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Nonlinear Schrödinger equation with spatiotemporal perturbations.
Franz G Mertens1, Niurka R Quintero, A R Bishop
1Physikalisches Institut, Universität Bayreuth, Bayreuth, Germany.
Summary
We studied driven solitons in the nonlinear Schrödinger equation (NLSE) with perturbations. A new theory predicts soliton stability based on momentum-velocity curves, confirmed by simulations.
Area of Science:
- Nonlinear Optics
- Mathematical Physics
- Soliton Dynamics
Background:
- Solitons in the cubic nonlinear Schrödinger equation (NLSE) are fundamental.
- Perturbations like driving and damping affect soliton behavior.
- Understanding stability is crucial for controlling soliton dynamics.
Purpose of the Study:
- Investigate the dynamics of driven solitons in the NLSE with specific perturbations.
- Develop a collective-coordinate theory to describe soliton behavior.
- Determine the stability criteria and estimate the lifetime of perturbed solitons.
Main Methods:
- Applied a collective-coordinate theory derived from Lagrangian or time-evolution methods.
- Solved resulting ordinary differential equations (ODEs) analytically and numerically.
- Analyzed the time evolution of soliton momentum and velocity to assess stability.
Main Results:
- Developed a set of ODEs for four collective coordinates describing the driven soliton.
- Found that solitons exhibit oscillations around a mean unidirectional trajectory.
- Identified a stability criterion based on the negative slope of the momentum-velocity curve, confirmed by simulations.
Conclusions:
- The collective-coordinate theory accurately describes perturbed NLSE soliton dynamics.
- The momentum-velocity curve provides a reliable predictor of soliton stability and lifetime.
- The study offers insights into controlling and predicting soliton behavior in complex systems.
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