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

We consider the Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equationi q t + q x x − i q 2q ¯ x + 1 2| q | 4 − q 0 4q = 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 ± |= q 0 >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 x t ‐plane. In the regions x < − 2 2q 0 2 t and x > 2 2q 0 2 t , the solution takes the form of a plane wave. In the region − 2 2q 0 2 t < x < 2 2q 0 2 t , the solution takes the form of a modulated elliptic wave.