,
1443generalized Schlichting problem, 912generalized selfsimilar solutions, 1436generalized separable solutions, 1464, 1620,
1652, 1654, 1711, 1714
Page: 1855
1856 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
generalized separable solutions (continued)by differentiation, nthorder nonlinear
PDE, 1652, 1711by splitting, 1653, 1712
generalized solution, 1364generalized travelingwave solutions, 1487generalized viscosity solutions and their
applications, 1382generalized/functional separation of variables
vs. differential constraints, 1561generalizing exact solutions, 1665generic system, 1754geometry description matrix, 1755Gibbons–Tsarev equation, 764global variables, 1739Goursat equation, 549Grad–Shafranov equation, 678, 691graphical user interface, 1736, 1748graphical user interface development
environment, 1738graphics system, 1736group generator, 1514, 1526group invariants, 1526
HHagen–Poiseuille flow, 1253Hamilton–Jacobi equation, 1379Hamiltonian, 1379Harry Dym equation, 878heat and mass transfer equations, 697, 730
in quiescent or moving media withchemical reactions, 390
with complicating factors, 734heat equations with powerlaw or exponential
temperaturedependent thermaldiffusivity, 409
HeleShaw cell, 984Helmholtz equation, 380, 397, 399, 420, 423,
558, 570, 601, 938, 1289Henry coefficients, 1231Hiemenz problem, 1264higherorder differential constraints, 1555Hill vortex, 1278hodograph transformation, 35, 212, 344,
1129, 1397Homann flow, 1302homogeneous Monge–Ampere equation, 775,
1430Hopf–Cole transformation, 187, 888, 1067,
1409, 1414Hopf equation, 5Hopf formula for generalized solution, 1366Hunter–Saxton equation, 745hydrodynamictype system of diagonal
form, 1236hydrodynamictype system of nondiagonal
form, 1236hydrodynamic boundary layer equation with
pressure gradient, 907hydrodynamic boundary layer equations, 903
hyperbolic equationfirst canonical form, 1391second canonical form, 1392
hyperbolic n × n systems of conservationlaws, 1614
hyperbolic sineGordon equation, 485hyperbolic system, 1607, 1608, 1615hypergeometric equation, 1012
Iideal (inviscid) fluid, 938, 1197ideal plasticity with von Mises yield
criterion, 1238implicit form of solution, 1648, 1708incorrect response, 1626, 1688, 1737inert functions, 1628infinitesimal generator, 1514infinitesimal operator, 1513, 1514, 1526initial value problem, 1383, 1611integral equation
Gelfand–Levitan–Marchenko, 863, 868,990, 1002
Gelfand–Levitan–Marchenko type, 1584integrals of motion, 1593integrating factor of Pfaffian equation, 1601introducing and removing additional arbitrary
constants, 1666invariance condition, 1516, 1527, 1595invariance of equations under scaling
transformations, 1431invariance of equations under translations,
1429invariance of solutions and equations under
translation transformations, 1430invariant, 1439, 1445, 1450, 1526
determination procedure, 1446of group, 1526of infinitesimal operator, 1514of operator, 1514, 1526of transformation, 1440
invariant condition, 1527invariant solution, 1431, 1435, 1451, 1513,
1520, 1527, 1602, 1620invariant surface condition, 1533inverse problems, 1455inverse scattering method, 990, 1002inverse scattering problem, 1587inviscid Burgers equation, 1674, 1763
explicit method, 1678, 1725initial value problem, 1762shock waves and rarefaction waves, 1757shock waves and weak solutions, 1759
irrotational gas flow, 1201isotherms of adsorption, 1231Ito equation, 1037
JJacobi elliptic function, 990Jacobi–Mayer bracket, 1377Jacobian elliptic cosine, 858
Page: 1856
INDEX 1857
Jeffery–Hamel flow, 1260Jeffery–Hamel problem, 1261Jeffery–Hamel solution, 1272jetsource problem, 1335jump condition, 1362, 1363, 1369
KKarman–Howarth equation, 253Kadomtsev–Petviashvili equation, 1587
in canonical form, 1000related equations, 1000
Kadomtsev–Petviashvili type equation, 1002Kaup–Kupershmidt equation, 1036Kawahara equation, 1032KdV equation, 857Kelvin functions, 1256Khokhlov–Zabolotskaya and related
equations, 749kinematic momentum, 1335kinematic viscosity, 1247kink, 1636, 1695KK equation, 1036Klein–Gordon equation, 1641, 1656, 1701,
1716k(n,n) equation, 891Kolmogorov–Petrovskii–Piskunov (KPP)
equation, 279, 1633, 1694Kolmogorov model, 1006Korteweg–de Vries–Burgers equation, 884Korteweg–de Vries equation, 857, 1574,
1580, 1583, 1638, 1698in canonical form, 857
Korteweg–de Vries type equationwith exponential nonlinearity, 874with logarithmic nonlinearity, 874with powerlaw nonlinearity, 872
Kovalevskaya–Painleve test, 1569KPPtype equation, 1633, 1694Kuramoto–Sivashinsky equation, 982
LLagrange–Charpit method, 1376Lagrangian, 1595Lagrangian coordinates, 1680, 1728laminar fluid flows, 1247Landau problem, 1012Landau solution, 1277Langmuir isotherm, 1231Laplace equation, 394, 409, 421, 556, 558,
579, 646, 938, 1392law of conservation of mass, 1610Lax condition, 1616, 1617Lax equation, 1035, 1068
with forcing term, 1037Lax method, 1678, 1726, 1763Lax pair, 1579leading terms of equation, 1567Legendre transformation, 122, 767, 768, 770,
1403, 1406with many variables, 1408
library functions, 1628, 1740Lie group, 1526limiting selfsimilar solution, 1454, 1456Lin–Reissner–Tsien equation, 756, 759linear determining equation, 1556, 1557linear heat equation, 807, 1252linear Schrodinger equation, 865, 1186linear subspaces invariant under nonlinear
operator, 1477linear transformations, 1396linear wave equation, 1392linearization of systems of gas dynamic type
by hodograph transformation, 1610Liouville equation, 144, 165, 470, 540, 1117,
1414lists, 1629, 1691local structure of generalized viscosity
solutions, 1383local transformations of derivatives, 1526local variables, 1739logical expression, 1739logical operators, 1628, 1689, 1738Loitsianskii decay law, 253Lorentz transformation, 1514Lotka–Volterra system, 1135
Mmfile, 1737Mach angle, 1203Mach number, 757main program, 1737Maple language, 1627Maple worksheets, 1626Maplet applications, 1627Maplet user interface, 1627Martin transformation, 775Mathematica language, 1689Mathematica objects, 1691mathematical constants, 1690mathematical function library, 1736Mathlink, 1688MATLAB language, 1738matrix, 1629, 1691, 1741mesh, 1677, 1725mesh points, 1677, 1725method
based on compatibility condition forsystems of linear equations, 1581
based on linear integral equations, 1584based on using Lax pairs, 1579classical, of separation of variables, 1651,
1711classical, of symmetry reductions, 1513Clarkson–Kruskal direct, 1503differentiation, 1492direct, of symmetry reductions and
differential constraints, 1562direct, of symmetry reductions of nonlinear
equations, 1503expfunction, 1643, 1703
Page: 1857
1858 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
method (continued)explicit, 1677, 1678, 1725, 1727explicit central difference, 1678, 1727,
1764finite element, 1748for constructing stable generalized
solutions, 1370forward difference, 1677, 1725, 1761forward time backward space, 1676general consistency for two equations,
1542inverse scattering, 990, 1002inverse scattering, Cauchy problem, 1587inverse scattering, Cauchy problem for
nonlinear equations, 1589Lagrange–Charpit, 1376Lax, 1678, 1726, 1763of argument elimination by test functions,
1772of characteristics, 32, 35, 1355–1357,
1647, 1649, 1707, 1708, 1756of characteristics, generalized, 1385, 1647,
1650, 1709of conversion to equivalent system of
equations, 1398of differential constraints, 1539of differential constraints, general
description, 1546of differential constraints for ordinary
differential equations, 1539of differentiation in independent variables,
1771of differentiation in parameter, 1769of functional separation of variables, 1487of generalized separation of variables,
1459of invariants, 1439of lines, 1722, 1743of separation of variables, 1374of symmetry reductions, nonclassical,
1533, 1562of vanishing viscosity, 1366, 1383similarity, 1431sinecosine, 1642, 1702specifying singlestage numerical, 1674,
1676specifying twostage numerical, startup
method, 1676spectral collocation, 1680, 1729splitting, 1496tanhcoth, 1641, 1701Titov–Galaktionov, 1477
methodsexact, for nonlinear partial differential
equations, 1353for constructing travelingwave solutions,
1641, 1700for solving firstorder nonlinear equations,
1373for solving firstorder quasilinear equa
tions, 1355
numerical and graphical, 1672numerical, embedded in Maple, 1674of inverse scattering problem (soliton
