The Gerdjikov-Ivanov type derivative nonlinear Schr\"odinger equation: Long-time dynamics of nonzero boundary conditions

Abstract

We consider the Gerdjikov--Ivanov type derivative nonlinear Schr\"odinger equation \berr \ii q_{t}+q_{xx}-\ii q^2\bar{q}_{x}+\frac{1}{2}(|q|^4-q_0^4)q=0 \eerr on the line. The initial value $q(x,0)$ is given and satisfies the symmetric, nonzero boundary conditions at infinity, that is, $q(x,0)\rightarrow q_\pm$ as $x\rightarrow\pm\infty$, and $|q_\pm|=q_0>0$. The goal of this paper is to study the asymptotic behavior of the solution of this initial-value problem as $t\rightarrow\infty$. The main tool is the asymptotic analysis of an associated matrix Riemann--Hilbert problem by using the steepest descent method and the so-called $g$-function mechanism. We show that the solution $q(x,t)$ of this initial value problem has a different asymptotic behavior in different regions of the $xt$-plane. In the regions $x<-2\sqrt{2}q_0^2t$ and $x>2\sqrt{2}q_0^2t$, the solution takes the form of a plane wave. In the region $-2\sqrt{2}q_0^2t<x<2\sqrt{2}q_0^2t$, the solution takes the form of a modulated elliptic wave.