Conservation Laws 1. Basic Definitions and Examples

S.11. Conservation Laws S.11.1. Basic Definitions and Examples

Consider a partial differential equation with two independent variables

∂x ∂t ∂x ∂x∂t ∂x 2

A conservation law for this equation has the form

∂x ∂t The left-hand side of the conservation law (2) must vanish for all (sufficiently smooth) solutions

∂x ∂t

of equation (1). In simplest cases, the substitution of relations (3) into the conservation law (2) followed by differentiation and elementary transformations leads to a relation that coincides with (1) up to a functional factor. The quantities T and X in (2) are called a density and a flow, respectively.

X on the interval a ≤ x ≤ b is equal to zero, i.e., X(a) = X(b), then the following “integral of motion” takes place:

If the total variation of the quantity

T dx = const

(for all t).

For nonstationary equations with n spatial variables x 1 , ...,x n , conservation laws have the form

Partial differential equations can have several (sometimes infinitely many) conservation laws or none at all.

Example 1. The Korteweg–de Vries equation

admits infinitely many conservation laws of the form (2). The first three are determined by

3 2 4 2 2 T 2 3 =2 w − w x , X 3 =9 w +6 w w xx − 12 ww x −2 w x w xxx + w xx , where the subscripts denote partial derivatives with respect to x.

Example 2. The sine-Gordon equation

∂ 2 w − sin w=0 ∂x∂t

also has infinitely many conservation laws. The first three are described by the formulas

3 = (2 4 − 24 T 2 w x w x w xx w x w xxx w xxx X w x w xx ) cos w. Example 3. The Monge–Amp`ere equation

where f (z) is an arbitrary function, admits the conservation law

∂x ∂x∂y

References for Subsection S.11.1: G. B. Whitham (1965), R. M. Miura, C. S. Gardner, and M. D. Kruskal (1968), M. D. Kruskal, R. M. Miura, C. S. Gardner, and N. J. Zabusky (1970), A. C. Scott, F. Y. Chu, and D. W. McLaughlin (1973), J. L. Lamb (1974), R. K. Dodd and R. K. Bullough (1977), P. J. Olver (1986), N. H. Ibragimov (1994), S. E. Harris (1996),

A. M. Vinogradov and I. S. Krasilshchik (1997), A. N. Kara and F. M. Mahomed (2002), B. J. Cantwell (2002).

S.11.2. Equations Admitting Variational Formulation. Noetherian

Symmetries

Here, we consider second-order equations in two independent variables, x and y, and an unknown function, w = w(x, y). We will deal with equations admitting the variational formulation of mini- mizing a functional of the form

Z[w] =

L(x, y, w, w x , w y ) dx dy.

The function L = L(x, y, w, w x , w y ) is called a Lagrangian. It is well known that a minimum of the functional (5) corresponds to the Euler–Lagrange equation

∂L

∂L

∂L

∂w

∂w x

∂w y ∂w y

A symmetry that preserves the differential form Ω = L(x, y, w, w x , w y ) dx dy is called a Noethe- rian symmetry of the Lagrangian L. In order to obtain Noetherian symmetries, one should find point transformations

x=f ¯ 1 ( x, y, w, ε), y=f ¯ 2 ( x, y, w, ε), w = g(x, y, w, ε) ¯ (7) such that preserve the differential form, ¯ Ω = Ω, i.e.,

(8) Calculating the differentials d ¯x, d ¯y and taking into account (7), we obtain

and therefore, relation (8) can be rewritten as

( L−¯ LD x f 1 D y f 2 ) dx dy = 0,

which is equivalent to

(9) Let us associate the point transformation (7) with the prolongation operator

L−¯ LD x f 1 D y f 2 = 0.

X = ξ∂ x + η∂ y + ζ∂ w + ζ 1 ∂ w x + ζ 2 ∂ w y , where the coordinates of the first prolongation, ζ 1 and ζ 2 , are defined by formulas (13) from

Subsection S.7.1. Then, by the usual procedure, from (9) one obtains the invariance condition in the form

(10) Noetherian symmetries are determined by (10).

X(L) + L(D x ξ+D y η) = 0.

Each Noetherian symmetry operator

X generates a conservation law,

Example 4. The equation of minimal surfaces

(1 + w 2 y ) w xx −2 w x w y w xy + (1 + w 2 x ) w yy =0

corresponds to the functional

Z[w] =

1+ w 2 x + w 2 y dx dy

p S with Lagrangian L=

1+ w 2 x + w 2 y . The admissible point operators

X 1 = ∂ x , X 2 = ∂ y , X 3 = x∂ x + y∂ y + w∂ w , X 4 = y∂ x − x∂ y , X 5 = ∂ w are found by the procedure described in detail in Section S.7.1-2. These operators determine Noetherian symmetries and

correspond to conservation laws: X 1 :

References for Subsection S.11.2: A. M. Vinogradov (1984), P. J. Olver (1986), J. A. Cavalcante and K. Tenenblat (1988), N. H. Ibragimov (1994), A. M. Vinogradov and I. S. Krasilshchik (1997).