Multisymplectic conservation laws for differential and differential-difference equations

Abstract

Many well-known partial differential equations can be written as multisymplectic systems. Such systems have a structural conservation law from which scalar conservation laws can be derived. These conservation laws arise as differential consequences of a 1-form ‘quasi-conservation law’, which is related to Noether's theorem. This paper develops the above framework and uses it to introduce a multisymplectic structure for differential-difference equations. The shallow water equations and the Ablowitz–Ladik equations are used to illustrate the general theory. It is found that conservation of potential vorticity is a differential consequence of two conservation laws; this surprising result and its implications are discussed.