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.