Conserved norms and related conservation laws for multi-peakon equations

TL;DR

This paper classifies multi-peakon equations within a broad nonlinear dispersive wave class based on their conserved quantities, identifying new families with specific conservation laws and solutions like solitary waves and peakons.

Contribution

It provides a comprehensive classification of multi-peakon equations with various conserved norms and laws, revealing new equation families with notable solution properties.

Findings

Identified two wide families of equations with conserved $H^1$ and $L^2$ norms.

Discovered a singular equation with both smooth and peakon solutions.

Provided a classification framework for conservation laws in multi-peakon equations.

Abstract

All nonlinear dispersive wave equations in the general class $m_t+f(u,u_x)m+(g(u,u_x)m)_x =0$ are known to possess multi-peakon weak solutions. A classification is presented for families of multi-peakon equations in this class that possess conserved momentum; conserved $H^1$ norm of $u$; conserved $H^2$ norm of $u$; conserved $L^2$ norm of $m$; related conservation laws. The results yield, among others, two interesting wide families of equations: $m_t + 2u_x h(u,u_x)m + u (h(u,u_x)m)_x=0$ for which the $H^1$ norm of $u$ is conserved; $m_t -\tfrac{1}{2}u_x h'(u)m + (h(u)m)_x =0$ for which the $L^2$ norm of $m$ is conserved. The overlap of these two families yields a singular equation which is nevertheless found to possess both smooth solitary wave solutions and peakon travelling wave solutions.