Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov–Kuznetsov and Kadomtsev–Petviashvili equations as examples.The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations.The code ConservationLawsMD.m has been applied to multi-dimensional versions of the Sawada–Kotera, Camassa–Holm, Gardner, and Khokhlov–Zabolotskaya equations.

► A direct method for finding conservation laws for nonlinear PDEs is presented.
► Only tools from the calculus of variations and linear algebra are used.
► The approach circumvents Noether’s theorem connecting symmetries to conservation laws.
► The method is algorithmic and has been implemented in Mathematica.
► The software is applied to several nonlinear PDEs in multiple space dimensions.