Infinite Symmetries and Infinite Conservation Lawsstaff.norbert/JNMP-Conference-2013/Rosenhaus.pdf · Equations that admit symmetries with arbitrary functions: Presence of arbitrary

Infinite Symmetries andInfinite Conservation Laws

V.Rosenhaus

California State University, Chico

Conference on Nonlinear Mathematical PhysicsNordfjordeid, June 4 - 14, 2013

V.Rosenhaus (California State University) Infinite Symmetries 1 / 54

Outline

1 Arbitrary functions in the generators of symmetry groups

2 Infinite symmetries and conservation laws

3 Infinite symmetries and Linear PDE.

4 Symmetries of hydrodynamic-type systemOpen problems


Equations that admit symmetries with arbitrary functions:

Presence of arbitrary functions in the generators ofsymmetry groups (pseudo-groups), history, different casesof arbitrary functions.

Infinite-dimensional symmetries and conservation laws forLagrangian differential systems. Essential conservationlaws.

Linear PDE and infinite-dimensional symmetries.

Hydrodynamic-type system and infinite symmetries.

Open problems.V.Rosenhaus (California State University) Infinite Symmetries 3 / 54

Arbitrary functions

Potential Kadomtsev-Petviashvili equation

uxxxx + 6uxuxx + 3s2uyy + 4uxt = 0, s2 = ∓1 (1)

(KP-I and KP-II). Lagrangian

L =u2

xx

2− u3

x − 2uxut −3s2u2

y

2, (2)


Arbitrary functions

Lie point symmetry group [David, Kamran, Levi, Winternitz,1986]:

Xm = m∂

∂x+[2xm′/3− 4s2y2m′′/9

] ∂∂u,

Xg = g∂

∂y−(2s2yg′/3

) ∂

∂x+[−4s2xyg′′/9 + 8y3f ′′′/81

] ∂∂u,

Xf = f∂

∂t+ (2yf ′/3)

∂

∂y+[xf ′/3− 2s2y2f ′′/9

] ∂∂x

(3)

+[−uf ′/3 + x2f ′′/9− 4s2xy2f ′′′/27 + 4y4f (IV )/243

] ∂∂u,

Xh = yh∂

∂u, Xl = l

∂

∂u,

where m(t),g(t), f (t),h(t), l(t) are arbitrary functions.V.Rosenhaus (California State University) Infinite Symmetries 4 / 54

Arbitrary functions and Conservation Laws

Infinite symmetry algebras with arbitrary functions and relationsto conservation laws have not been studied very extensively.

f (x1, x2, ..., xn) do not lead to conservation laws.Noether II: Infinite variational symmetries with arbitraryfunctions of all independent variables lead to a certainrelation between equations of original system (the originalsystem is underdetermined) .f (x1, x2, ..., xk ), k < n lead to a finite number of essentialconservation laws. f (t).Arbitrary functions of not all independent variables lead to afinite number of essential conservation laws, and eachessential conservation law is determined by a specific formof boundary conditions. [Rosenhaus, JMP, 2002]


Arbitrary functions of independent variables

Infinite symmetries and corresponding essential conservationlaws

Transonic gas flows” [J. Dyn. Sys. Geom. Theor., 2002],“Short waves” equation, [Rep. Math. Phys., 2003]Potential Zabolotskaya–Khokhlov equation, [JNMP, 2006]Potential Kadomtsev–Petviashvili equation, [J. Phys. A,2006],The Davey-Stewartson equation” [V.R. and M.Gandarias, J.Phys. A, 2010].


Arbitrary functions of dependent variables

Infinite symmetry algebras with arbitrary functions of dependentvariables - radically different situation leading to an infinite setof essential conservation laws (local conserved densities)[V.R. TMP 2005, 2007].Example: equations of Liouville type: [Zhiber, Shabat, Dokl.Akad. Nauk SSSR, 1979], [Zhiber, Ibragimov, Shabat Dokl.Akad. Nauk SSSR, 1979], orDarboux - integrable diff.eq. [Sokolov, Zhiber, Phys. Lett. A,1995], [Bryant, Griffiths, Hsu, Selecta Math., 1995], [Anderson,Kamran, Duke Math. J., 1997], [Juras, Anderson I.M., DukeMath. J., 1997].Another situation - hydrodynamic type equations.



