Towards the classification of scalar nonpolynomial evolution equations: Quasilinearity

Abstract

We prove that, for m ≥ 7, scalar evolution equations of the form u t = F( x, t, u, …, u m ) which admit a nontrivial conserved density of order m + 1 are linear in u m . The existence of such conserved densities is a necessary condition for integrability in the sense of admitting a formal symmetry, hence, integrable scalar evolution equations of order m ≥ 7, admitting nontrivial conserved densities are quasilinear.