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Published on: April 4, 2016
Soliton interactions in perturbed nonlinear Schrödinger equations
J A Besley1, P D Miller, N N Akhmediev
1Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australian Capital Territory, Australia. James.Besley@rrl.co.uk
Summary
We studied how solitons in the nonlinear Schrödinger equation interact with a modified potential. Perturbation theory reveals how soliton behavior, like binding energy and ejection velocity, changes due to these modifications.
Area of Science:
- Nonlinear Dynamics
- Mathematical Physics
- Soliton Theory
Background:
- Solitons in the cubic nonlinear Schrödinger equation are fundamental in various physical systems.
- Understanding soliton interactions is crucial for predicting system behavior.
- The influence of perturbations on soliton dynamics requires detailed investigation.
Purpose of the Study:
- To analyze the interaction of multiple solitons under a modified nonlinear potential.
- To develop a theoretical framework for perturbed soliton dynamics.
- To quantify the effects of perturbations on soliton properties like binding energy and velocity.
Main Methods:
- Multiscale perturbation theory applied to the nonlinear Schrödinger equation.
- Inverse scattering transform to analyze soliton solutions.
- Derivation of Newton's equations for soliton center-of-mass motion.
- Reduction of a two-soliton problem to a one-dimensional system using symmetries.
Main Results:
- A force law for soliton interactions derived from an integral formula.
- Calculation of binding energy and oscillation frequency for attractive perturbations.
- Determination of asymptotic ejection velocity for unstable cases.
- Validation of perturbative calculations through numerical experiments.
Conclusions:
- The study provides a robust method for analyzing perturbed soliton systems.
- Perturbations significantly alter soliton interaction dynamics, leading to stable bound states or unstable ejections.
- The findings offer insights into the behavior of nonlinear systems with modified potentials.
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