The matrix Lax representation of the generalized Riemann equations and its conservation Laws

TL;DR

This paper introduces a new matrix Lax representation for the generalized Riemann equations, revealing a class of explicit time-dependent conserved densities and generating numerous non-polynomial conservation laws, advancing understanding of integrable systems.

Contribution

It presents a novel matrix and scalar Lax representation for the multicomponent Hunter-Saxton equation and an algorithm to generate extensive non-polynomial conservation laws.

Findings

New matrix and scalar Lax representations are derived.

A method to generate a large class of conservation laws is developed.

A series of new conservation laws for the Hunter-Saxton equation is obtained.

Abstract

It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter - Saxton equation. New matrix and scalar Lax representation is presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.