TheoremEquations with infinite variational symmetry algebrasparametrized by arbitrary functions of dependent variablespossess infinite sets of essential conservation laws.


Essential Conservation Laws

DefinitionBy a conservation law for a differential system

ωa(x ,u,u(1),u(2), . . . ) = 0, a = 1, . . . ,n,

is meant a divergence expression

DiKi(x ,u,u(1),u(2), . . . ).

= 0,

vanishing for all solutions of the original system ( .=).x = (x1, x2, . . . , xm+1), xm+1 = t , u = (u1,u2, . . . ,un),u(r) - r th-order derivatives of u, ωa and Ki are differentialfunctions: smooth functions of x , u, and u(r).



Definition

Two conservation laws K and K are equivalent if they differ bya trivial conservation law [Olver].

DefinitionA conservation law DiPi

.= 0 is trivial if a linear combination of

two kinds of triviality is taking place:1. (m+1)-tuple P vanishes on the solutions of the originalsystem: Pi

.= 0.

2. The divergence identity is satisfied in the whole space:DiPi = 0.



DefinitionBy an essential conservation law , we mean such non-trivialconservation law DiKi

.= 0, which gives rise to a non-vanishing

conserved quantity

Dt

∫DKt dx1dx2 · · · dxm .

= 0, x ∈ D ⊂ Rm+1, Kt ˙6= 0. (4)


The approach

We consider functions ua = ua(x) defined on a region D of(m+1)-dimensional space-time. Let

S =∫

DL(x ,u,u(1), . . . ) dm+1x

be the action functional, where L is the Lagrangian density.Then the equations of motion are

Ea(L) ≡ ωa(x ,u,u(1),u(2), . . . ) = 0, (5)

whereEa =

∂

∂ua −∑

iDi

∂

∂uai

+∑i6j

DiDj∂

∂uaij

+ · · · , (6)

is the Euler–Lagrange operator.


The approachConsider an infinitesimal (one-parameter) transformation withthe canonical infinitesimal operator

Xα = αa ∂

∂ua +∑

i(Diα

a)∂

∂uai

+∑i6j

(DiDjαa)

∂

∂uaij

+ · · · , (7)

αa = αa(x ,u,u(1), . . . ).

Variation of the functional S under the transformation withoperator Xα is

δS =∫

DXαL dm+1x . (8)

Xα is a variational (Noether) symmetry if

XαL = DiMi , (9)

where Mi = Mi(x ,u,u(1), . . . ) are smooth functions.V.Rosenhaus (California State University) Infinite Symmetries 13 / 54

The approachThe Noether identity. Relates the operator Xα to Ea,

Xα = αaEa + DiRαi , (10)

Rαi = αa ∂

∂uai

+

∑k>i

(Dkαa)− αa

∑k6i

Dk

∂

∂uaik

+ · · · . (11)

Applying the Noether identity to L we obtain

Di(Mi − RαiL) = αaωa, (12)

On the solution manifold (ω = 0, Diω = 0, . . . ) is

Di(Mi − RαiL).

= 0, (13)

Thus, any variational one-parameter symmetry transformationXα gives rise to a conservation law (13) - First Noether Theorem.


Arbitrary functions of independent variables

Consider an infinite variational symmetry with a characteristic αof the form

αa = αa0p(x) + αaiDip(x) +∑i6jαaijDiDjp(x) + · · · , (14)

where p(x) are smooth functions, and αa0, αai , αaij , . . . aredifferential functions.

p(x) is an arbitrary function of all base variables Accordingto Noether II a consequence of an infinite symmetry (11) offunctional S is not a conservation law but a certain relationbetween the original differential equations [Noether 1918].p(x) is an arbitrary function of not all base variables[Rosenhaus JMP 2002].


Boundary conditionsFor a Noether symmetry operator Xα we have

δS =∫

DδL dm+1x =

∫D

XαL dm+1x =∫

DDiMi dm+1x = 0. (15)

Therefore, the following conditions for Mi (Noether boundaryconditions) should be satisfied

Mi(x ,u, . . . )∣∣∣∣x

i→ ∂D= 0, i = 1, . . . ,m + 1, (16)

where i→ denotes the limit along the i th axis. Equations (16) areusually satisfied for a regular asymptotic behavior: ua → 0 andua

i → 0 at infinity, or for periodic solutions.Integrating (13) over the space Ω (x1, x2, . . . , xm) we get∫

Ωdx1 . . . dxmDt(Mt − RαtL)

.=∫

Ωdx1 . . . dxm

m∑i=1

Di(RαiL−Mi).

