Cauchy problem and multi-soliton solutions for a two-component short pulse system

TL;DR

This paper investigates the well-posedness and multi-soliton solutions of a two-component short pulse system, deriving explicit solutions and analyzing their interactions through Darboux transformations and Lax pairs.

Contribution

It introduces a novel $N$-fold Darboux transformation for the system, enabling explicit construction of multi-soliton solutions and detailed analysis of their dynamics.

Findings

Existence and uniqueness of solutions with analytic lifespan estimate.

Explicit multi-soliton solutions including loops and breathers.

Detailed analysis of soliton interactions and dynamics.

Abstract

In this paper, we study the Cauchy problem and multi-soliton solutions for a two-component short pulse system. For the Cauchy problem, we first prove the existence and uniqueness of solution with an estimate of the analytic lifespan, and then investigate the continuity of the data-to-solution map in the space of analytic function. For the multi-soliton solutions, we first derive an $N$-fold Darboux transformation from the Lax pair of the two-component short pulse system, which is expressed in terms of the quasideterminant.Then by virtue of the $N$-fold Darboux transformation we obtain multi-loop and breather soliton solutions. In particular, one-, two-, three-loop soliton, and breather soliton solutions are discussed in details with interesting dynamical interactions and shown through figures.