Long-time asymptotics for a fourth-order dispersive nonlinear Schr\"{o}dinger equation with nonzero boundary conditions

TL;DR

This paper analyzes the long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with nonzero boundary conditions, using Riemann-Hilbert problem techniques and the nonlinear steepest descent method.

Contribution

It develops a Riemann-Hilbert framework and applies the nonlinear steepest descent method to derive the asymptotic behavior of solutions with nonzero boundary conditions.

Findings

Derived the asymptotic stage of modulation instability.

Constructed explicit solutions via Riemann-Hilbert analysis.

Analyzed the effect of nonzero boundary conditions on long-time dynamics.

Abstract

In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schr\"{o}dinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic Riemann-Hilbert problem associated with nonzero boundary conditions, we analysis direct scattering problem. Then we deform the corresponding matrix Riemann-Hilbert problem to explicitly solving models via using the nonlinear steepest descent method and employing the $g$-function mechanism to eliminate the exponential growths of the jump matrices. Finally, we obtain the asymptotic stage of modulation instability for the fourth-order dispersive nonlinear Schr\"{o}dinger equation.