Abstract
An asymptotic algorithm presented in a previous paper is applied to investigate the possible structures of evolution equations ut=uM+K(u,...,uM−1), M=3,5, which could be compatible with the existence of a conserved density ρ0(u), depending only on u, and with the existence as well of conserved densities with arbitrarily high-order derivatives. For M=3 it is shown that the Calogero–Degasperis–Fokas equation is essentially the only nonpolynomial equation of that type. For the case ut=D[u4+Q], with Q(u,...,u3) a polynomial, we find a very narrow class of admissible structures for Q, typified by the few particular examples known up to date. Actually, there is in this case an essentially unique structure, modulo a Miura transformation.
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