Darboux transformation and solitonic solution to the coupled complex short pulse equation

TL;DR

This paper develops a Darboux transformation approach for the coupled complex short pulse equation, enabling the explicit construction of various solitonic solutions and analyzing their stability and properties.

Contribution

The paper introduces a novel Darboux transformation method for the CCSP equation and constructs explicit bright, dark, breather, and rogue wave solutions.

Findings

Constructed explicit soliton solutions including breathers and rogue waves.

Performed inverse scattering analysis for vanishing boundary conditions.

Confirmed rogue wave existence conditions via modulational instability analysis.

Abstract

The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis to understand the soliton solution better. Breather and rogue wave solutions are constructed in case of non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution.