Designable integrability of the variable coefficient nonlinear Schr\"odinger equation

TL;DR

This paper introduces the concept of designable integrability for the variable coefficient nonlinear Schrödinger equation, allowing coefficients to be analytically designed and leading to novel soliton behaviors in optical and BEC systems.

Contribution

It presents a method to transform VCNLSE into the standard NLSE, enabling analytical design of coefficients and potentials, with applications to soliton dynamics.

Findings

Coefficients can be designed analytically via transformation.

Soliton profiles are variable and trajectories are non-linear.

Applications include optical super-lattices and multi-well potentials.

Abstract

The designable integrability(DI) of the variable coefficient nonlinear Schr\"odinger equation (VCNLSE) is first introduced by construction of an explicit transformation which maps VCNLSE to the usual nonlinear Schr\"odinger equation(NLSE). One novel feature of VCNLSE with DI is that its coefficients can be designed artificially and analytically by using transformation. A special example between nonautonomous NLSE and NLSE is given here. Further, the optical super-lattice potentials (or periodic potentials) and multi-well potentials are designed, which are two kinds of important potential in Bose-Einstein condensation(BEC) and nonlinear optical systems. There are two interesting features of the soliton of the VCNLSE indicated by the analytic and exact formula. Specifically, its the profile is variable and its trajectory is not a straight line when it evolves with time $t$.