A geometric method to determine PDE conservation laws

Vol. 30 (1991)

REPORTS

A GEOMETRIC

METHOD

M. Dipartimento

di Matematica,

ON MATHEMATICAL

TO DETERMINE D’APRILE

Universita

and V.

della Calabria,

No. 2

PHYSICS

PDE CONSERVATION

LAWS*

MARINO 87036 Arcavacata

di Rende

(CS), Italy

(Received August 9, 1990) It is proved within the framework of a generalized Backlund problem that, given a formally integrable partial differential equation Ek, there exist a PDE K’ whose solutions are the conservation laws for Ek. Moreover, given a conservation law fi for Ek there exists a PDE A,,, whose solutions lead to @.

1. Introduction

Differential geometry plays an important role in the theory of Partial Differential Equations (PDEs). It is in fact possible to interpret a given PDE Eli as a submanifold of a suitable jet-derivative space on a fibre bundle YT: W -+ hf. The conservation laws of Ek in this case are particular differential (n - 1)-forms which lead to elements of the cohomology group H”-‘(nQ of the base manifold llI(n = dim Al) while the set of conservation laws of Ek can be regarded as a particular term of the C-spectral sequences [11]: namely the term EF,npl. Geometrical and analytical methods have both been used to find the conservation laws associated to a given E,+ (Noether, [5, 6, 10, 111); in some cases, however, it is preferable to calculate them one by one. The purpose of this paper is to study this problem within the framework of a generalized Backlund problem. The use of fiber bundle techniques enables K’ to be obtained, i.e. the particular PDE whose solutions represent the conservation laws for a given PDE Ek. These techniques can also be used to select the particular solutions leading to these same conservation laws ($3). The paper also contains a preliminary paragraph describing the notation used; it also provides a brief overview of the theory of PDEs as geometrical objects. The authors are indebted to Prof. A. Prastaro for suggesting the problem. 2. Formal PDE theory For details x : W + Al bundle [9, lo]. 7rTT,,, : JD”(W) For any section section: D’s E * Research

of this theory see [l, 2, 4, 71. Let A1 be an n-dimensional manifold, a fibre bundle and 7Fk : JD”(W) -+ M its Ic-jet derivative associated For any natural integers T, s, with s > T, there exist canonical projections --f JD’(W); let us define ?Q,O: JD”(W) -+ W, such that nk = 7r o Xk,a. s of the bundle x, i.e. for s E C”(W), let us denote by D’s the prolonged C”(JD”(W)).

partially

supported

by M.U.R.S.T.

W91

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M.D'APRILE and V.MARINO

The difirentiul equation (PDE) Ek c JD”(W) is a fibred solution of Ek is a local section s of ?r, over an open set U contained in Ek. The set of all solutions to Ek is denoted by The r-prolongation Ek+r of E,+ is the subset of JD”+‘(W) &+r

submanifold of JD”(W); a c Al, such that D”s(U) is Sol(Ek). defined by

= JD”+r(W) n JD’(Ek).

Let vTW and vT& respectively represent the vertical bundles tangent to W and Ek and S$U’ the bundle of symmetric tensors of type (0, Ic) on M. The symbol of Ek is the family of the vector spaces over Ek g,+=

VTEk

n (s,$f

@‘EA. VTw)

.

A PDE Ek is forma& integrable if, for any r 2 0, gk +r+l is a vector bundle over Ek and

the following projection is surjective [2]: rk+r+l,k+r

’

Ek+r+l

+

Ek+r .

Spencer cohomology [l] represents a basic tool in studying formal integrability [l, 2, 71. The following theorems provide some useful criteria for the same: THEOREM [2].If Ek is a PDE in JD”(W) such that gk+l is a vector bundle over Ek, gk is 2-acyclic and the map ?‘tk+l.k: ,?&+I -+ El, is surjective, then Ek is formally integrable.

To check the surjectivity of the map rk+l,k it is possible to use: THEOREM [7].If Ek is a PDE in JDk(W) and ifgk+l is a vector bundle over Ek, then there exist a vector bundle FI over Ek and a section k of this vector bundle such that the following sequence is exact:

E/=%k

k

3

FI .