theory), 1579Miura transformation, 862, 868, 1410mKdV equation, 867model equation for dynamics of nonlinear
strings, 534model equation of gas dynamics, 5, 25,
1360model equation of nonlinear waves with
damping, 6modification of coupled KdV equation, 1172modified k(n,n) equation, 892modified Bessel function, 1260modified Burgers equation, 191modified Burgers–Korteweg–de Vries
equation, 886modified Harry Dym equation, 891modified Korteweg–de Vries–Burgers
equation, 886modified Korteweg–de Vries equation, 862,
866, 869, 1583in canonical form, 867
module, 1629, 1691module system, 1741Monge–Ampere equation, 774, 1393Monge cone, 1647, 1706movable singularities of solutions, 1565
of ordinary differential equations, 1565mth prolongation of group generator, 1527multipeakon solutions, 968multiple statements in line, 1739multiplicative separable solution, 1374, 1459,
1651, 1710
Nndimensional modified Schrodinger equation
with cubic nonlinearity, 431ndimensional Schrodinger equation with
cubic nonlinearity, 430nsoliton solution, 859, 1591
of Korteweg–de Vries equation, 1591Nadai multiplicative separable solution, 1238Nadai solution, 1238name of symbol, 1689, 1690Navier–Stokes equations, 1247
general properties, 1249in Boussinesq approximation, 1329nonstationary, 1013stationary hydrodynamic, 1003
nested functions, 1741nested lists, 1691Newell–Whitehead equation, 1534Newtonian fluid, 1247Nijenhuis tensor, 1237ninthorder KdVtype equation, 1041noslip condition, 1253Noetherian symmetry, 1595
Page: 1858
INDEX 1859
nonNewtonian fluid, 911, 932, 933boundary layer equations, 915of general form, 915
nonclassical hodograph transformation, 345,1399
nonclassical method of symmetry reductions,1533
nonclassical method of symmetry reductionsand differential constraints, 1562
nonclassical selfsimilar solution, 500, 685,689, 1435
nonclassical symmetries, 1533nonclassical travelingwave solution, 679,
690, 706noninvariant selfsimilar solution, 1435nonlinear diffusion boundary layer equations,
402nonlinear diffusion equation, 1645, 1705
with cubic source, 1577nonlinear dispersive nondissipative waves,
857nonlinear elliptic PDEs (geometrical
models), 1754nonlinear elliptic PDEs in two space
dimensions, 1748nonlinear equation
adiabatic, 1262BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian
fluid, 917boundary layer, for powerlaw fluid with
pressure gradient, 913Boussinesq, 368, 987, 1504Boussinesq, in canonical form, 987Boussinesqtype, 1028breaking soliton, 1030Burgers, 186, 1409, 1414, 1572, 1637,
1673, 1677, 1697, 1723, 1725, 1743,1761
Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400Camassa–Holm, 968Chaplygin, 765characteristic, 1389, 1393, 1647, 1706Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968diffusion, 1645, 1705diffusion, with cubic source, 1577double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392
Emden–Fowler, 248, 442, 444, 447, 462,464, 647, 648, 655
Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,
346fast diffusion, 211firstorder, 1355, 1373first Painleve, 861, 989Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouville type, 1030fourth Painleve, 988Fujita–Storm, 212Gardner, 871Gauss, 1772general fifthorder KdV, 1034generalized Burgers–Korteweg–de
Vries, 1045, 1503generalized Burgers, 288generalized Calogero, 748generalized Emden–Fowler, 327, 477, 511,
512, 513, 516, 527, 528, 697, 698generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878homogeneous Monge–Ampere, 775, 1430Hopf, 5Hunter–Saxton, 745hydrodynamic boundary layer, with
pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392hypergeometric, 1012inviscid Burgers, 1674, 1763inviscid Burgers, explicit method, 1678,
1725inviscid Burgers, initial value problem,
1762inviscid Burgers, shock waves and
rarefaction waves, 1757inviscid Burgers, shock waves and weak
solutions, 1759Ito, 1037k(n,n), 891Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical
form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036
Page: 1859
1860 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
nonlinear equation (continued)Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749KK, 1036Klein–Gordon, 499, 1494, 1641, 1656,
1701, 1716Kolmogorov type, 377, 398Kolmogorov–Petrovskii–Piskunov
(KPP), 279, 1633, 1694Korteweg–de Vries, 857, 1574, 1580,
1583, 1638, 1698Korteweg–de Vries, in canonical form, 857Korteweg–de Vries–Burgers, 884Korteweg–de Vries type, with exponential
nonlinearity, 874Korteweg–de Vries type, with logarithmic
nonlinearity, 874Korteweg–de Vries type, with powerlaw
nonlinearity, 872KPPtype, 1633, 1694Kuramoto–Sivashinsky, 982Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759Liouville, 144, 165, 470, 540, 1117, 1414mKdV, 867model, for dynamics of nonlinear
strings, 534model, of gas dynamics, 5, 25, 1360model, of nonlinear waves with damping,
6modification of coupled KdV, 1172modified Burgers, 191modified Burgers–Korteweg–de Vries, 886modified Harry Dym, 891modified k(n,n), 892modified Korteweg–de Vries, 862, 866,
869, 1583modified Korteweg–de Vries–Burgers, 886modified Korteweg–de Vries, in canonical
form, 867Monge–Ampere, 774, 1393of axisymmetric steady hydrodynamic
boundary layer, 1508of Burgers hierarchy, 1067of complete Korteweg–de Vries hierarchy,
1069of helical surface, 8of Korteweg–de Vries hierarchy, 1068of light rays, 63of minimal surfaces, 768, 1597of modified Korteweg–de Vries hierarchy,
1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756onesoliton solution, mKdV, 867onesoliton solution, potential KdV, 876Ostrovsky, 968ndimensional modified Schrodinger, with
cubic nonlinearity, 431ndimensional Schrodinger, with cubic
nonlinearity, 430Newell–Whitehead, 1534ninthorder KdVtype, 1041nthorder Liouville type, 1109palindromic, 1442Pfaffian, 1600Pfaffian, condition for integrability by
single relation, 1601Plebanski first heavenly, 802Plebanski second heavenly, 803Poisson, 1679, 1727, 1750Poisson–Boltzmann, 1755porous medium, 210potential, of onedimensional flow of
compressible gas, 768potential Korteweg–de Vries (KdV), 876potential modified Korteweg–de Vries
(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262Riccati, 140, 142, 162, 164, 423, 425, 453,
482, 520, 1012, 1277Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, 348, 425, 1582Schrodinger, of general form, 358, 360,
363, 364Schrodinger, with cubic nonlinearity, 348,
350, 351, 352, 355Schrodinger, with powerlaw nonlinearity,
351, 353Schwarzian Korteweg–de Vries, 870second Painleve, 861, 867, 988, 989seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,
1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649telegraph, 494, 538, 583, 585thin film, 985thirdorder Liouville type, 972Thomas, 544threedimensional Khokhlov–Zabolotskaya,
752threedimensional Schrodinger, of general
form, 429threedimensional Schrodinger, with cubic
nonlinearity, 428tractrix, 253Tricomi, 659twodimensional Khokhlov–Zabolotskaya,
749twodimensional Schrodinger, of general
form, 427twodimensional Schrodinger, with cubic
nonlinearity, 425
Page: 1860
INDEX 1861
nonlinear equation (continued)twodimensional Schrodinger, with
powerlaw nonlinearity, 427Tzitzeica, 541unnormalized Boussinesq, 990unnormalized Burgers, 187unnormalized Kadomtsev–Petviashvili,
1002unnormalized Korteweg–de Vries, 865unnormalized modified Korteweg–de
Vries, 870variant of modified Burgers–Korteweg–de
Vries, 886vector Burgers, 417wave, 1400, 1519, 1536, 1673, 1678, 1723,
1727, 1764Weber, 1193Yermakov, 309, 518
nonlinear equationsadmitting variational form, 1595algebraic, 1441algebraic, with even powers, 1441axisymmetric steady laminar hydrodynamic
boundary layer, 909axisymmetric unsteady laminar boundary
layer, 929basic steps of Painleve test, 1570Bellman type, 805Benney moment, 764boundary layer (Prandtl equations), 1333evolution, canonic Backlund transforma
tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231cubic in derivatives, 134diffusion boundary layer, 402elliptic, geometrical models, 1754elliptic, in two space dimensions, 1748Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197evolution, nonlinear in second derivative,
346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder with three or more independent
variables, 125firstorder with two independent variables
of general form, 99firstorder with two independent variables
quadratic in derivatives, 43for convective fluid motions, 1329fourthorder, 977functional, containing complex argument,
1776functional, reducible to bilinear, 1775functional, solutions, 1496functionaldifferential, 1776
heat and mass transfer, 697heat and mass transfer, in quiescent
or moving media with chemicalreactions, 390
heat and mass transfer, with complicatingfactors, 734
heat, with powerlaw or exponentialtemperaturedependent thermaldiffusivity, 409
hydrodynamic boundary layer, 903hydrodynamic type, order reduction, 1422hyperbolic, canonical form, 1390hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392in highest derivatives, 404in two independent variables, 1392invariance under scaling transformations,
1431invariance under translations, 1429involving arbitrary differential operators,
1101involving arbitrary functions, 77, 279, 390,
499, 546, 682involving arbitrary functions of derivatives,
113involving arbitrary functions of four
variables, 123involving arbitrary functions of indepen
dent variables, 107involving arbitrary functions of one
variable, 19, 113involving arbitrary functions of three
variables, 120involving arbitrary functions of two
variables, 30, 116, 168involving arbitrary linear differential
operators, 1065involving arbitrary parameters, 43, 99, 540involving arbitrary powers of derivatives,
102, 136involving cosine, 276involving cotangent, 278involving derivatives in radicands, 101involving exponential nonlinearities, 587involving first derivative in t, 811, 857,
977involving first derivative in t and linear in
highest derivative, 1031involving fourth powers of derivatives, 99involving hyperbolic cosine, 268involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,
279involving mixed derivatives, 1000, 1103involving one arbitrary power of derivative,
107involving powerlaw functions of
derivatives, 156involving powerlaw nonlinearities, 583
Page: 1861
1862 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
nonlinear equations (continued)involving products of derivatives, 133involving products of derivatives with
respect to different variables, 131involving roots and moduli of derivatives,
136involving secondorder mixed derivatives,
949involving second derivative in t, 896, 987,
1088involving sine, 277involving squares of derivatives, 133involving squares of one or two deriva
tives, 125involving squares of three derivatives, 130involving tangent, 278involving thirdorder mixed derivatives,
958involving two or more second derivatives,
839involving two or three arbitrary powers of
derivatives, 111linear in mixed derivative, 745, 847mathematical physics, Painleve test, 1565mathematical physics, transformations,
1395Navier–Stokes, 1247Navier–Stokes, general properties, 1249Navier–Stokes, in Boussinesq approxima
tion, 1329Navier–Stokes, in various coordinate
systems, 1247nonstationary, 942nonstationary, hydrodynamic (Navier–
Stokes equations), 1013nonstationary, of motion of viscous
incompressible fluid, 1013of atmospheric circulation in equatorial
region, 1225of breeze and monsoons, 1223of Burgers hierarchies, 1067of diffusion (thermal) boundary layer, 346of dynamic convection in the sea, 1227of flows in the baroclinic layer of the
ocean, 1229of general form, 150, 169, 1392of general form, involving arbitrary
functions of single argument, 358of general form, involving arbitrary
functions of two arguments, 362of general form, involving first derivative
in t, 1070of heat and mass transfer in anisotropic
media, 412of higher orders, 1031of ideal incompressible fluid, 942of ideal plasticity, 1238of Korteweg–de Vries hierarchies, 1067of mass transfer in quiescent or moving
media with chemical reactions, 406of motion of ideal fluid (Euler equations),
938, 1197of sixth to ninthorder, 1039of stationary transonic planeparallel gas
flow, 1122of thermal (diffusion) boundary layer, 346of vibrations, of stratified medium, 1131of viscous incompressible fluid, 1003ordinary differential, 1445ordinary differential, Painleve test, 1566Painleve, 1566parabolic, 1677, 1725parabolic, canonical form, 1390passing Painleve test, 1572Pfaffian, 1600Pfaffian, not satisfying integrability
condition, 1602Plebanski heavenly, 802quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772reducible to Korteweg–de Vries equation,
875secondorder, classification, 1389secondorder, elliptic, with three or more
space variables, 713secondorder, elliptic, with two space
variables, 641secondorder, evolution, 1547secondorder, hyperbolic, 1551secondorder, hyperbolic, with one space
variable, 433secondorder, hyperbolic, with two or more
space variables, 553secondorder, involving mixed derivatives
and some other, 745secondorder, involving real functions of
real variables, 1165secondorder, nonlinear, for laser systems,
1170secondorder, of general form, 811, 1553secondorder, parabolic, with one space
variable, 175secondorder, parabolic, with two or more
space variables, 367semilinear, in two independent variables,
1389shallow water, 1124simple separation of variables, 1459stationary, 938stationary hydrodynamic (Navier–Stokes
equations), 1003steady boundary layer, for nonNewtonian
fluids, 911steady hydrodynamic boundary layer, for
Newtonian fluid, 903steady hydrodynamic boundary layer,
symmetries, 1528steadystate hydrodynamic boundary
layer, 903telegraph, with two space variables, 583thirdorder, 857three and ndimensional, 428
Page: 1862
INDEX 1863
nonlinear equations (continued)threeargument functional, 1497threedimensional, 1240threedimensional, involving arbitrary
functions, 730threedimensional, of ideal plasticity, 1240twodimensional, 425, 1238twodimensional Euler, for incompressible
ideal fluid (plane flows), 1198unsteady boundary layer, for Newtonian
fluid, 917unsteady boundary layer, for non
Newtonian fluids, 930von Karman (fourthorder elliptic
equations), 1192with arbitrary dependence on derivatives,
148with arbitrary number of independent
variables, 39with arbitrary number of variables,
involving arbitrary functions, 162with arbitrary number of variables,
involving arbitrary parameters, 158with coefficients involving exponential
functions, 11with coefficients involving hyperbolic
functions, 14with coefficients involving logarithmic
functions, 16with coefficients involving powerlaw
functions, 3with coefficients involving trigonometric
functions, 17with cubic nonlinearities involving
arbitrary functions, 355with exponential nonlinearities, 254, 469,
662with four independent variables, 154with hyperbolic nonlinearities, 268, 485,
675with logarithmic nonlinearities, 271, 388,
486, 677with n independent variables, 739, 853with n space variables, 417with powerlaw nonlinearities, 175, 433,
641, 808with powerlaw nonlinearity in derivatives,
146, 161, 167with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables,
involving arbitrary functions, 624with three space variables, involving
arbitrary parameters, 604with three space variables, involving
exponential nonlinearities, 722with three space variables, involving
powerlaw nonlinearities, 713with three variables, involving arbitrary
functions, 139
with three variables, quadratic in derivatives, 125
with trigonometric nonlinearities, 276, 389,680
with two independent variables, 1355with two independent variables, involving
arbitrary functions, 19with two independent variables, involving
arbitrary parameters, 3with two independent variables, nonlinear
in two or more highest derivatives, 849with two space variables, involving
arbitrary functions, 589with two space variables, involving
exponential nonlinearities, 384, 574with two space variables, involving
powerlaw nonlinearities, 367, 553nonlinear firstorder partial differential
equation, 1373nonlinear functional equations containing
complex argument, 1776nonlinear functional equations reducible to
bilinear equations, 1775nonlinear Klein–Gordon equation, 499, 1494nonlinear Kolmogorov type equation, 377,
398nonlinear parabolic equations, 1677, 1725nonlinear partial differential equations, 1650,
1709with Maple, 1625with Mathematica, 1687with MATLAB, 1735
nonlinear Poisson–Boltzmann equation, 1755nonlinear Poisson equation, 1679, 1727,
1750nonlinear problems of suspension transport in
porous media, 1604nonlinear Schrodinger equation, 348, 425,
1582nonlinear Schrodinger equations and related
equations, 348nonlinear systems
of firstorder PDEs, 1659, 1719, 1720of many equations involving first
derivatives with respect to t, 1348of partial differential equations, 1599of secondorder PDEs, 1660, 1661, 1720,
1721of two equations involving first derivatives
