ISSN:
1432-0916

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics
,
Physics

Notes:
Abstract Letu=u(x, t) be a function ofx andt, andu i =D i u,D=d/dx,i=0, 1, 2, ..., be its derivatives with respect tox. Denote byW n the set {f|f=f(u,u 1, ...,u n ), (∂/∂u n )f≠0}, wheref(u, ...,u n ) are polynomials ofu i with constant coefficients. To any $$f \in W = \mathop \cup \limits_{n = 2}^\infty W_n $$ , we relate it with an operator . In this paper we prove that: ℳ(f) commutes with ℳ(g) if they commute respectively with ℳ(h), providedf,g,hεW. Relating to this commutativity theorem, we prove that, if an evolution equationu t =f(u, ...,u n ) possesses nontrivial symmetries (or conservation laws for a class of polynomialsf), thenf=Cu n +f 1(u, ...,u r ), whereC=const, andr〈n. In the end of this paper, we state a related open problem whose solution would be of much value to the theory of soliton.

Type of Medium:
Electronic Resource

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