Recent Titles in This Series

167 V . V. Lychagin, Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l Equation s

166 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volum e II I

165 Yu . Ilyashenk o an d S. Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . Bokut' , M . Hazewinkel , an d Yu. G . Reshetnyak , Editors , Thir d Siberia n Schoo l "Algebr a

and Analysis "

162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m

161 Katsum i Nomizu , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s

160 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d Differentia l

Geometry

159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society , Volum e I I

158 A . K . Kelmans , Editor , Selecte d Topic s i n Discret e Mathematics : Proceeding s o f th e Mosco w

Discrete Mathematic s Seminar , 1972-199 0

157 M . Sh . Birman , Editor , Wav e Propagation . Scatterin g Theor y

156 V . N. Gerasimov , N . G . Nesterenko , an d A. I . Valitskas , Thre e Paper s o n Algebra s an d Thei r

Representations

155 O . A . Ladyzhenskay a an d A. M . Vershik , Editors , Proceeding s o f th e St . Petersbur g Mathematica l

Society, Volum e I

154 V . A. Artamonov e t al. , Selecte d Paper s i n K -Theory

153 S . G . Gindikin , Editor , Singularit y Theor y an d Som e Problem s o f Functiona l Analysi s

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151 LA . Aleksandrov , L . A . Bokut' , an d Yu . G . Reshetnyak , Editors , Secon d Siberia n Winte r Schoo l

"Algebra an d Analysis "

150 S . G . Gindikin , Editor , Spectra l Theor y o f Operator s

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148 A . D . Aleksandrov , O . V . Belegradek, L . A . Bokut' , an d Yu . L . Ershov , Editors , Firs t Siberia n

Winter Schoo l "Algebr a an d Analysis "

147 I . G . Bashmakov a e t al. , Nin e Paper s fro m th e Internationa l Congres s o f Mathematicians , 198 6

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143 V . A. Belonogo v e t al. , Eigh t Paper s Translate d fro m th e Russia n

142 M . B . Abalovic h e t al. , Te n Paper s Translate d fro m th e Russia n

141 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s

140 V . I. Berni k e t al. , Eleve n Paper s Translate d fro m th e Russia n

139 A . Ya . ATzenshta t e t al. , Ninetee n Paper s o n Algebrai c Semigroup s

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137 V . I. Arnol d e t al. , Fourtee n Paper s Translate d fro m th e Russia n

136 L . A . Aksent'ev e t al. , Fourtee n Paper s Translate d fro m th e Russia n

135 S . N . Artemo v e t al. , Si x Paper s i n Logi c

134 A . Ya . ATzenshta t e t al. , Fourtee n Paper s Translate d fro m th e Russia n

133 R . R . Sunchelee v e t al. , Thirtee n Paper s i n Analysi s

132 I . G . Dmitrie v e t al. , Thirtee n Paper s i n Algebr a

131 V . A. Zmorovic h e t al. , Te n Paper s i n Analysi s

130 M . M . Lavrent'ev , K . G . Reznitskaya , an d V . G. Yakhno , One-dimensiona l Invers e Problem s o f

Mathematical Physic s

129 S . Ya . Khavinson , Tw o Paper s o n Extrema l Problem s i n Comple x Analysi s (See th e AM S catalo g fo r earlie r titles )

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The Interplay betwee n Differential Geometr y an d Differential Equation s

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 16 7

Advances in the Mathematical Sciences—24 {Formerly Advances in Soviet Mathematics)

The Interpla y betwee n Differential Geometr y an d Differential Equation s

V. V. Lychagi n Editor

American Mathematical Societ y Providence, Rhode Island

^/VDED

http://dx.doi.org/10.1090/trans2/167

ADVANCES IN TH E MATHEMATICAL SCIENCE S EDITORIAL COMMITTE E

V I . ARNOL D S. G . GINDIKI N V. P . MASLO V

1991 Mathematics Subject Classification. Primar y 17B37 , 35Axx , 58Axx , 58Gxx .

ABSTRACT. Th e purpos e o f th e boo k i s t o emphasiz e th e advantag e o f algebrai c geometr y approac h to nonlinea r differentia l equations , includin g application s o f symplecti c method s an d th e discussio n o f quantization problems . On e of the common feature s for the majority o f papers in the book is the systematic use o f geometry o f jet spaces . Th e boo k i s useful t o researcher s an d graduat e student s wh o ar e intereste d in nonlinea r differentia l equations , differentia l geometry , quantu m groups , an d thei r applications .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-0428- 6

ISSN 0065-929 0

Copying an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r educationa l an d scientific purposes without fee or permission with the exception o f reproduction b y services that collect fee s for deliver y o f documents an d provide d tha t th e customary acknowledgmen t o f th e sourc e i s given. Thi s consent doe s no t exten d t o othe r kind s o f copying fo r genera l distribution , fo r advertisin g o r promotiona l purposes, o r fo r resale . Request s fo r permissio n fo r commercia l us e o f materia l shoul d be addresse d t o the Assistant Directo r o f Production, America n Mathematica l Society , P . O. Box 6248, Providence, Rhod e Island 02940-6248 . Request s can also be made by e-mail t o reprint-permiss ionQmath. am s . org.

