Persistence Properties and Unique Continuation for a Dispersionless Two-Component Camassa-Holm System with Peakon and Weak Kink Solutions

TL;DR

This paper investigates the decay and continuation properties of solutions to a dispersionless two-component Camassa-Holm system, establishing conditions for decay at infinity and demonstrating unique continuation under specific initial data constraints.

Contribution

It provides new results on the decay behavior and unique continuation of solutions, including an optimal decay index for the momentum in the system.

Findings

Solutions decay at infinity if initial data decay

Unique continuation holds for non-negative initial momentum

Optimal decay index of the momentum is established

Abstract

In this paper, we study the persistence properties and unique continuation for a dispersionless two-component system with peakon and weak kink solutions. These properties guarantee strong solutions of the two-component system decay at infinity in the spatial variable provided that the initial data satisfies the condition of decaying at infinity. Furthermore, we give an optimal decaying index of the momentum for the system and show that the system exhibits unique continuation if the initial momentum $m_0$ and $n_0$ are non-negative.