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

• Published in: Mathematical methods in the applied sciences 2019-09, Vol.42 (14), p.4839-4861
• Main Authors: Guo, Boling, Liu, Nan
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Boundary conditions; Boundary value problems; Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation; long‐time asymptotics; nonlinear steepest descent method; Plane waves; Riemann‐Hilbert problem; Schrodinger equation; Steepest descent method
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5698

We consider the Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation iqt+qxx−iq2q¯x+12|q|4−q04q=0 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 x22q02t, the solution takes the form of a plane wave. In the region −22q02t