The Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation: Long-time dynamics of nonzero boundary conditions

2018/12/08 by Boling Guo, Nan Liu, Guo, Boling +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1812.03257

arxiv created 2018/12/08 · arxiv updated 2018/12/11

Abstract

We consider the Gerdjikov--Ivanov type derivative nonlinear Schrödinger equation \berr \ii q_t+q_xx-\ii q²q_x+(1)/(2)(|q|⁴-q₀⁴)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)→ q_± as x→±∞, and |q_±|=q₀>0. The goal of this paper is to study the asymptotic behavior of the solution of this initial-value problem as t→∞. 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√(2)q₀²t and x>2√(2)q₀²t, the solution takes the form of a plane wave. In the region -2√(2)q₀²t<x<2√(2)q₀²t, the solution takes the form of a modulated elliptic wave.

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• Rogue wave statistics and integrable turbulence in the Gerdjikov-Ivanov equation

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