Nonlinear self-adjointness of a class of third order nonlinear dispersive equations

•A nonlinear self-adjoint classification with differential substitutions of a general class of third order evolution equations is carried out.•Several sub-families of nonlinearly self-adjoint equations are obtained, some of them admitting differential substitutions.•Conservation laws are established via differential substitutions.

In this paper we carry out a nonlinear self-adjoint classification with differential substitutions of a class of dispersive equations with three arbitrary functions. As a particular case, we obtain the sub families of the C1(m,a+b)C1(m,a+b) equations which are nonlinearly self-adjoint. Conservation laws for these last families are established using the techniques for finding conserved vectors proposed by Ibragimov.