Soliton resolution for the coupled complex short pulse equation

TL;DR

This work analyzes the long-time behavior of the Cauchy problem for the integrable coupled complex short pulse equation on the line with decaying Schwartz initial data that may support solitons. By applying the bar{\partial} generalization of the Deift–Zhou steepest descent method to a 4×4 matrix Lax pair, the authors derive precise two-region asymptotics in a new $\hat{\zeta}=\zeta/t$ scaling: in the far-left region $\hat{\zeta}< -\varepsilon$ the solution splits into self-symmetric solitons/composite breathers and radiation with a $t^{-1/2}$ decay, while in the far-right region $\hat{\zeta}>\varepsilon$ the solution decays as $O(t^{-1})$. The analysis constructs a sequence of Riemann–Hilbert problems, performs contour deformations, and employs outer and local model RH problems governed by parabolic cylinder functions to capture the soliton/breather content, along with a $\bar{\partial}$-problem that yields a small, decaying error. The results establish a soliton-resolution-type description for the ccSP equation and provide detailed reconstruction formulas for $q_1$ and $q_2$, highlighting the interplay between discrete spectrum and radiation. This work extends the nonlinear steepest-descent framework to higher-rank, matrix Lax pairs and offers a robust approach for future investigations of multi-component integrable systems with solitons.

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 $(ζ,t)$. The solution of the ccSP equation decays as a speed of $O(t^{-1})$ in the region $ζ/t>\varepsilon$ with any $\varepsilon>0$; while in the region $ζ/t<-\varepsilon$, 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 $O(t^{-1}\ln t)$.

Figures (7)

• Figure 1: Sign distribution of Im$\theta$ in the $k$-plane in the case $\hat{\zeta}<-\varepsilon$.
• Figure 2: The open sets $\{\Omega_l\}_{l=1}^{10}$ and the contours $\{\Gamma_l\}_{l=1}^{12}$ in the complex $k$-plane.
• Figure 3: The contour $\Upsilon$, (a) Re$k_n>0$, (b) Re$k_n=0$. $J^{(out)}$ decays exponentially on red contours.
• Figure 4: The contours $\Gamma^{\epsilon1}\cup\Gamma^{\epsilon2}$ in the complex $k$-plane.
• Figure 5: The oriented contour $\tilde{\Gamma}^{(k_0)}$.

Theorems & Definitions (37)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Remark 2.1
• Proposition 2.4
• Proposition 2.5
• Theorem 2.1
• proof
• Theorem 2.2