Infinite sets of polynomial conserved densities for nonlinear evolution equations

The infinite sets of polynomial conserved densities which have been found for the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the Sine-Gordon equation, and the classical nonlinear shallow-water equations, are investigated using Noether's theorem. These sets are identified as energy or momentum densities of sets of higher-order integro-differential equations. These higher-order equations are obtained by operating n times on the evolution equation under consideration, with a nonlinear integro-differential operator, and hence their solution sets contain that of the evolution equation under consideration. The technique has possibilities for predicting the existence of infinite sets of polynomial conserved densities for other nonlinear evolution equations