(17)V.Rosenhaus (California State University) Infinite Symmetries 16 / 54

Boundary conditionsApplying the Noether boundary condition and requiring the LHSto vanish on the solution manifold leads to the “strict”boundary conditions

Rα1L∣∣∣∣x

1→ ∂Ω= Rα2L

∣∣∣∣x

2→ ∂Ω= · · · = RαmL

∣∣∣∣x

m→ ∂Ω= 0. (18)

In the case L = L(x ,u,u(1)), the strict boundary conditions takea simple form

αa ∂L∂ua

l

∣∣∣∣x

l→ ∂Ω= 0, l = 1, . . . ,m. (19)

Thus, Noether and strict boundary conditions are necessary forthe existence of essential conservation laws. In case ofarbitrary functions of independent variables these conditionsallow only a finite number of essential conservation laws.


Arbitrary functions of dependent variablesConsider infinite symmetries whose characteristics contain anarbitrary smooth function f of dependent variables and theirderivatives, i.e.

αa = αa0f + αas∂sf + · · · ,where s numerates arguments of f = f (u,u(1), . . . ), and αa0, αas,. . . are some differential functions. The conservation law (17)then has the form

Dl(Ml − RαlL) + Dt(Mt − RαtL).

= 0, l = 1, . . . ,m, (20)

where

Mi = M0i f + Ms

i ∂sf + · · · ,RαiL = P0

i f + Psi ∂sf + · · ·

(21)

for some differential functions M0i , Ms

i , . . . , and P0i , Ps

i , . . . .V.Rosenhaus (California State University) Infinite Symmetries 18 / 54


In order for the system to possess (Noether) local conservedquantities, both Noether (16) and strict boundary conditions (18)have to be satisfied. Let

f (u,u(1), . . . ) = g(ξ(u,u(1), . . . )), (22)

where g is some smooth function. Assuming regular boundaryconditions ua → 0, ua

i → 0, . . . at infinity, the Noether boundaryconditions take the form

Mi (g(ξ(0, . . . ,0),g′(ξ(0, . . . ,0), . . . )−Mi (g(ξ(0, . . . ,0),g′(ξ(0, . . . ,0), . . . ) = 0.

(23)


Arbitrary functions of dependent variablesThese conditions are satisfied for any smooth function g and∣∣∣ξ(0, . . . ,0)

∣∣∣ <∞. (24)

The strict boundary are also satisfied if conditions (24) are met.Thus, in the case of infinite symmetries with arbitrary functionsof dependent variables αa = f a(u,u(1), . . . ) unlike the case withindependent variables, there are no serious restrictions forfunctions f a to lead to local conservation laws, and thecontinuity equation provides an infinite number of essentialconservation laws [Rosenhaus, TMP, 2007]. Thecorresponding Noether conserved quantities

Dt

∫Ω

dx1dx2 . . . dxm (Mt − RαtL).

= 0. (25)


f(u)1. Which equations would admit infinite variational symmetrieswith an arbitrary function of u: f (u)?

α = pf (u) + qf ′(u) + rf ′′(u), (26)

where p,q, r = gj(x , t ,u,ux ,ut), j = 1,2,3. We look forequations with Lagrangians of the first order

L = L(u,ux ,ut). (27)

Requiring Xα to be a variational symmetry

XαL = DxMx + DtMt ,

Mx = Af (u) + Bf ′(u) + Cf ′′(u), (28)Mt = Ef (u) + Ff ′(u) + Gf ′′(u),

(A,B,C,E ,F ,G = hj(x , t ,u,ux ,ut)) we obtain:V.Rosenhaus (California State University) Infinite Symmetries 21 / 54

f(u)

L(u,ux ,ut) = R(ux

ut,u)

+ ut S(ux

ut,u), (29)

where R and S are arbitrary functions. [V.R.TMP 2007]

Any equations with Lagrangians of the form (29) have an infinitenumber of essential conservation laws given by expression (13).


f (ux ,ut)

2. Look for equations admitting infinite variational symmetrieswith an arbitrary function of the first derivatives f (ux ,ut)

α = pf (ξ) + qf ′(ξ) + rf ′′(ξ). (30)

