Soliton resolution for the coupled complex short pulse equation

TL;DR

This paper analyzes the long-time behavior of solutions to the coupled complex short pulse equation, revealing decay rates and soliton structures using advanced asymptotic methods for integrable systems.

Contribution

It develops a $ar{ ext{D}}$-steepest descent method for the ccSP equation, providing detailed asymptotic descriptions in different space-time regions.

Findings

Solution decays as O(t^{-1}) in certain regions

Presence of multi-soliton and breather solutions in others

Precise asymptotic error estimates including logarithmic terms

Abstract

We address the long-time asymptotics of the solution to the Cauchy problem of ccSP (coupled complex short pulse) equation on the line for decaying initial data that can support solitons. The ccSP system describes ultra-short pulse propagation in optical fibers, which is a completely integrable system and posses a $4\times4$ matrix Wadati--Konno--Ichikawa type Lax pair. Based on the $\bar{\partial}$-generalization of the Deift--Zhou steepest descent method, we obtain the long-time asymptotic approximations of the solution in two kinds of space-time regions under a new scale $(\zeta,t)$. The solution of the ccSP equation decays as a speed of $O(t^{-1})$ in the region $\zeta/t>\varepsilon$ with any $\varepsilon>0$; while in the region $\zeta/t<-\varepsilon$, the solution is depicted by the form of a multi-self-symmetric soliton/composite breather and $t^{-1/2}$ order term arises from self-symmetric soliton/composite breather-radiation interactions as well as an residual error order $O(t^{-1}\ln t)$.