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Moving solitons in the discrete nonlinear Schrödinger equation
1Department of Maths and Applied Maths, University of Cape Town, Rondebosch 7701, South Africa. Oliver.Oxtoby@uct.ac.za
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 13, 2007
Summary
Small-amplitude solitons in discrete nonlinear systems generally emit radiation, losing energy. However, in saturable nonlinearities, specific "sliding velocities" can suppress this radiation, allowing solitons to move without decay.
Area of Science:
- Nonlinear Physics
- Soliton Dynamics
- Condensed Matter Theory
Background:
- Discrete nonlinear Schrödinger equation (DNLS) models various physical systems.
- Solitons are particle-like waves that maintain their shape.
- Radiation emission from moving solitons affects their stability and dynamics.
Purpose of the Study:
- To investigate the radiation amplitude of moving small-amplitude solitons in the DNLS.
- To analyze the influence of cubic and saturable nonlinearities on soliton radiation.
- To identify conditions for radiation suppression and understand soliton deceleration mechanisms.
Main Methods:
- Application of the asymptotics beyond all orders method.
- Analytical evaluation of radiation amplitude for moving solitons.
- Analysis of soliton behavior under different nonlinearity types and velocities.
Main Results:
- For cubic nonlinearity, solitons always radiate, irrespective of velocity; no radiation-free motion exists.
- For saturable nonlinearity, radiation is suppressed at specific 'sliding velocities'.
- Solitons generally decelerate radiatively, eventually pinning to the lattice or locking onto sliding velocities in the saturable case.
Conclusions:
- Small-amplitude solitons in DNLS with cubic nonlinearity are unstable due to continuous radiation.
- Saturable nonlinearities offer a mechanism for stable, radiation-free soliton propagation at discrete sliding velocities.
- Despite radiative deceleration, small-amplitude solitons can travel for exponentially long times due to extremely slow energy loss.
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