The non-polynomial conservation laws and integrability analysis of generalized Riemann type hydrodynamical equations

TL;DR

This paper investigates the integrability of generalized Riemann type hydrodynamical equations using the gradient-holonomic algorithm, revealing infinite conservation laws, Lax representations, and bi-Hamiltonian structures, especially for N=2,3,4.

Contribution

It introduces a systematic analysis of non-polynomial conservation laws and integrability structures for generalized hydrodynamical equations, including new results for N=2,3,4.

Findings

Constructed infinite hierarchies of conservation laws

Analyzed Lax type representations and bi-Hamiltonian structures

Identified a new set of non-polynomial conservation laws for Hunter-Saxton equation

Abstract

Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation $%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ The infinite hierarchies of polynomial and non-polynomial conservation laws, both dispersive and dispersionless are constructed. Special attention is paid to the cases $%N=2,3$ and N=4 for which the conservation laws, Lax type representations and bi-Hamiltonian structures are analyzed in detail. We also show that the case N=2 is equivalent to a generalized Hunter-Saxton dynamical system, whose integrability follows from the results obtained. As a byproduct of our analysis we demonstrate a new set of non-polynomial conservation laws for the related Hunter-Saxton equation.