0 lc

is called the curvature of Ek.

CRITERION OF FORMAL INTEGRABILITY [2]:Given a PDE Ek in JDk(W), there exist an integer ho > k depending only on n, k and the dimension of W such that Eke is involutive and if for 0 5 j < ho - k the symbol gk+j+l is a vector bundle over Ek and rk+l+j,k+j : Ek,j + 1 -+ Ek,j is surjective, then Ek is formally integrable. Since the curvature of a PDE Ek will be used in the following as a basic tool, in the appendix we provide an example to show that the condition K - 0 produces a PDE K involving derivatives of the coefficients of Ek.

3. Conservation laws of PDEs Let Ek c JD”(W) be a PDE; let 7~: A~~,JD”(W) + JD”(W) be the fibre bundle of the differential (n - 1)-forms on JD”(W). From the set of all the differential (n - l)forms p E C@‘(AE_,JD”(W)), we only want to select those satisfying the particular equation d((Dks)*/3) = 0,

(1)

171

AMETHODTODETERMINEPDECONSERVATIONLAWS

for some s belonging to the solution set of the given PDE Ek. If there exist a solution of Ek and a differential form /3 which satisfy eq. (l), then p is called conservation law for Ek. It must be remembered that a conservation law ,Ois called trivial if p is the exterior differential of a differential (n - 2)-form n : JD”(W) 4 A~_,JD”(W). We now deal with the problem of finding conservation laws within the framework of the generalized Backlund problem [S]. Let us prove the following THEOREM. Let El, c JD”(W) be a formally integrable PDE. There exist an integer q > k and a PDE K’ c JDq+l(A~_-lJDk(W)) such that the set of conservation laws associated to EI, coincides with the set SoI( Proof: Let us consider the fibre product on JD”(W) p : P = JD”(W)

x A;-1 JD”(W)

+ JDk(W),

whose fibre on X E JD”(W) is isomorphic to TT-r(X). Let p’ : P --f JD”(W) and p* : P + A0n_l JDk(W) be the two canonical projections. From the definitions we also obtain, for any integer h, the canonical identifications JDh(P) = JDh(JDk(W)) x x JDh(&, JD”(W)) with projections pk : JDh(P)

--+ JDh(JDk(W))

and

pi : JDh(P)

+ JDh(Af_,

JD”(W)).

Let Cr be the submanifold of the first jet-derivative space of the fibre bundle p, which is described by eq. (l), where (Dks,p) are the variables in P. Let Ek+l be the first prolongation of the formally integrable PDE Ek. We then consider the following differential correspondence of order one on P: BI = (&+I

x JD(AO,_,JD”(W)))

n Cl

that is the system involving all the equations of Ek+l and Ci. Any solution to Bi is a local section s of p, s : x + (X,P(X)), with X E JD”(W), such that Ds(U) = (DX(U), D/3(U)) c BI, for some open set U c JDk(W). The actual construction can be represented by the following diagram:

JD”+‘(W) =k+l,k

JD(&,

I/

P'

JDk(W)

P

Xl .o

\I I

Id \

P2

P

JD”(W)

JD”(W))

&JDk(W)

J

71

Let p be an arbitrary section of n; we consider the pull-back Al of Bi by the D,O: Al = (DP)*&

.

172

M.D'APRILE andV.MARINO

Let us consider

Ct = pf(B1). > Al -----‘I31

JD”+‘(IV)

n1+1,b1

lp:

JD”(W) -cl When JD"(W),

section

,0 is a solution

c JD(P) c JD(flO,_, JD”(W))

04

to Cl, i.e. when Da(U) c Ct for some open set U c of JD”+t(W). In fact, the fibre of Al on with the fibre on DO(X) of

AI proves to be a fibred submanifold

X E J@(W)

can be identified

P?lB : B, + JD(nO,_,JD”(w)). Although the PDE Al is not formally integrable, it is possible to find an integer q such that Al+, is an involutive PDE (because of the integrability criterion). Moreover, to obtain a formally integrable PDE we must impose the condition that 7rq+2,q+l : A4+2 t A,+1 is surjective. We thus obtain the integrability condition K(At+,) which is a PDE K in JD’J+‘(&,JD”(W)).