with respect to t, 1337of two equations involving second
derivatives with respect to t, 1344nonlinear telegraph equations with two space
variables, 583nonlinear wave equation, 1400, 1519, 1536,
1673, 1678, 1723, 1727, 1764nonlocal transformation, 1412nonstationary equations, 942
of motion of viscous incompressiblefluid, 1013
Page: 1863
1864 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
nonstationary hydrodynamic equations(Navier–Stokes equations), 1013
notation, 1630, 1692nthorder chemical reaction, 1142nthorder Liouville type equation, 1109number
Grashov, 1329Mach, 757, 1203Prandtl, 347, 1329Reynolds, 253, 1013, 1021, 1260, 1275,
1291, 1310–1312Reynolds, modified, 1280
numerical and graphical solutions, 1673,1723
by default methods, 1672specifying singlestage numerical method,
1674specifying singlestage numerical method
and numerical boundary condition(NBC), 1676
specifying twostage numerical method,startup method, and NBC, 1676
numerical approximations, 1627, 1689numerical methods embedded in Maple,
1674numerical solutions and their visualizations,
1672, 1722numerical solutions via predefined functions,
1741
OOlver–Rosenau solution, 1028onedimensional barotropic flows of ideal
compressible gas, 1127onedimensional case, 1205onedimensional polytropic ideal gas
flow, 1125onedimensional rotation fluid flows, 1257onedisk problem, 1290onemode solution, 1683, 1731oneparameter Lie groups of point transfor
mations, 1526oneparameter transformations, 1513
local properties, 1513onesoliton solution, 869, 1592, 1638, 1698
of mKdV equation, 867of potential KdV equation, 876
online help system, 1626, 1737operator
infinitesimal, 1513, 1514, 1526infinitesimal, invariant, 1514invariant, 1514, 1526second prolongation, 1516
operatorsarithmetic, 1627, 1738assignment/unassignment, 1627basic arithmetic, 1689commutator, 1579logical, 1628, 1689, 1738range, 1627relational, 1627, 1738, 1739
operators := and =, 1627order reduction of hydrodynamic type
equations, 1422order reduction procedure for equations with
n ≥ 2 (reduction to solvable form withn = 1), 1446
ordinary differential equationAiry, 861Emden–Fowler, 248, 442, 444, 447, 462,
464, 647, 648, 655Emden–Fowler, generalized, 327, 477, 511,
512, 513, 516, 527, 528, 697, 698hypergeometric, 1012Painleve, first, 861, 989Painleve, second, 861, 867, 988, 989Riccati, 140, 142, 162, 164, 423, 425, 453,
482, 520, 1012, 1277Yermakov, 309, 518
ordinary differential equations, 1445oscillatory motion
of flat plate, 1252of flat porous plate, 1265
Ostrovsky equation, 968overdetermined systems
of firstorder equations in one unknown,1599
of two equations, 1599
Ppackages (subpackages), 1627Painleve equations, 1566Painleve property, 1566Painleve test, 1569
for nonlinear equations of mathematicalphysics, 1565
for ordinary differential equations, 1566for systems of ordinary differential
equations, 1569palettes, 1627, 1688, 1691, 1737palindromic equation, 1442palindromic polynomial, 1442partial differential equation
adiabatic, 1262BBM, 958Bernoulli, 394, 421Born–Infeld, 766, 1401boundary layer, for nonNewtonian
fluid, 917boundary layer, for powerlaw fluid with
pressure gradient, 913Boussinesq, 368, 987, 1504Boussinesq, in canonical form, 987Boussinesqtype, 1028Burgers, 186, 1409, 1414, 1572, 1637,
1673, 1677, 1697, 1723, 1725, 1743,1761
Burgers–Huxley, 189Burgers–Korteweg–de Vries, 884Calogero, 745, 746Calogerotype, 1400
Page: 1864
INDEX 1865
partial differential equation (continued)Camassa–Holm, 968Chaplygin, 765Clairaut, 118, 152, 172, 1375Clairaut, in implicit form, 122combined KdVmKdV, 871complete Lin–Reissner–Tsien, 757continuity, 1197, 1247, 1610cylindric Burgers, 191cylindrical Korteweg–de Vries, 866Degasperis–Procesi, 968determining, 1555double sineGordon, 492double sinhGordon, 485Dym, 878elliptic, 1392Enneper, 491Euler–Lagrange, 1595evolution, nonlinear in second derivative,
346fast diffusion, 211firstorder quasilinear, 1355Fisher, 176, 1639, 1699, 1744FitzHugh–Nagumo, 179, 1746fKdV, 1034Fokas–Yortsos, 237fourthorder, Liouville type, 1030fourth Painleve, 988Fujita–Storm, 212Gardner, 871Gauss, 1772general fifthorder KdV, 1034generalized Burgers, 288generalized Burgers–Korteweg–de
Vries, 1045, 1503generalized Calogero, 748generalized Kadomtsev–Petviashvili, 1029generalized Khokhlov–Zabolotskaya, 751generalized Korteweg–de Vries, 871generalized Kuramoto–Sivashinsky, 983generalized Landau–Ginzburg, 360generalized Liouville, 1108Gibbons–Tsarev, 764Goursat, 549Grad–Shafranov, 678, 691Hamilton–Jacobi, 1379Hamilton–Jacobi, Cauchy problem, 1380Harry Dym, 878Helmholtz, 380, 397, 399, 420, 423, 558,
570, 601, 938, 1289homogeneous Monge–Ampere, 775, 1430Hopf, 5Hunter–Saxton, 745hydrodynamic boundary layer, with
pressure gradient, 907hyperbolic, first canonical form, 1391hyperbolic, second canonical form, 1392inviscid Burgers, 1674, 1763inviscid Burgers (explicit method), 1678,
1725inviscid Burgers (initial value problem),
1762
inviscid Burgers (shock waves andrarefaction waves), 1757
inviscid Burgers (shock waves and weaksolutions), 1759
Ito, 1037k(n,n), 891Kadomtsev–Petviashvili, 1000, 1587Kadomtsev–Petviashvili, in canonical
form, 1000Kadomtsev–Petviashvili type, 1002Karman–Howarth, 253Kaup–Kupershmidt, 1036Kawahara, 1032KdV, 857Khokhlov–Zabolotskaya, 749KK, 1036Klein–Gordon, 1641, 1656, 1701, 1716Kolmogorov–Petrovskii–Piskunov
(KPP), 279, 1633, 1694Korteweg–de Vries, 857, 1574, 1580,
1583, 1638, 1698Korteweg–de Vries, in canonical form, 857Korteweg–de Vries–Burgers, 884Korteweg–de Vries type, with exponential
nonlinearity, 874Korteweg–de Vries type, with logarithmic
nonlinearity, 874Korteweg–de Vries type, with powerlaw
nonlinearity, 872KPPtype, 1633, 1694Kuramoto–Sivashinsky, 982Laplace, 394, 409, 421, 556, 558, 579,
646, 938, 1392Lax, 1035, 1068Lax, with forcing term, 1037Lin–Reissner–Tsien, 756, 759linear heat, 807, 1252linear Schrodinger, 865, 1186linear wave, 1392Liouville, 144, 165, 470, 540, 1117, 1414mKdV, 867model, for dynamics of nonlinear
strings, 534model, of gas dynamics, 5, 25, 1360model, of nonlinear waves with damping,
6modification of coupled KdV, 1172modified Burgers–Korteweg–de Vries, 886modified Burgers, 191modified Harry Dym, 891modified k(n,n), 892modified Korteweg–de Vries, 862, 866,
869, 1583modified Korteweg–de Vries–Burgers, 886modified Korteweg–de Vries, in canonical
form, 867Monge–Ampere, 774, 1393ndimensional modified Schrodinger, with
cubic nonlinearity, 431ndimensional Schrodinger, with cubic
nonlinearity, 430
Page: 1865
1866 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
partial differential equation (continued)Newell–Whitehead, 1534ninthorder KdVtype, 1041nonlinear diffusion, 1645, 1705nonlinear diffusion, with cubic source,
1577nonlinear firstorder, 1373nonlinear Klein–Gordon, 499, 1494nonlinear Kolmogorov type, 377, 398nonlinear Poisson, 1679, 1727, 1750nonlinear Poisson–Boltzmann, 1755nonlinear Schrodinger, 348, 425, 1582nonlinear wave, 1400, 1519, 1536, 1673,
1678, 1723, 1727, 1764nthorder Liouville type, 1109of axisymmetric steady hydrodynamic
boundary layer, 1508of Burgers hierarchy, 1067of complete Korteweg–de Vries hierarchy,
1069of helical surface, 8of Korteweg–de Vries hierarchy, 1068of light rays, 63of minimal surfaces, 768, 1597of modified Korteweg–de Vries hierarchy,
1069of steady transonic gas flow, 657of surface of bodies of revolution, 8of unsteady transonic gas flows, 756Ostrovsky, 968Plebanski first heavenly, 802Plebanski second heavenly, 803Poisson, 379, 397, 423, 580, 588, 938,
1008porous medium, 210potential, of onedimensional flow of
compressible gas, 768potential Korteweg–de Vries (KdV), 876potential modified Korteweg–de Vries
(mKdV), 877Rayleigh–Zababakhin–Plesset, 1262Sawada–Kotera, 1035Sawada–Kotera, with forcing term, 1038Schrodinger, of general form, 358, 360,
363, 364Schrodinger, with cubic nonlinearity, 348,
350, 351, 352, 355Schrodinger, with powerlaw nonlinearity,
351, 353Schwarzian Korteweg–de Vries, 870secondorder nonlinear, 1392secondorder quasilinear, 1393seventhorder KdVtype, 1040Sharm–Tasso–Olver, 887, 1067sineGordon, 485, 490, 543, 1117, 1635,
1639, 1646, 1695, 1698, 1706sinhGordon, 485, 542, 1580SK, 1035spherical Korteweg–de Vries, 866stationary heat, with nonlinear source, 682stationary Khokhlov–Zabolotskaya, 649
telegraph, 494, 538, 583, 585thin film, 985thirdorder Liouville type, 972Thomas, 544threedimensional Khokhlov–Zabolotskaya,
