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.

Multidimensional conservation laws and integrable systems II

“…Here we used the notation 𝑢 𝑗 = 𝜕 𝑗 𝑥 𝑢. In the previous paper 1 we proved that the system (48) admits infinitely many quasi-local three-dimensional conservation laws, 5 where the first three of them are (we remind that here 𝑤 is a potential function such that under the substitution 𝑢 = 𝑤 𝑥 and 𝑣 = 𝑤 𝑡 the Mikhalëv system (48) comes back to Equation (1)):…”

“…In the previous paper 1 we discovered a new phenomenon in the theory of integrable systems. We showed that commuting pairs of two-dimensional integrable hydrodynamic chains (the Benney moment chains associated with the dispersionless limit of the Kadomtsev-Petviashvili equation and the linearly degenerate hydrodynamic chains associated with the Mikhalëv equation) possess infinitely many three-dimensional conservation laws.…”

Lax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebras

“…λ yields an infinite-dimensional Lax representation for (1). This infinite-dimensional Lax representation can be used to study nonlocal symmetries [1] and nonlocal conservation laws [8] of equation (1). In section 3 we expand the new Lax representation into the Taylor series and find explicit expressions for the coefficients of the obtained infinite-dimensional Lax representation.…”