A commutativity theorem of partial differential operators

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.