Requiring Xα to be a variational symmetry

XαL = DxMx + DtMt ,

Mx = Af (ξ) + Bf ′(ξ) + Cf ′′(ξ), (31)Mt = Ef (ξ) + Ff ′(ξ) + Gf ′′(ξ),

(A,B,C,E ,F ,G,p,q, r = hj(x , t ,u,ux ,ut)). We obtain

L = ut P(ux

ut,u)

(32)

L = Q(ux ,ut). (33)

with arbitrary functions P,QV.Rosenhaus (California State University) Infinite Symmetries 23 / 54

f (ux ,ut)

The function ξ(z,w) should satisfy

ξw2Lzz − 2rξwξzLzw + ξz

2Lww = 0. (34)

To ensure the existence of real solutions for ξ the followingcondition has to hold.

L2zw − LzzLww 0. ux ≡ z,ut ≡ w). (35)

Any equations with Lagrangians of the form (32) or (33) have aninfinite number of essential conservation laws given byexpression (13)


f (ux ,ut)

An interesting example of the class (33) is Born-Infeld equation

(1− u2t )uxx + 2utuxuxt − (1 + u2

x )utt = 0, (36)

with the Lagrangian density

L =(1 + u2

x − u2t

)1/2. (37)

Born–Infeld Lagrangian is related to classical relativistic stringLagrangian in 4-d space. Infinite set of conservation laws forBorn–Infeld equation and its infinite group of contacttransformations [Koiv, Rosenhaus, Algebras Groups Geom.,1986].


f (ux ,ut)

Another interesting example is hyperbolic Fermi–Pasta–Ulamequation [Juras, J. Diff. Eq., 2000]

utt − ku2x uxx = 0, (38)

with the Lagrangian density

L =u2

t

2− ku3

x

3. (39)


f (ux ,ut ,uxx ,uxt ,utt)

3. Look for equations possessing infinite variational symmetrieswith an arbitrary function of the dependent variables and itsfirst and second derivatives. We look for equations withfirst-order Lagrangians, L = L(u,ux ,ut), possessing Xα as avariational symmetry for each characteristic α of the form

α = Pf (ξ) + Qf ′(ξ) + Rf ′′(ξ), (40)

where f is an arbitrary smooth functions of a single argument, ξis a smooth function of u and its first and second derivatives:ξ = ξ(u,u(1),u(2)), and P, Q and R are some differentialfunctions. Then we have [V.R. 2013]

Dtξ.

= 0. (41)

(or Dxξ.

= 0). Equations of Liouville type.V.Rosenhaus (California State University) Infinite Symmetries 27 / 54

f (ux ,ut ,uxx ,uxt ,utt)

The most common equation of this class is the Liouville equation

uxt = eu with L =uxut

2+ eu, (42)

for which we have

α = ux f ′(ξ) + Dx f ′(ξ) = ux f ′(ξ) + ξx f ′′(ξ),

where

ξ = uxx −u2

x

2.

Indeed,Dtξ = uxxt − uxuxt = −Dxω + uxω

.= 0

where ω = eu − uxt .


gauge transformations

In the field theory the presence of arbitrary functions ofindependent variables g(x) in the generators of a symmetrygroup is gauge invariance. However, the total symmetry groupincludes an arbitrary function of dependent variables as well[Kiiranen, Rosenhaus, 1989]. For example, for Maxwellequations with the action functional

S =∫∫

FµνFµν d4x (43)

Fµν = Aν,µ − Aµ,ν , Aµ,ν =∂Aµ

∂xν, µ, ν = 1, . . . ,4,



a variational symmetry transformation

Xα = αµ∂

∂Aµ+∑µ,ν

(Dναµ)

∂

∂Aµ,ν

+ · · ·

takes a formαµ = Dµg (x ,A,A,ν , . . .) (44)

Dµ =∂

∂xµ+ Aν

, µ

∂

∂Aν+ · · ·

g is an arbitrary function of all xν as well as 4-potentials Aµ andtheir derivatives. This is an extended form of the gauge(gradient) transformation for Maxwell equation.



Similarly, for the Yang-Mills equations (non-abelian case) a localgauge transformation can also be extended to include arbitraryfunctions g of potentials A and their derivatives :

g = g(x ,A,A,ν , . . .).


