Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged

TL;DR

This paper presents a method for real-time control of solitary wave motion in nonlinear systems using adiabatic management of dispersion and gain/loss, applicable to various physical contexts.

Contribution

It introduces a novel approach for adiabatic soliton management that maintains wave structure while controlling speed and direction in real time.

Findings

Effective control of KdV solitons demonstrated

Manipulation of wavefronts in Fisher-KPP equations shown

Method applicable to multiple nonlinear wave systems

Abstract

We describe how to control the motion -- both speed and propagation direction -- of a nonlinear traveling wave in real time via soliton management with time-varying dispersion/diffusion and loss/gain terms. When carried out subject to certain parameter constraints we derive, the approach allows for real-time manipulation of wave motion without compromising the structure or stability of the wave, maintaining strict adiabaticity in time. This makes our approach a promising tool for physical systems which admit a degree of control through adjustable dispersion and loss/gain terms, including problems arising in nonlinear optics, atomic physics, Bose-Einstein condensates, and thermally driven chemical reactions, where one is interested in modifying the position or speed of a wave while still maintaining adiabaticity. We illustrate both the utility and simplicity of the method through several canonical examples, demonstrating how the motion of KdV solitons, Fisher-KPP solitary wavefronts, BBM solitons, nonlinear Klein-Gordon equations, and interacting nonlinear waves in a vector NLS system can be controlled in real time.