A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems

TL;DR

This paper introduces a coordinate-free method to construct and analyze a broad class of integrable hydrodynamic-type systems using a special tensor, leading to new systems with explicit solutions via the generalized hodograph method.

Contribution

The authors develop a coordinate-free framework based on a (1,1)-tensor with zero Nijenhuis torsion to generate and transform integrable hydrodynamic systems, expanding the class of solvable models.

Findings

Construction of seed systems using tensor L

Derivation of conservation laws and reciprocal transformations

Generation of new integrable systems with explicit solutions

Abstract

Using a (1,1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for the seed systems. The reciprocal transformations constructed from these conservation laws yield a considerably larger class of hydrodynamic-type systems from the seed systems, and we show that these new systems are again defined in a coordinate-free manner, using the tensor L alone, and, moreover, are weakly nonlinear and semi-Hamiltonian, so their general solution can be obtained by means of the generalized hodograph method of Tsarev.