Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions

TL;DR

This paper investigates an integrable two-component Camassa-Holm system with cubic nonlinearity, describing peakon solutions, analyzing local well-posedness, and establishing new blow-up criteria for strong solutions.

Contribution

It introduces explicit peakon solutions for the system and provides a novel blow-up scenario based on initial data analysis.

Findings

Explicit formulas for peakon and two-peakon solutions

Local well-posedness of the Cauchy problem established

New blow-up result with respect to initial data

Abstract

This paper is devoted to an integrable two-component Camassa-Holm system with cubic nonlinearity, which includes the cubic Camassa-Holm equation (also called the Fokas-Olver-Rosenau-Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data.