Perfectly Matched Layers for Coupled Nonlinear Schr\"{o}dinger Equations with Mixed Derivatives

TL;DR

This paper develops stable perfectly matched layers for 2D coupled nonlinear Schrödinger equations with mixed derivatives, enabling effective absorption of waves in simulations of nonlinear periodic structures.

Contribution

It introduces a PML construction in Laplace Fourier space for coupled NLS equations with mixed derivatives, demonstrating stability and effectiveness in both linear and nonlinear regimes.

Findings

PML is stable if absorption is below a threshold

Numerical tests show good convergence and performance

PML performs well in nonlinear simulations

Abstract

This paper constructs perfectly matched layers (PML) for a system of 2D Coupled Nonlinear Schr\"odinger equations with mixed derivatives which arises in the modeling of gap solitons in nonlinear periodic structures with a non-separable linear part. The PML construction is performed in Laplace Fourier space via a modal analysis and can be viewed as a complex change of variables. The mixed derivatives cause the presence of waves with opposite phase and group velocities, which has previously been shown to cause instability of layer equations in certain types of hyperbolic problems. Nevertheless, here the PML is stable if the absorption function $\sigma$ lies below a specified threshold. The PML construction and analysis are carried out for the linear part of the system. Numerical tests are then performed in both the linear and nonlinear regimes checking convergence of the error with respect to the layer width and showing that the PML performs well even in many nonlinear simulations.