= 0,

(2)

L e t us consider

the q-prolongation

Cl+,

of

Cl and the submanifold K’ = KnCI+, c JP+‘(A~_,JD”(W)). Let p’ be a solution of the PDE K’ and let A;,, be the corresponding formally integrable PDE obtained by the previous method as the such that X = Dks and is a solution to the PDE (n - l)-form /3’ satisfies

q-prolongation of A’, = (DP’)*Bj, We can choose X E J@(W) s belongs to the set Sol(A;+,) c Sol(Ek). Then the pair (X>p’) Bt, i.e., there exists a solution s to Ek for which the differential the PDE Cl:

d((D”s)*P’) = 0, i.e. p’ E Sol(K’)

is a conservation

Remark: If Sol(K’)

= 0 there

law associated

to the PDE Ek.

are no conservation

laws associated

q.e.d. to the PDE Ek.

COROLLARY: Let El, c JD” (W) be a formally integrable PDE. If p’ is a conservation law for Ek, there exists a PDE A\,, c JDq(JD’“+‘(W)) whose solutions are the solutions to Ek which lead to p’.

Proof From the construction used in proving the theorem, it can be seen that there exists a suitable prolongation A{+, of the PDE A’, = (D/Y)*Bl which is formally integrable. The set Sol(A’,+ ) is contained in Sol(Ek), thus any s E Sol(Ai+,) leads to the same conservation law @‘. Appendix This example is largely inspired by [7], Chap. 2, n. 4. Let us consider the trivial vector bundle R2 x R, whose coordinates are (x1, x2; y). We want to investigate a second order PDE on it, i.e., a fibred manifold E2 in JD2(R2 x R). Let the coordinates of the bundle JD2(R2 x R) on R2 be: (x1, x2; Y,Yl,

Y2, Yll,

Y12, Y22)

A METHOD

TO DETERMINE

PDE CONSERVATION

LAWS

173

Let E2 be defined by the following equations (here H, G are coefficients depending on the base variables zr, x2): @’ ~2 *. :

{

~22 - H(h~2)(~11)3

= 0,

y12 - G(d,~~)(y~~)~

= 0.

This system has constant rank, hence E2 is a fibred manifold on R2 x R, whose fibre dimension is 6 - 2 = 4. The trivial vertical tangent bundle vT(JD2(R2 x R)) has coordinates (x’ >x2,

y, Yl

3

Y2, Yll, Yl2r y22; 21,v,

‘u2, ‘ull, v2, u22).

these coordinates, $(R2) @a R is identified x R)) which is defined by the equations

By considering vT(JD2(R2

v=vt

with the subbundle

of

=?J2=0.

The symbol of El is the subbundle g2 of Si(R2) @ R which is given by the equations: ~22 - ~H(YII)~~JII = 0, i v12 - 2Gyllvlr = 0.

The fibre dimension of g2 is therefore 1. We now consider the bundle Fa and the families of vector spaces on E2 g2+t, FI which are defined by the following exact sequences: 0 --f vT( E2) + vT(JD2(R2

x R)) -+ F. + 0,

0-tg~4’;(R2)@vT(R2xR)+Fo+0,

(1) (2)

0 + g2+l --f S;+,(R2) 8 vT(R2 x R) + T* @ Fo -+ F, + 0.

(3)

As the fibre dimension of St(R2) @ R is 3, we find from the exact sequence (2) that FO is a rank 2 vector bundle; let the coordinates on the fibre for FO be called (f’, f2). The rank of JD3(R2 x R) is 10 and its coordinates are: (& x2; Y,Yl,

y2, Yll, Yl2, Y22r Ylll,

Yll2, Yl22r Y222).

The first prolongation Es of E2 is obtained by adding the following equations to those of E2 (here HI indicates the partial derivative of H(z’, x2) with respect to 2’ and so on):

(

~122 - fh(~rl)”

- 3H(~ll)~y111

= 0,

~222 - H2(yllj3

- 3H(~11)~~112

= 0,

~112 - Gl(~11)~

- 2G~11~111 = 0,

~122 - G2(y11J2 - 2G~11~112 = 0.