Infinite symmetries and Linear PDE

The presence of arbitrary functions in symmetry generators andcomponents of a continuity equation is not unusual for linearequations. The wave equation

uxx − utt = 0, (45)

has infinite conservation laws:

Dt (f (ut + ux ) + g(ut − ux )) + Dx (−f (ut + ux ) + g(ut − ux )).

= 0.(46)

with arbitrary functions f and g.



Similarly, for a parabolic equation

R2uxx + 2Ruxt + utt = 0, R = const . (47)

infinite conservation laws are given by

Dt (f (ut + Rux )) + Dx (f (ut + Rux )).

= 0. (48)

with an arbitrary function f .


Infinite symmetries and Linear PDELet us look at a connection between equations with infinitesymmetries and linear equations. Consider the class (33) ofequations with symmetries containing an arbitrary function ofthe first derivatives f (ux ,ut). The Lagrangian function (33)

L = Q(ux ,ut), (49)

leads to the equation

Dx (Lux ) + Dt (Lut ) = 0. (50)

In the notations ux ≡ z,ut ≡ w we will get the following system

uxxLzz + 2uxtLzw + uttLww = 0, (51)zt = wx .



Using hodograph transformation:

z(x , t), w(x , t)→ x(z,w), t(z,w)

the system (51) will take a form

twLzz − (xw + tz)Lzw + xzLww = 0, (52)xw = tz ,

which is a linear system.



Consider the class (29) of equations possessing symmetrieswith an arbitrary function of u

L = R(ux

ut,u), (53)

where function R is arbitrary.



Corresponding Euler-Lagrange equation will have a form

−u2t uxx + 2uxutuxt − u2

x utt = u4t f(ux

ut,u), (54)

where

f(ux

ut,u)

= −Ru

(uxut,u)

Rξξ

(uxut,u) , (55)

andξ =

ux

ut.



Applying a hodograph transformation again u(x , t)→ t(x ,u) theequation (54) can be given a following form

txx (x ,u) = f (u,−tx (x ,u)) . (56)

The equation (56) in general, is not linear although it is linearwith respect to second derivatives.


Symmetries of hydrodynamic-type system

Joint work with Jonathan Roy.Systems of hydrodynamic type were introduced by Dubrovinand Novikov [Sov. Math. Dokl. 1983] as quasi-linear systemsof first order PDE which possessed Hamiltonian structure.Hydrodynamic-type systems describe various physicsphenomena; hydrodynamics, gas dynamics, MHDmagnetohydrodynamics, nonlinear elasticity, nonlinear plasticitymodels, and other.



We will not require a Hamiltonian structure and consider ageneral n-component homogeneous system of first order PDEin the form

u it =

n∑j=1

vj(u1,u2, . . . ,un, x , t) u jx , (57)

where u i = u i(x , t), i = 1,2, . . . ,n: with n functions u i thatdepend on time t and one space variable x . We will furtherassume that the system is diagonalizable (in terms of itsRiemann invariants) and the coefficients v i do not depend on xor t explicitly:

u it = v i(u1,u2, . . . ,un) u i

x , i = 1,2, . . . ,n. (58)


Symmetries of hydrodynamic-type systemMany papers have been written on systems ofhydrodynamic-type , incl. work related to invariance propertiesof this system: Tsarev, Ferapontov, Rogers, Sheftel, others.

Symmetries of hydrodynamic-type systems [Sheftel 2002],[Grundland, Sheftel, Winternitz J.Phys. A 2000]. Theyconsidereda) Strictly hyperbolic systems: v i 6= v j for i 6= j . We will notassume the condition of non-degeneracy of the spectrum.b) Coefficients v i had explicit dependence on t .c) Looking for hydrodynamic flows commuting with the flowsdetermined by the original system:

u iτ =

n∑j=1

Aij(u

1,u2, . . . ,un, x , t) u ix , i = 1,2, . . . ,n. (59)



(Sheftel, Grundland, Winternitz) called these commuting flowshydrodynamic symmetries. These symmetries are not classicalLie point or tangent symmetries.

