Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time

TL;DR

This paper analyzes the long-time behavior of solutions to the focusing Fokas-Lenells equation, revealing soliton and dispersive components in specific space-time regions using advanced Riemann-Hilbert techniques.

Contribution

It provides the first detailed long-time asymptotic description of the focusing Fokas-Lenells equation in the solitonic region, accounting for spectral singularities and stationary phase points.

Findings

Solution decomposes into soliton and dispersive parts.

Dispersive part decays as |t|^{-1/2}.

Residual error decays as |t|^{-3/4}.

Abstract

We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation $$ u_{xt}+\alpha\beta^2u-2i\alpha\beta u_x-\alpha u_{xx}-i\alpha\beta^2|u|^2u_x=0 \label{cs} $$ with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at $k=0$. (ii) four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone $$\mathcal{C}\left(x_{1}, x_{2}, v_{1}, v_{2}\right)=\left\{(x, t) \in \mathbb{R}^{2} | x=x_{0}+v t, x_{0} \in\left[x_{1}, x_{2}\right], v \in\left[v_{1}, v_{2}\right]\right\},$$ the long-time asymptotic behavior of the solution $u(x,t)$ for the focusing FL equation can be characterized with an $N(\mathcal{I})$-soliton on discrete spectrums and a leading order term $\mathcal{O}(|t|^{-1/2})$ on continuous spectrum up to a residual error order $\mathcal{O}(|t|^{-3/4})$. The main tool is the $\overline{\partial}$ nonlinear steepest descent method and the $\overline{\partial}$-analysis.