752threedimensional nonlinear Schrodinger,
of general form, 429threedimensional Schrodinger, with cubic
nonlinearity, 428Tricomi, 659twodimensional Khokhlov–Zabolotskaya,
749twodimensional linear heat, 1256twodimensional nonlinear Schrodinger, of
general form, 427twodimensional Schrodinger, with cubic
nonlinearity, 425twodimensional Schrodinger, with
powerlaw nonlinearity, 427Tzitzeica, 541unnormalized Boussinesq, 990unnormalized Burgers, 187unnormalized Kadomtsev–Petviashvili,
1002unnormalized Korteweg–de Vries, 865unnormalized modified Korteweg–de
Vries, 870variant of modified Burgers–Korteweg–de
Vries, 886vector Burgers, 417Weber, 1193
partial differential equationsadmitting variational form, 1595axisymmetric steady laminar hydrodynamic
boundary layer, 909axisymmetric unsteady laminar boundary
layer, 929Bellman type, 805Benney moment, 764boundary layer (Prandtl equations), 1333elliptic, canonical form, 1392evolution, canonic Backlund transforma
tions, 1420Chaplygin, 1202Charpit, 1647, 1706chromatography, 1231cubic in derivatives, 134Euler, 938, 1197, 1681, 1729Euler, for barotropic gas flow, 1201Euler, in Gromeka–Lamb form, 1201Euler, in various coordinate systems, 1197evolution, nonlinear in second derivative,
346exact methods, 1353explicitly dependent on x and/or t, 266fifthorder, 1031firstorder with three or more independent
variables, 125firstorder with two independent variables
quadratic in derivatives, 43
Page: 1866
INDEX 1867
partial differential equations (continued)firstorder with two independent variables
of general form, 99firstorder quasilinear, 3for convective fluid motions, 1329fourthorder, 977Gelfand–Levitan–Marchenko type, 1584heat, with powerlaw or exponential
temperaturedependent thermaldiffusivity, 409
heat and mass transfer, 697heat and mass transfer, in quiescent
or moving media with chemicalreactions, 390
heat and mass transfer, with complicatingfactors, 734
hydrodynamic boundary layer, 903hydrodynamic type, order reduction, 1422hyperbolic, canonical form, 1390invariance under scaling transformations,
1431invariance under translations, 1429involving arbitrary differential operators,
1101involving arbitrary functions, 77, 279, 390,
499, 546, 682involving arbitrary functions of derivatives,
113involving arbitrary functions of four
variables, 123involving arbitrary functions of indepen
dent variables, 107involving arbitrary functions of one
variable, 19, 113involving arbitrary functions of three
variables, 120involving arbitrary functions of two
variables, 30, 116, 168involving arbitrary parameters, 43, 540involving arbitrary powers of derivatives,
102, 136involving cosine, 276involving cotangent, 278involving derivatives in radicands, 101involving exponential nonlinearities, 587involving first derivative in t, 811, 857,
977involving first derivative in t and linear in
highest derivative, 1031involving fourth powers of derivatives, 99involving hyperbolic cosine, 268involving hyperbolic cotangent, 270involving hyperbolic sine, 269involving hyperbolic tangent, 270involving inverse trigonometric functions,
279involving mixed derivatives, 1000, 1103involving one arbitrary power of derivative,
107involving powerlaw functions of
derivatives, 156
involving powerlaw nonlinearities, 583involving products of derivatives, 133involving products of derivatives with
respect to different variables, 131involving roots and moduli of derivatives,
136involving second derivative in t, 896, 987,
1088involving secondorder mixed derivatives,
949involving sine, 277involving squares of derivatives, 133involving squares of one or two deriva
tives, 125involving squares of three derivatives, 130involving tangent, 278involving thirdorder mixed derivatives,
958involving two or more second derivatives,
839involving two or three arbitrary powers of
derivatives, 111linear in mixed derivative, 745, 847Navier–Stokes, 1247Navier–Stokes, general properties, 1249Navier–Stokes, in Boussinesq approxima
tion, 1329Navier–Stokes, in various coordinate
systems, 1247nonlinear, diffusion boundary layer, 402nonlinear, elliptic, geometrical models,
1754nonlinear, elliptic, in two space dimen
sions, 1748nonlinear, in highest derivatives, 404nonlinear, in two independent variables,
1392nonlinear, involving arbitrary linear
differential operators, 1065nonlinear, involving arbitrary parameters,
99nonlinear, of general form, 150, 169, 1392nonlinear, of thermal (diffusion) boundary
layer, 346nonlinear, parabolic, 1677, 1725nonlinear, secondorder, 1516nonlinear, simple separation of variables,
1459nonlinear, telegraph, with two space
variables, 583nonlinear, with arbitrary number of
variables involving arbitrary functions,162
nonlinear, with arbitrary number ofvariables involving arbitrary parameters,158
nonlinear, with four independent variables,154
nonlinear, with three variables involvingarbitrary functions, 139
Page: 1867
1868 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
partial differential equations (continued)nonlinear, with three variables quadratic in
derivatives, 125nonstationary, 942nonstationary, hydrodynamic (Navier–
Stokes equations), 1013nonstationary, of motion of viscous
incompressible fluid, 1013of atmospheric circulation in equatorial
region, 1225of breeze and monsoons, 1223of Burgers and Korteweg–de Vries
hierarchies, 1067of dynamic convection in the sea, 1227of flows in the baroclinic layer of the
ocean, 1229of general form, involving arbitrary
functions of single argument, 358of general form, involving arbitrary
functions of two arguments, 362of general form, involving first derivative
in t, 1070of heat and mass transfer in anisotropic
media, 412of higher orders, 1031of ideal incompressible fluid, 942of ideal plasticity, 1238of mass transfer in quiescent or moving
media with chemical reactions, 406of motion of ideal fluid (Euler equations),
938, 1197of nonlinear vibrations stratified medium,
1131of sixth to ninthorder, 1039of stationary transonic planeparallel gas
flow, 1122of viscous incompressible fluid, 1003Plebanski heavenly, 802parabolic, canonical form, 1390passing Painleve test, 1572quadratic in derivatives, 139, 154, 158, 162quadratic in highest derivatives, 772quasilinear, 1393, 1647, 1649, 1707, 1708quasilinear, discontinuous solutions, 1360quasilinear, in conservative form, 1368quasilinear, of general form, 1368quasilinear, qualitative features, 1360quasilinear, with n independent variables,
1356reducible to Korteweg–de Vries equation,
875secondorder, classification, 1389secondorder, elliptic, with three or more
space variables, 713secondorder, elliptic, with two space
variables, 641secondorder, evolution, 1547secondorder, hyperbolic, 1551secondorder, hyperbolic, with one space
variable, 433secondorder, hyperbolic, with two or more
space variables, 553secondorder, involving mixed derivatives
and some other, 745secondorder, involving real functions of
real variables, 1165secondorder, nonlinear, for laser systems,
1170secondorder, of general form, 811, 1553secondorder, parabolic, with one space
variable, 175secondorder, parabolic, with two or more
space variables, 367semilinear, in two independent variables,
1389shallow water, 1124stationary, 938stationary, hydrodynamic (Navier–Stokes
equations), 1003steady boundary layer, for nonNewtonian
fluids, 911steady hydrodynamic boundary layer, for
Newtonian fluid, 903steady hydrodynamic boundary layer,
symmetries, 1528steadystate hydrodynamic boundary
layer, 903thirdorder, 857three and ndimensional, 428threeargument functional, 1497threedimensional, 1240threedimensional, involving arbitrary
functions, 730threedimensional, of ideal plasticity, 1240transformations, 1395twodimensional, 425, 1238twodimensional Euler, for incompressible
ideal fluid (plane flows), 1198with arbitrary dependence on derivatives,
148with arbitrary number of independent
variables, 39with coefficients involving exponential
functions, 11with coefficients involving hyperbolic
functions, 14with coefficients involving logarithmic
functions, 16with coefficients involving powerlaw
functions, 3with coefficients involving trigonometric
functions, 17with cubic nonlinearities involving
arbitrary functions, 355with exponential nonlinearities, 254, 469,
662with hyperbolic nonlinearities, 268, 485,
675with logarithmic nonlinearities, 271, 388,
486, 677with n independent variables, 739, 853with n space variables, 417
Page: 1868
INDEX 1869
