Multi‐dimensional conservation laws and integrable systems

Abstract In this paper, we introduce a new property of two‐dimensional integrable hydrodynamic chains—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Infinitely many local three‐dimensional conservation laws 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 Mikhalëv equation and the dispersionless limit of the Kadomtsev‐Petviashvili equation are investigated. All known local and infinitely many new quasilocal three‐dimensional conservation laws are presented. Also four‐dimensional conservation laws are considered for couples of three‐dimensional integrable quasilinear systems and for triplets of corresponding hydrodynamic chains.