On the equivalence of linearization and formal symmetries as integrability tests for evolution equations

Abstract

Gurses' integrability test (1991) consists of the compatibility of the linearized equation with an eigenvalue equation and leads to the recursion operator. This test is applied to quasilinear fifth-order equations and the same classification as the 'formal symmetry' method of Mikhailov et al (1990) is obtained. The same classification for polynomial equations is obtained using Fokas' test (1987), i.e. the existence of one higher-order symmetry. It is shown that the recursion operators of a specific form can be constructed using symmetries and conserved covariants and the recursion operators for polynomial equations are obtained with this method.