Conservation laws of the generalized Riemann equations at $N=2,3,4$

TL;DR

This paper identifies infinitely many conserved quantities for generalized Riemann equations at N=2,3,4, revealing new conservation laws and their connections to other nonlinear equations.

Contribution

It introduces new conserved densities and laws for the generalized Riemann equations at N=2,3,4, including those involving arbitrary functions, and links them to well-known equations.

Findings

Infinite conserved densities for N=2,3,4

New conservation laws with arbitrary functions at N=2

Connections to Hunter-Saxton and Monge-Ampere equations

Abstract

In this paper, we present infinitely many conserved densities satisfying particular conservation law $F_{t}=(2uF)_{x}$ for the generalized Riemann equations at $N=2,3,4$. In the $N=2$ case, we also construct conserved densities corresponding to new conservation laws containing an arbitrary smooth function. In virtue of reductions and/or changes of variables, related conserved densities are obtained for two component Hunter-Saxton equation, Hunter-Saxton equation, Gurevich-Zybin equation and Monge-Ampere equation.