On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity

TL;DR

This paper studies the Cauchy problem for a modified, integrable Camassa-Holm equation with cubic nonlinearity, analyzing well-posedness, blow-up scenarios, and solution properties.

Contribution

It establishes local well-posedness in Besov spaces, describes blow-up mechanisms, and proves nonexistence of smooth traveling waves for this equation.

Findings

Equation is locally well-posed in Besov spaces

Blow-up scenario and lower bound of maximal existence time identified

Nonexistence of smooth traveling wave solutions proven

Abstract

Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa-Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. In addition, the persistence properties of the strong solutions for the equation are obtained.