On the existence of conservation law multiplier for partial differential equations

•A sufficient and necessary condition for the existence of multipliers without involving derivatives is presented.•A necessary condition for the existence of multipliers involving derivatives is given for the evolution PDEs.•Applications of the results to nonlinear telegraph equations and a class of Korteweg-de Vries type equations are given.

In this paper, we use the property of nonlinear self-adjointness with differential substitution to study the existence of conservation law multiplier for partial differential equations (PDEs). Firstly, we give a sufficient and necessary condition for the existence of the multipliers involving only independent and dependent variables, which is the nonlinear self-adjointness of the studying PDEs. Secondly, a necessary condition for the existence of the multipliers involving derivatives is given for the general evolution PDEs, which is the nonlinear self-adjointness with differential substitution. Finally, applications of multiplier and nonlinear self-adjointness with differential substitution methods to nonlinear telegraph equations and a class of Korteweg-de Vries (KdV) type equations are performed and different types of conservation laws are constructed.