A new class of exact solitary wave solutions of one dimensional Gross-Pitaevskii equation

TL;DR

This paper derives a broad family of exact solitary wave solutions for the one-dimensional Gross-Pitaevskii equation with variable parameters, linking them to quantum systems and matching experimental phenomena in Bose-Einstein condensates.

Contribution

It introduces a novel analytical method connecting soliton solutions to linear Schrödinger eigenvalue problems, enabling explicit solutions under time-varying conditions.

Findings

Exact soliton solutions for variable parameters

Analytical description of soliton trains and dynamics

Modeling of experimental phenomena like collapse and revival

Abstract

We present a large family of {\it{exact}} solitary wave solutions of the one dimensional Gross-Pitaevskii equation, with time-varying scattering length and gain/loss, in both expulsive and regular parabolic confinement regimes. The consistency condition governing the soliton profiles is shown to map on to a {\it{linear}} Schr\"odinger eigenvalue problem, thereby enabling one to find analytically the effect of a wide variety of temporal variations in the control parameters, which are experimentally realizable. Corresponding to each solvable quantum mechanical system, one can identify a soliton configuration. These include soliton trains in close analogy to experimental observations of Strecker {\it{et al.,}} [Nature {\bf{{417}}{150}{2002}], spatio-temporal dynamics, solitons undergoing rapid amplification, collapse and revival of condensates and analytical expression of two-soliton bound states, to name a few.