The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

Abstract

An invariant differential-geometric approach to the integrability of (2 + 1)-dimensional systems of hydrodynamic type,is developed. We prove that the existence of special solutions known as 'double waves' is equivalent to the diagonalizability of an arbitrary matrix of the two-parameter family (kE + A) −1 (lE + B).Since the diagonalizability can be effectively verified by calculating the Haantjes tensor, this provides a simple necessary condition for integrability.MSC: 35L40, 35L65, 37K10.

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

“…Both transformations preserve the Poisson bracket specified by (13). Hence, they can be used to simplify the Hamiltonian.…”

“…Remark. The apparent similarity of the cases (28) and (33) is not accidental and manifests an important reciprocal invariance of the class of Hamiltonian chains (13). Recall that the conservation of momentum reads…”

“…Before proceeding with a detailed analysis of the equations (29), let us make a digression on conservation laws of the chains (13). First of all, any Hamiltonian chain (13) possesses three conservation laws of the form u…”

A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential

“…The system (4.43) is written in the form (1.11) and can not be written in the form (1.3). In particular, it does not belong to the class of the systems studied in the paper [3].…”

“…A general theory of such systems was developed in the papers [1,2,3]. This theory is based on the existence of sufficiently many of the hydrodynamic reductions [4,1] which has been proposed as the definition of integrability.…”