Let (v~~~,v~~~,vI~~,v~~~) be the coordinates on the fibre of Si(R2). The prolongation of the symbol is then given by the equations: ‘~122- 3H(~ll)~vlll

= 0 >

~222 - 3H(~ll)~vll2

= 0,

‘~112- 2G~11~111 = 0, ~122 - 2G~11~112 = 0.

gs

174

M. D’APRILE

and V. MARINO

When

(4)

3H = 4G’, the bundle

gs is not trivial on the submanifold E’ = E2\{yYll = 0} .

Since the ranks of S_y(R2) @ R and T* @ Fo are both 4 when (4) is not satisfied, infered from the exact sequence (3) that

it can be

Fl =O.

Then,

from the sequence

it can be deduced that K is identically 0. The interesting case is therefore represented by that when (4) holds. Then, the dimension of the fibres of g3 on E’ is 1, hence the dimension of those of Fl is also 1. Let 4 be the coordinate on the fibre for FI and (f:, ff, fi : f,“) be the coordinates on the fibre of T’ @ F 0. From sequence (1) for bundles restricted to E’, the following exact sequence can be deduced on E’: 0 + T* @ vT(E’)

+ T* @ vT(JD2(R2

x R)) + T* @ F. + 0,

where the last surjective morphism can be represented by the following are some of the coordinates in T” @ uT(JD2(R2 x R))):

equations

(5) (uZ,jk

.f: =

~1,22 - 4G2(~,d2~,,~1, f?’ = v2,22 - 4G2(~1~)2~2,11,

f? = u1,12 - ~GYIIW,II. i f;

where yrr is different from 0. The last arrow in sequence

= ~2.12 - 2Gy1~2,11.

(3) can be described

f;.

q = .f: -2Gy,,.fWe can identify

T* @ uT(JD2(R’

by

x R)) with JD’(JD2(R2 Ui,kh

=

x R)) by setting

yz,kh :

where i, Ic, h = 1, 2, and k, h are symmetric indices. Finally, by combining the last three morphisms, the coordinate obtained: 4 = y1,22 - ~GYI~YI,I~ - y2,12 + By considering the equations and only when the functions

~GYIIY~,H

representation

.

of E’, we can conclude that the curvature H, G satisfy the differential equation Hlyll

- 2GG1yI,

- GZ = 0.

of K is

K is zero when

A METHOD TO DETERMINE

PDE CONSERVATION LAWS

175

This equation coincides with the condition ensuring that the system E3 is compatible; it can be deduced by imposing on the equations of E3 the dependence conditions given by d@

-_2Gy,,$$ dxl

I

In conclusion, the condition K = 0 produces coefficients H, G of E2.

=(). 2

a PDE K involving derivatives of the

REFERENCES 111 H. Goldschmidt: Ann. of Math. 86 (1967), 246. PI H. Goldschmidt: J. DifJ: Geometry 1 (1967), 269. [31 D. Husemoller: F&e Bundles, Springer-Verlag, New York, 196.5. [41 I. S. Krasil’shchik, V. V. Lychagin, A. M. Vinogradov: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach Science Publishers, New York, 1986. PI V. Marino, A. Prastaro: Lect. Notes Math. 1209 (1986), 222. P-d V. Marino, A. Prastaro: Repons on Math. Phys. 26 (1988), 211. [71 J. F. Pommaret: Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach Science Publishers, New York, 1978. ISI J. F. Pommaret: Differential Galois Theory, Gordon and Breach Science Publishers, New York, 1983. [91 A. Prastaro: Boll. U.M.I. (5) 17-B (1980), 704; Boll. U.M.I. (5) suppl. FM (1981), 69; Boll. U.M.I. (5) suppl. FM (1981), 107. A. Prastaro: Riv. Nuovo Cimenro 5 (1982), 1. WI A. M. Vinogradov: .I. Math. Anal. Appl. 100 (1984), 1. WI