partial differential equations (continued)with powerlaw nonlinearities, 175, 433,
641, 808with powerlaw nonlinearity in derivatives,
161, 167with power nonlinearity in derivatives, 146with quadratic nonlinearities, 805, 896with three independent variables, 35, 838with three or more space variables, 406with three or more space variables,
involving arbitrary functions, 624with three space variables, involving
arbitrary parameters, 604with three space variables, involving
exponential nonlinearities, 722with three space variables, involving
powerlaw nonlinearities, 713with trigonometric nonlinearities, 276, 389,
680with two independent variables, 1355with two independent variables, involving
arbitrary functions, 19with two independent variables, involving
arbitrary parameters, 3with two independent variables, nonlinear
in two or more highest derivatives, 849with two space variables involving
arbitrary functions, 589with two space variables involving
exponential nonlinearities, 384, 574with two space variables involving
powerlaw nonlinearities, 367, 553unsteady boundary layer, for Newtonian
fluid, 917unsteady boundary layer, for non
Newtonian fluids, 930von Karman (fourthorder elliptic
equations), 1192pattern, 1689, 1739performing Painleve test and truncated
expansions for studying some nonlinearequations, 1572
persistent variables, 1739Pexider equation, 1771Pfaffian equations, 1600
not satisfying the integrability condition,1602
their solutions, 1600phase diagram, 1640, 1700phase portraits, 1639, 1699phase trajectories, 1640, 1700plane jet, 1335plane radially symmetric unsteady fluid
motion, 1261Plebanski heavenly equation
first, 802second, 803
Poincare phase plane, 1640, 1700point transformations: overview and
examples, 1395Poisson equation, 379, 397, 423, 580, 588,
938, 1008polar coordinates, 1198, 1248porous medium equation, 210potential, 1587potential equation of onedimensional flow of
compressible gas, 768potential flow, 1334potential gas flow, 1201potential Korteweg–de Vries (KdV) equation,
876potential modified Korteweg–de Vries
(mKdV) equation, 877powerlaw nonNewtonian fluid, 911power isotherm, 1234Prandtl–Meyer solutions, 1203Prandtl solution, 1238predefined constants, 1628, 1740predefined functions, 1628, 1740pressure flow in rotating layer, 1283pressure gradient, 1253previous results, 1626, 1688, 1737probability integral, 187problem
Blasius, 904, 1333Blasius, reversed statement, 1334boundary value, 1679, 1727Cauchy, 1357, 1379, 1611Cauchy, classical, 1647, 1707, 1709, 1756Cauchy, for Hamilton–Jacobi equation,
1380Cauchy, procedure of solving, 1358Cauchy, qualitative features of solutions,
1611Cauchy, solution, 1360Cauchy, solution by inverse scattering
problem method, 1587Cauchy, solution for nonlinear equations
by inverse scattering problem method,1589
direct scattering, 1587Falkner–Skan, 907, 1334first Stokes, 1251generalized Blasius, 912generalized Cauchy, 1649, 1650, 1708,
1709generalized Falkner–Skan, 914generalized Schlichting, 912Hiemenz, 1264initial value, 1383, 1611initial value, for inviscid Burgers equation,
1762inverse problem, 1455inverse scattering, 1587inverse scattering, methods of solving,
soliton theory, 1579Jeffery–Hammel, 1261jetsource, 1335Landau, 1012onedisk, 1290of motion of material point, 126of motion of rod, 129of propagation of signal, 1367
Page: 1869
1870 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
problem (continued)of suspension transport in porous media,
1604on motion of two point bodies, 126without initial data, 1252, 1256Riemann, 1611Riemann, qualitative features of solutions,
1611Schlichting, 905second Stokes, 1252terminal value, 1383, 1386twobody, in celestial mechanics, 65twodisk, 1291
problemsboundary layer, selfsimilar solutions, 1333Cauchy, solutions, 1756Riemann, solutions, 1618simple inverse, determination of form of
equations from their properties, 1455procedure
for constructing exact solutions, 1450for constructing invariant solutions, 1520of solving Cauchy problem, 1358
prolongation of group generator, 1527prompt symbol, 1626, 1736pseudodifferential equations, 1484, 1483pure functions, 1690purely radial fluid motions, 1260
Qqualitative features and discontinuous
solutions of quasilinear equations, 1360quasiplane flow, 1285quasiplane flows (with fluid velocity
components independent of z), 1285quasilinear equations, 35, 1393, 1647, 1649,
1707, 1708discontinuous solutions, 1360firstorder, 3, 1355firstorder, hyperbolic systems, 1607firstorder, methods for solving, 1355hyperbolic systems, 1607in conservative form, 1368of general form, 1368PDEs, 1647, 1649, 1707, 1708qualitative features, 1360secondorder, 1393semiHamiltonian system, 1236with n independent variables, 1356
quotes, 1740
Rrange operator, 1627Rankine–Hugoniot condition, 1126, 1128,
1759Rankine–Hugoniot jump condition, 1363,
1369, 1615rarefaction wave, 6, 26, 1360, 1360, 1611,
1758
rational solution, 858Rayleigh–Zababakhin–Plesset equation, 1262reciprocal equation, 1442reciprocal polynomial, 1442reduction of systems to canonical form,
1611reduction to standard functional equation,
1496redundant solution, 1662reflectance, 1588reflection factor, 1588regular expression, 1739regular point of generalized solution, 1383related equations, 935relational operators, 1627, 1738, 1739remarks on Painleve test, 1567removing redundant exact solutions, 1665removing redundant solutions via predefined
functions, 1667resonance, Painleve test, 1570reversed statement of Blasius problem, 1334Reynolds number, 1021, 1260RF pairs, 1413
their use for constructing Backlundtransformations, 1415
Riccati equation, 140, 142, 162, 164, 423,425, 453, 482, 520, 1012, 1277
Riemann invariants, 1231, 1237, 1609, 1611,1612
Riemann problem, 1611qualitative features of solutions, 1611
Riemann simple waves, 1125, 1127, 1129rotation transformation, 1396rotationally symmetric motions of fluid,
1200rotationally symmetric motions of general
form, 1297round jet, 1277Ryzhov–Shefter solution, 759
SSawada–Kotera equation, 1035
with forcing term, 1038scalar nonlinear PDEs in one space
dimension, 1742scaling transformation, 1396, 1645, 1705,
1770scattering data, 1588scheme for constructing exact solutions by
nonclassical method, 1533Schlichting–Bickley problem, 1335Schlichting problem, 905Schrodinger equation
of general form, 358, 360, 363, 364with cubic nonlinearity, 348, 350, 351,
352, 355with powerlaw nonlinearity, 351, 353
Schwarzian Korteweg–de Vries equation,870
script, 1737, 1741mfiles, 1737
Page: 1870
INDEX 1871
second hodograph transformation, 1400secondorder differential constraints, 1553
for PDEs, 1553secondorder elliptic equations
with three or more space variables, 713with two space variables, 641
secondorder equationsBacklund transformations, 1413classification, 1389elliptic, 1191elliptic, with three or more space variables,
713elliptic, with two space variables, 641evolution, 1547hyperbolic, 1551hyperbolic, with one space variable, 433hyperbolic, with two or more space
variables, 553involving secondorder mixed derivatives,
949involving mixed derivatives and some
other, 745involving real functions of real variables,
1165Klein–Gordon type, 1173nonlinear, 1516nonlinear, classification, 1389, 1565nonlinear, for laser systems, 1170nonlinear partial differential, 1392nonlinear, symmetries, 1516of general form, 811, 1553of reactiondiffusion type, 1620parabolic, with one space variable, 175parabolic, with two or more space
variables, 367quasilinear, 1393
secondorder evolution equations, 1547secondorder hyperbolic equations, 1551
with one space variable, 433with two or more space variables, 553
secondorder nonlinear equations of lasersystems, 1170
secondorder parabolic equationswith one space variable, 175with two or more space variables, 367
secondorder quasilinear partial differentialequation, 1393
second Painleve equation, 861, 867, 988,989
second prolongation of operator, 1516second Stokes problem, 1252selfsimilar continuous solutions, 1607selfsimilar solutions, 1429, 1431, 1452
of some boundary layer problems, 1333selfsimilar variables, 1645, 1705semiHamiltonian system of quasilinear
equations, 1236semilinear equations in two independent
variables, 1389separation constant, 1459separation of variables, 1651, 1710
sequences, 1629set formula, 1755sets, 1629, 1691seventhorder KdVtype equation, 1040shallow water equations, 1124Sharm–Tasso–Olver equation, 887, 1067Shcheprov solution, 1278shock wave, 6, 26, 1675, 1126, 1128, 1362,
