Multi-Dimensional Conservation Laws and Integrable Systems

In this paper we introduce a new property of two-dimensional integrable systems -- existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional commuting flows. Infinitely many three-dimensional local conservation laws for the Korteweg de Vries pair of commuting flows and for the Benney commuting hydrodynamic chains are constructed. As a by-product we established a new method for computation of local conservation laws for three-dimensional integrable systems. The Mikhalev equation and the dispersionless limit of the Kadomtsev--Petviashvili equation are investigated. All known local and infinitely many new quasi-local three-dimensional conservation laws are presented. Also four-dimensional conservation laws are considered for couples of three-dimensional integrable quasilinear systems and for triples of corresponding hydrodynamic chains.