Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

摘要

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