1615, 1616, 1758similarity and invariant solutions, 1646similarity method, 1431similarity reductions in equations with three
or more independent variables, 1510similarity transformation, 1645, 1705simple inverse problems (determination
of form of equations from theirproperties), 1455
simple nonlinear point transformations, 1396simple Riemann waves, 1609simple scheme for studying nonlinear partial
differential equations, 1570simple separation of variables in nonlinear
partial differential equations, 1459simple transformations, 1446simplest model of chemical reactor with
secondorder kinetic functions, 1115simplest plane purely radial fluid motion,
1260simplified scheme for constructing general
ized separable solutions, 1465simplifying exact solutions, 1665Simulink, 1736sinecosine method, 1642, 1702sineGordon equation, 490, 543, 1117, 1635,
1639, 1646, 1695, 1698, 1706singleprecision floating point, 1740singlesoliton solution, 491singular integral, 1373, 1374singular point, 1640, 1700
of generalized solution, 1383sinhGordon equation, 485, 542, 1580SK equation, 1035Slezkin solution, 1276slow diffusion, 251soliton, 857, 1636, 1695solution
additive separable, 1374, 1459by reduction to equations with quadratic
(or power) nonlinearities, 1487complete, 1647, 1706Couette–Poiseuille type, 1283general, firstorder PDEs, 1355generalized, 1364generalized, regular points, 1383generalized, singular points, 1383generalized, stability condition, 1369invariant, 1431, 1435, 1451, 1513, 1520,
1527, 1602, 1620Jeffery–Hamel, 1272Landau, 1277limiting selfsimilar, 1454, 1456methods, 1373
Page: 1871
1872 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
solution (continued)Mises, 768multiplicative separable, 1374, 1459, 1651,
1710nsoliton, 859, 1591nsoliton, of Korteweg–de Vries equation,
1591Nadai, 1238Nadai, multiplicative separable, 1238nonclassical selfsimilar, 500, 685, 689,
1435nonclassical travelingwave, 679, 690, 706noninvariant selfsimilar, 1435of Cauchy problem, 1360of Cauchy problem by inverse scattering
problem method, 1587of Cauchy problem for nonlinear equations
by inverse scattering problem method,1589
of determining equations in form ofpolynomials in λ, 1582
of functionaldifferential equations bydifferentiation, 1467
of functionaldifferential equations bysplitting, 1471
Olver–Rosenau, 1028onemode, 1683, 1731onesoliton, 869, 1592, 1638, 1698onesoliton, mKdV equation, 867onesoliton, potential KdV equation, 876Prandtl, 1238rational, 858redundant, 1662Ryzhov–Shefter, 759selfsimilar, 1429, 1431, 1452Shcheprov, 1278singlesoliton, 491Slezkin, 1276threesoliton, KdV equation, 859travelingwave, 1429, 1602, 1615, 1620,
1637, 1659, 1697, 1720twosoliton, 491, 869twosoliton, KdV equation, 858twosoliton, periodic, 492viscosity, 1365, 1366Volosov, 179von Mises, 768weak, 1364, 1675, 1758
solutionsadditive separable, 1459along characteristics, constructing
method, 1647, 1706analytical, 1629, 1691analytical, nonlinear systems, 1659, 1719analyticalnumerical, 1680, 1728classical selfsimilar, 1435classical travelingwave, 1431describing shock waves, 1618discontinuous, quasilinear equations, 1360exact, using nonclassical method, 1533exact, using symmetries of equations, 1520
exact, verifying, 1667for fixed singularities, 1565for Pfaffian equations, 1600for Riemann problem, 1618functional separable, structure, 1487generalized separable, structure, 1464generalized separable, simplified scheme
for constructing, 1465generalized travelingwave, 1487generalized viscosity, 1382generalized viscosity, local structure, 1383generalizing exact, 1665invariant, 1646multipeakon, 968multiplicative separable, 1459numerical, their visualizations, 1672, 1722numerical, via predefined functions, 1741numerical and graphical, 1673, 1723numerical and graphical, by default
methods, 1672numerical and graphical, specifying
singlestage numerical method, 1674numerical and graphical, specifying single
stage numerical method and numericalboundary condition (NBC), 1676
numerical and graphical, specifying twostage numerical method, startup method,and NBC, 1676
of nonlinear equations that fail Painlevetest, using truncated expansions, 1577
of nonlinear functional equations and theirapplications, 1496
of partial differential equations withmovable pole, 1569
of simple functional equations and theirapplication, 1472
oneparameter, using symmetries ofequations, 1520
removing redundant, exact, 1665removing redundant, via predefined
functions, 1667selfsimilar continuous, 1607selfsimilar, for some boundary layer
problems, 1333similarity, 1646simplifying exact, 1665special functional separable, 1487stable generalized, method for construct
ing, 1370steadystate, 1329symbolic and numerical, with Maple,
Mathematica and MATLAB, 1623travelingwave, constructing method, 1637,
1659, 1697, 1719twodimensional, in cylindrical coordinates
(plane flows), 1270twodimensional, in rectangular Cartesian
coordinates (plane flows), 1263unsteady, 1331via predefined functions, method for
constructing, 1692
Page: 1872
INDEX 1873
solutions (continued)via transformations, constructing method,
1635, 1694viscosity, based on test functions and
differential inequalities, 1383viscosity, based on use of parabolic
equation, 1382with linear dependence of velocity
components on one space variable, 1323with linear dependence of velocity
components on two space variables,1303, 1317
with movable singularities, 1565with movable singularities, for ordinary
differential equations, 1565with one nonzero component of fluid
velocity, 1251with spiral symmetry, 1328with three nonzero fluid velocity com
ponents dependent on three spacevariables, 1302
with three nonzero fluid velocity components dependent on two spacevariables, 1285
with two nonzero components of fluidvelocity, 1263
solving Cauchy problems, 1756solving nonlinear ODEs, 1664some systems depending on arbitrary
parameters, 1150spatial discretization, 1743special form for symmetry reduction, 1505special functional separable and generalized
travelingwave solutions, 1487special functional separable solutions, 1487special functions, 1740spectral collocation method, 1680, 1729spherical Korteweg–de Vries equation, 866spherical radially symmetric fluid motion,
1261spherical radially symmetric unsteady fluid
motion, 1262spherical vortex, 1278splitting in derivatives, 1516splitting method, 1496splitting procedure, 1527stability condition, 1369
for generalized solution, 1369statement separator, 1628, 1739statements, 1628, 1739stationary anisotropic heat (diffusion)
equation, 702stationary equations, 938stationary heat equation with nonlinear
source, 682stationary hydrodynamic equations (Navier–
Stokes equations), 1003stationary Khokhlov–Zabolotskaya equation,
649steady boundary layer equations for non
Newtonian fluids, 911
steady hydrodynamic boundary layerequations for Newtonian fluid, 903
steady laminar hydrodynamic boundarylayer, 903
steadystate hydrodynamic boundary layerequations, 903
steadystate solutions, 1329stream function, 903, 909, 911, 915, 917,
929, 930, 932, 938, 1003, 1013, 1333strictly hyperbolic system, 1608, 1615string, 1628, 1690string variable, 1739strip condition, 1651, 1710structure, 1737
of functional separable solutions, 1487of generalized separable solutions, 1464
subfunctions, 1741surface tension coefficient, 1262symbol, 1689symbolic and numerical solutions of nonlin
ear PDEs with Maple, Mathematica, andMATLAB, 1623
symbolic math toolbox, 1736symmetric bivariate polynomial, 1444symmetries, 1439symmetries, of equations of steady hydrody
namic boundary layer, 1528of nonlinear secondorder equations, 1516of systems of equations of mathematical
physics, 1527symmetry reductions based on generalized
separation of variables, 1507system
algebraic, 1441characteristic, 1355generic, 1754graphics, 1736hydrodynamictype, diagonal form, 1236hydrodynamictype, nondiagonal form,
1236hyperbolic, 1607, 1608, 1615Lotka–Volterra, 1135module, 1741nonlinear, of firstorder PDEs, 1659, 1719,
1720nonlinear, of secondorder PDEs, 1660,
1720of coordinate functions, simplified scheme
for constructing solutions, 1465of gas dynamic type equations, 1403of nonlinear elliptic equations, 1753of two PDEs, 1402semiHamiltonian, of quasilinear equations,
1236strictly hyperbolic, 1608, 1615
systemsbasic relations used in symmetry analysis,
1527depending on arbitrary parameters, 1150describing fluid flows in atmosphere, seas
