Towards the classification of scalar nonpolynomial evolution equations: Quasilinearity

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. (c) 2005 Elsevier Ltd. All rights reserved.