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© Copyrigh t 199 5 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

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Contents

Foreword V. LYCHAGI N i x

Modeling Integro-Differentia l Equation s an d a Metho d fo r Computin g their Symmetrie s an d Conservatio n Law s V. N . CHETVERIKO V an d A . G . KUDRYAVTSE V 1

Braiding o f th e Li e Algebra si (2) J. DONI N an d D . GUREVIC H 2 3

Poisson-Lie Aspect s o f Classica l ^-Algebra s B. ENRIQUEZ , S . KHOROSHKIN , A . RADUL , A . ROSLY , an d V . RUBTSO V 3 7

On Symmetr y Subalgebra s an d Conservatio n Law s fo r th e k - e Turbulence Mode l an d th e Navier-Stoke s Equation s N. G . KHOR'KOV A an d A . M . VERBOVETSK Y 6 1

Graded Frolicher-Nijenhui s Bracket s an d th e Theor y o f Recursio n Operators fo r Supe r Differentia l Equation s R H . M . KERSTE N an d I . S . KRASIL'SHCHI K 9 1

Symplectic Geometr y o f Mixe d Typ e Equation s A. KUSHNE R 13 1

Homogeneous Geometri c Structure s an d Homogeneou s Differentia l Equations V. LYCHAGI N 14 3

Geometry o f Quantize d Supe r PDE' s AGOSTINO PRASTAR O 16 5

Symmetries o f Linea r Ordinar y Differentia l Equation s ALEXEY V . SAMOKHI N 19 3

Foliations o f Manifold s an d Weightin g o f Derivative s N. A . SHANANI N 20 7

Higher Symmetr y Algebr a Structure s an d Loca l Equivalence s o f Euler -Darboux Equation s VALERY E . SHEMARULI N 21 7

Hyperbolicity an d Multivalue d Solution s o f Monge-Amper e Equation s D. V . TUNITSK Y 24 5

viii CONTENT S

Singularities o f Solutions o f the Maxwell-Dirac Equatio n L. ZiLBERGLEI T 26 1

Characteristic Classe s of Monge-Ampere Equation s L. ZiLBERGLEI T 27 9

Foreword

This collection presents work concentrated mainly around the differential geom -etry approach to the theory of nonlinear differentia l equations .

Actually, differentia l geometr y an d differentia l equation s ar e s o closely relate d that i t i s practically impossibl e t o dra w a clea r delimitin g lin e betwee n thes e tw o branches of mathematics.

Thus, whereas the connections betwee n linea r differentia l equation s an d differ -ential geometry were few and far between, as were the connections between analytic geometry (i n the elementary sens e of the term) an d geometr y in the sense of Feli x Klein, in contrast the theory of nonlinear differential equation s is clearly a geometric theory, based on the special geometry o f jet spaces .

Moreover, differentia l geometry , in its turn, may be presented a s the part o f the theory of nonlinear differential equations that studies differential equations that have no solutions. Fro m thi s point o f view, all the basic notions of differential geometr y acquire their natural meaning. Fo r example, various curvatures and torsion arise as obstructions to the solvability of the appropriate differential equations . I n this sense the theory of differential equation s in its classical understanding is the geometry of flat objects .

The papers comprising this collection may be conditionally subdivided into three groups.

The first group consists of papers dealing with general questions of the geometric theory o f differentia l equation s an d th e method s o f differentia l geometr y use d i n their study (I . Krasilshchik and P. Kersten, A. Kushner, A. Prastaro. A . Samgkhin , N. Shananin, D . Tunitsky, L. Zilbergleit).

The second group deals with applications of the geometric theory o f differentia l equations t o th e stud y o f specifi c problem s (A . Khor'kov a an d A . Verbovetsky , V. Chetverikov and A. Kudryavtsev, V. Shemarulin, L . Zilbergleit).

Finally, th e thir d grou p i s devote d t o quantizatio n problem s (I . Doni n an d D. Gurevich, B . Enriques, S. Khoroshkin, A. Radul, A. Rosly, and V. Rubtsov).

The appearance of this last group of papers in a collection mainly concerned with differential equation s understood in the classical sense, and not in the quantum one, nevertheless seems quite logical to me, since the true understanding of the problems of differential equation s must unavoidably pass through the quantum domain .

V. Lychagin Moscow, August 199 4

Translated by A. B. SOSSINSKY

IX

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