Conserved densities for nonlinear evolution equations. I. Even order case

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Abstract
This paper is a first attempt to analyze in detail the number and structure of nontrivial polynomial conserved densities for a nonlinear evolution equation ut=P, P an arbitrary polynomial in the spatial derivatives of u. Our attention is here focused on the even order case, where stronger conclusions can be derived. Several criteria for nonexistence of conserved densities are afforded. The coexistence of conserved densities is shown to severely restrict both the evolution equation and the functional structure of such densities. Finally for the case of second order equations the problem is completely worked out.