and oceans, 1223
Page: 1873
1874 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
systems (continued)firstorder hydrodynamic, involving three
or more equations, 1197firstorder hyperbolic, of quasilinear
equations, 1607gas dynamic type, linearizable with
hodograph transformation, 1122, 1610gas dynamics, canonical form, 1611hydrodynamictype, 1236hyperbolic n × n, of conservation
laws, 1614in form of conservation laws, 1607involving arbitrary functions, 1117involving arbitrary parameters, 1115involving thirdorder evolution equations,
1172mathematical physics, symmetries, 1527nonlinear, analytical solutions, 1659, 1719nonlinear, of many equations involving
first derivatives with respect to t, 1348nonlinear, of partial differential equations,
1599nonlinear, of secondorder PDEs, 1660,
1661, 1720, 1721nonlinear, of two equations, 1347nonlinear, of two equations involving first
derivatives with respect to t, 1337nonlinear, of two equations involving
second derivatives with respect tot, 1344
nonlinear, Painleve test, 1576of algebraic equations symmetric with
respect to permutation of arguments,1444
of conservation laws of gas dynamictype, 1607
of firstorder equations describingconvective mass transfer with volumereaction, 1602
of general form, 1337of nonlinear elliptic PDEs in two space
dimensions, 1752of nonlinear equations, 1557of nonlinear PDEs in one space dimension,
1745of secondorder equations of reaction
diffusion type, 1620of two elliptic equations, 1185of two equations, 1122of two firstorder partial differential
equations, 1115of two parabolic equations, 1133of two quasilinear equations, 1165, 1607of two secondorder elliptic equations,
1191of two secondorder Klein–Gordon type
hyperbolic equations, 1173ordinary differential equations, Painleve
test, 1569overdetermined, in one unknown, 1600overdetermined, of firstorder equations in
one unknown, 1599overdetermined, of two equations, 1599reduction to canonical form, 1611reducible to ordinary differential equations,
1603various coordinate, Euler equations, 1197various coordinate, Navier–Stokes, 1247
Ttable, 1629, 1691tanhcoth method, 1641, 1701telegraph equation, 494, 538, 583, 585tensor, 1691terminal value problem, 1383, 1386thin film equation, 985thirdorder equations, 857, 949
evolution, 1172involving thirdorder mixed derivatives,
958examples of finding exact solutions, 1465
thirdorder Liouvilletype equation, 972Thomas equation, 544three and ndimensional equations, 428threeargument functional equations of
special form, 1497threedimensional equations, 1240
involving arbitrary functions, 730of ideal plasticity, 1240
threedimensional Khokhlov–Zabolotskayaequation, 752
threedimensional nonlinear Schrodingerequation of general form, 429
threedimensional Schrodinger equation withcubic nonlinearity, 428
threedimensional stagnationpoint typeflows, 1302
threesoliton solution, KdV, 859Titov–Galaktionov method, 1477toolboxes, 1738toroidal vortex, 1278tractrix equation, 253transformation
autoBacklund, 681, 1413, 1635, 1695Boussinesq, 1289classical hodograph, 1400Cole–Hopf, 1573Cole–Hopf, for Burgers equation, 1637,
1696Crocco, 906, 909, 1422Euler, 122, 346, 769, 957, 1403, 1407first hodograph, 1399Galileo, 1514hodograph, 35, 212, 344, 1129, 1397hodograph, for gas dynamic type systems,
1122, 1610hodograph, nonclassical, 345, 1399hodograph, second, 1400Hopf–Cole, 187, 888, 1067, 1409, 1414Legendre, 122, 767, 768, 770, 1403, 1406Legendre, with many variables, 1408Lorentz, 1514
Page: 1874
INDEX 1875
transformation (continued)Martin, 775Miura, 862, 868, 1410nonclassical hodograph, 345, 1399nonlocal, 1412rotation, 1396scaling, 1396, 1645, 1705, 1770second hodograph, 1400similarity, 1645, 1705translation, 1396von Mises, 761, 906, 909, 916, 937, 971,
1106, 1409, 1410transformations
Backlund, 1413, 1635, 1695Backlund, canonic for evolution equations,
1420Backlund, for secondorder equations,
1413based on conservation laws, 1425contact, 1403contact, for ordinary differential equations,
1403contact, for partial differential equations,
1405for constructing solutions, 1635, 1694in plane, 1514linear, 1396local, for derivatives, 1526of equations of mathematical physics, 1395oneparameter, 1513oneparameter, local properties, 1513oneparameter, point, Lie groups, 1526point, general form, 1395point, oneparameter Lie groups, 1526point, overview and examples, 1395point, simple, 1396preserving form of equations, 1439, 1445,
1450scaling, invariance of equations, 1431simple, 1446simple, nonlinear point, 1396translation, invariance of equations, 1430translation, invariance of solutions, 1430
transient fluid motion for whirlwind, 1258transient motion of flat plate, 1251translation transformation, 1396transversality condition, 1380travelingwave solutions, 1429, 1602, 1615,
1620, 1637, 1659, 1697, 1720ansatz methods, 1641, 1700classical, 1431functional equation, 1431general form, 1429generalized, 1487generalized, general scheme for construct
ing, 1487methods for constructing, 1641, 1700nonclassical, 679, 690, 706
Tricomi equation, 659truncated expansions, 1571twobody problem in celestial mechanics, 65
twodimensional equations, 425, 1238twodimensional Euler equations for
incompressible ideal fluid (planeflows), 1198
twodimensional Khokhlov–Zabolotskayaequation, 749
twodimensional linear heat equation, 1256twodimensional nonlinear Schrodinger
equation of general form, 427twodimensional Schrodinger equation with
cubic nonlinearity, 425twodimensional Schrodinger equation with
powerlaw nonlinearity, 427twodimensional solutions in cylindrical
coordinates (plane flows), 1270twodimensional solutions in rectangular
Cartesian coordinates (plane flows),1263
twodisk problem, 1291twosoliton periodic solution, 492twosoliton solution, 491, 869
KdV equation, 858two classes of functions, 1690two control structures, 1629, 1691types of brackets, 1628, 1690, 1739types of equations, 1389types of numbers, 1628, 1690, 1740types of quotes, 1628, 1690Tzitzeica equation, 541
Uunassignment of definitions, 1690unidirectional flows in tubes of various
crosssections, 1253unidirectional plane flows, 1251universal invariant, 1526unnormalized Boussinesq equation, 990unnormalized Burgers equation, 187unnormalized Kadomtsev–Petviashvili
equation, 1002unnormalized Korteweg–de Vries equation,
865unnormalized modified Korteweg–de Vries
equation, 870unsteady boundary layer equations
for Newtonian fluid, 917for nonNewtonian fluids, 930
unsteady hydrodynamic boundary layer, 917unsteady solutions, 1331upwind explicit finite difference scheme,
1762usage of several differential constraints,
1557userdefined function, 1628, 1691, 1737,
1740using integral relations for determining
generalized solutions, 1364using symmetries of equations for construct
ing oneparameter solutions, 1520using symmetries of equations for finding
exact solutions, 1520
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1876 Polyanin & Zaitsev, Handbook of Nonlinear PDEs, 2nd Edition, 2012
VVshaped body, 1334variable name, 1627, 1738variant of modified Burgers–Korteweg–de
Vries equation, 886various reserved keywords, 1627, 1738vector, 1629, 1691, 1741vector and array indices, 1741vector Burgers equation, 417verifying exact solutions, 1667viscosity solutions, 1365, 1366
based on test functions and differentialinequalities, 1383
based on use of parabolic equation, 1382viscous incompressible fluid, 1247Volosov solution, 179von Karman equations (fourthorder elliptic
equations), 1192von Karman flow, 1290von Karmantype rotationally symmetric
flows, 1316von Karmantype rotationally symmetric
motions, 1290von Mises solution, 768von Mises transformation, 761, 906, 909,
916, 937, 971, 1106, 1409, 1410von Mises variables, 1106von Mises yield criterion, 1238, 1239, 1240
Wwave
breaking, 1360breaking effect, 26centered rarefaction, 1359cnoidal, 858linear equation, 1392nonlinear, model equation with damping, 6nonlinear dispersive nondissipative, 857nonlinear equation, 1400, 1519, 1536,
1673, 1678, 1723, 1727, 1764overturn, 1360rarefaction, 6, 26, 1360, 1360, 1611, 1758rarefaction, for inviscid Burgers equation,
1757Riemann simple, 1125, 1127, 1129shock, 6, 26, 1675, 1126, 1128, 1362,
1615, 1616, 1758shock, for inviscid Burgers equation, 1757simple Riemann, 1609
weak solution, 1364, 1675, 1758Weber equation, 1193Weierstrass elliptic function, 176, 885, 988,
1122Whitham rule of equal areas, 1363
YYermakov equation, 309, 518
Page: 1876