We will look for point and tangent transformation group for thesystem (58), and our main interest is the presence of infinitesymmetries with arbitrary functions. Start with the simplest case

n = 1.We have

ω = ut − v(u)ux = 0. (60)



We are looking for a symmetry operator

Xα = α(u, ux )∂

∂u+ (Dx α(u, ux ))

∂

∂ux+ (Dt α(u, ux ))

∂

∂ut, (61)

such thatXα ω

.= 0. (62)

We will get the following result:



v ′(u) 6= 0 α(u, ux ) = C(u)ux , C(u) is arbitrary.v ′(u) = 0 α(u, ux ) is arbitrary.

Thus, n = 1 hydrodynamic systems admit infinite-dimensionalsymmetry with arbitrary functions either of u or both u andux .



n = 2.ωi = u i

t − v i(u1,u2)u ix = 0. (63)

The symmetry operator will take a form

Xα = α1(u1,u2,u1x ,u

2x )

∂

∂u1 + α2(u1,u2,u1x ,u

2x )

∂

∂u2 + . . . . (64)

The specific form of symmetry admitted by the system willdepend on the form of functions v i(u1,u2).



1. v1(u1,u2) 6= v2(u1,u2). in this case

α1 = α11(u1,u2) + α1

2(u1,u2)u1x ,

α1 = α21(u1,u2) + α2

2(u1,u2)u2x . (65)

Many sub-cases leading to various symmetries; wherecoefficients αi

k (u1,u2) include arbitrary functions l(u1),m(u2) orS(u1,u2).



2. v1(u1,u2) = v2(u1,u2) = v(u1,u2).

2a. vu1 = vu2 = 0 (66)

α1(u1,u2,u1x ,u2

x ), α2(u1,u2,u1x ,u2

x ) are completely arbitraryfunctions of four variables.

2b. v = v(u1) (67)

The coefficient v(u1,u2) is a function of a single variable.



α1(u1,u2,u1x ,u

2x ) = C1

(u1,u2,

u2x

u1x

)u1

x , (68)

α2(u1,u2,u1x ,u

2x ) = C1

(u1,u2,

u2x

u1x

)u2

x + C2

(u1,u2,

u2x

u1x

). (69)

with arbitrary functions C1, C2.

Symmetry vectors have arbitrary functions of threevariables.



2c. vu1 , vu2 6= 0. (70)

The coefficient v(u1,u2) is a function of two variables. In thiscase we obtain

α1(u1,u2,u1x ,u

2x ) = f

(u1,u2,

u2x

u1x

)u1

x − vu2 g(

u1,u2,u2

x

u1x

), (71)

α2(u1,u2,u1x ,u

2x ) = f

(u1,u2,

u2x

u1x

)u2

x + vu1 g(

u1,u2,u2

x

u1x

). (72)

Again, the symmetry group (pseudo-group) has arbitraryfunctions of three variables.



Corollary

In degenerate case v1 = v2 the symmetry of 2x2 system ofhydrodynamic-type drastically changes to include arbitraryfunctions of the derivatives u1

x ,u2x . The same effect is taking

place for cases of higher n.



n = 3.

1. v1(u1,u2,u3) 6= v2(u1,u2,u3) 6= v3(u1,u2,u3).

αi = αi1(u i) + αi

2(u1,u2,u3)u ix , i = 1,2,3 (73)

Many sub-cases leading to various symmetries; coefficients αik

include arbitrary functions l(u1),m(u2),n(u3),p(u1,u2),q(u1,u3),or r(u2,u3).



2. v2 = v3. (74)

α1 = α11(u1) + α1

2(u1,u2,u3)u1x ,

α2 = α21(u2,u3) + α2

2(u1,u2,u3)u2x + α2

4

(u2,u3,

u3x

u2x

),

α3 = α31(u2,u3) + α3

2(u1,u2,u3)u2x + α3

4

(u2,u3,

u3x

u2x

)(75)

In the degenerate case arbitrary functions of derivatives show

up.



3. v1 = v2 = v3. (76)

αi have arbitrary functions C i(u1,u2,u3, u2

xu1

x, u3

xu1

x

).

More degeneracy - more degrees of freedom in the derivatives.

Relations between infinite symmetries and conservation laws forhydrodynamic-type (non-Lagrangian) systems


Open problems

Physical meaning of infinite number of conservation laws.

Relations between systems with infinite-dimensionalsymmetries and known integrable systems


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Infinite Symmetries and Infinite Conservation Lawsstaff.norbert/JNMP-Conference-2013/Rosenhaus.pdf · Equations that admit symmetries with arbitrary functions: Presence of arbitrary