Double waves in multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

TL;DR

This paper develops a differential-geometric method to determine the integrability of multi-dimensional hydrodynamic systems by analyzing the existence of special solutions called double waves, providing a practical necessary condition for integrability.

Contribution

It introduces an invariant geometric approach linking double waves to matrix diagonalizability, offering a new criterion to assess integrability of (2+1)-dimensional systems.

Findings

Existence of double waves is equivalent to matrix diagonalizability.

Diagonalizability can be verified via differential geometry.

Provides a simple necessary condition for integrability.

Abstract

An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type u_t+A(u)u_x+B(u)u_y=0 is developed. It is proved 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 differential-geometric means, this provides a simple necessary condition for integrability.