The Nonlinear Schr\"{o}dinger Equation in the Finite Line

TL;DR

This paper numerically investigates the nonlinear Schrödinger equation on a finite line with various boundary conditions, revealing soliton-boundary interactions, boundary layer formations, and phase diagram behaviors similar to the Duffing equation.

Contribution

It introduces a numerical analysis of the NLS equation with different boundary conditions, including a novel method of images for Dirichlet and Neumann cases.

Findings

Solitons behave as in infinite domains even after boundary collisions.

Boundary layers trap and delay solitons under Robin conditions.

Phase diagrams exhibit recurrent patterns similar to the Duffing equation.

Abstract

A numerical study of the nonlinear Schr\"odinger (NLS) equation subject to homogeneous Dirichlet, Neumann and Robin boundary conditions in the finite line is presented. The results are compared with both the exact analytical ones for the initial-value problem (IVP) of the NLS equation and the numerical ones for periodic boundary conditions. It is shown that initial solutions obtained by truncating the exact N-soliton solution of the IVP of the NLS equation into a finite interval develop solitary waves that behave as solitons, even after collisions with the boundaries. For periodic and homogeneous Dirichlet and Neumann boundary conditions, it is observed that the interaction between solitons and boundaries is equivalent to the collision between solitons in IVP or quarterplane problems. It is shown that for homogeneous Robin boundary conditions, boundary layers that trap and delay the soliton are formed at the boundaries. Phase diagrams for the soliton amplitude at the boundary points and for the soliton's maximum amplitude show a recurrent phenomenon, and are similar to those of the cubic Duffing equation. It is also shown that the phase diagrams are strong functions of the parameter that defines the Robin boundary conditions. A method of images, similar to the one used in potential theory, is developed for the NLS equation in the quarterplane with homogeneous Dirichlet and Neumann boundary conditions at the finite boundary.