New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order

In a recent communication Ibragimov introduced the concept of nonlinearly self-adjoint differential equation [Ibragimov NH. Nonlinear self-adjointness and conservation laws. J Phys A Math Theor 2011;44:432002 (8pp.)]. In this paper a nonlinear self-adjoint classification of a general class of fifth-order evolution equation with time dependent coefficients is presented. As a result five subclasses of nonlinearly self-adjoint equations of fifth-order and four subclasses of nonlinearly self-adjoint equations of third-order are obtained. From the Ibragimov’s theorem on conservation laws [Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333:311–28] conservation laws for some of these equations are established.

► The concept of nonlinearly self-adjoint equations is applied to a general class of fifth-order equations. ► Five classes of fifth-order nonlinear self-adjoint equations are obtained. ► Four classes of third-order nonlinear self-adjoint equations are obtained. ► Conservation laws for some of these equations are established.