Higher order nonlinear Schr\"odinger equation in domains with moving boundaries

TL;DR

This paper studies a higher order nonlinear Schrödinger equation in domains with moving boundaries, proving existence, uniqueness, stability of solutions, and introducing a stable numerical method with illustrative examples.

Contribution

It establishes the well-posedness and stability of solutions for HNLS in moving boundary domains and develops a convergent finite difference numerical scheme.

Findings

Proved existence and uniqueness of global weak solutions.

Established stability of solutions in the $L^2$-norm.

Developed a stable, convergent numerical method with numerical illustrations.

Abstract

The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for a higher order nonlinear Schr\"odinger (HNLS) equation is considered. Existence and uniqueness of global weak solutions are proved as well as the stability of the solution. Additionally, a conservative numerical method of finite differences is introduced that also verifies stability properties with respect to the $L^2$-norm, and along with proving its convergence, some interesting numerical examples are shown that illustrate the behavior of the solution.