A Quantum Mechanics Analogy for the Nonlinear Schr\"{o}dinger Equation in the Finite Line

TL;DR

This paper uses a quantum mechanics analogy to analyze soliton dynamics governed by the nonlinear Schrödinger equation in finite domains with various boundary conditions, revealing boundary-specific effects on energy and force densities.

Contribution

It introduces a quantum mechanics analogy approach to study soliton boundary interactions under different boundary conditions, highlighting dynamic behaviors and energy variations.

Findings

Energy densities are nearly constant for periodic boundaries.

Large variations in energy and force densities occur with homogeneous boundary conditions.

The quantum momentum changes sign but recovers after boundary collisions.

Abstract

A quantum mechanics analogy is used to determine the forces acting on and the energies of solitons governed by the nonlinear Schr\"odinger equation in finite intervals with periodic and with homogeneous Dirichlet, Neumann and Robin boundary conditions. It is shown that the energy densities remain nearly constant for periodic, while they undergo large variations for homogeneous, boundary conditions. The largest variations in the force and energy densities occur for the Neumann boundary conditions, but, for all the boundary conditions considered in this paper, the magnitudes of these forces and energies recover their values prior to the interaction of the soliton with the boundary, after the soliton rebound process is completed. It is also shown that the quantum momentum changes sign but recovers its original value after the collision of the soliton with the boundaries. The asymmetry of the Robin boundary conditions shows different dynamic behaviour at the left and right boundaries of the finite interval.