A new technique for finding conservation laws of partial differential equations

Zhi-Yong Zhang

Published 2014-09-22Version 1

We propose a new technique to construct conservation laws of partial differential equations (PDEs), where the whole computations can be fully implemented on a computer. The result is obtained by proving that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and vice versa. Consequently, each symmetry of PDEs corresponds to a conservation law via a formula if the PDEs are nonlinearly self-adjoint with differential substitution, which extends the results of Noether' theorem. As a byproduct, we find that the set of differential substitutions includes the set of conservation law multipliers as a subset. With the help of the computer algebra system Mathematica, applications to the illustrated examples are performed and new conservation laws are constructed.