Integrability of the Gross-Pitaevskii Equation with Feshbach Resonance management

TL;DR

This paper investigates the conditions under which the Gross-Pitaevskii equation with Feshbach resonance management is integrable, providing a transformation linking it to the standard nonlinear Schrödinger equation, thus enabling systematic analysis of soliton solutions.

Contribution

It establishes the integrability condition for the Feshbach-resonance-managed Gross-Pitaevskii equation and introduces a transformation to the standard nonlinear Schrödinger equation, connecting nonautonomous and canonical solitons.

Findings

Identifies a specific integrability condition consistent with prior work.

Derives a transformation linking the Gross-Pitaevskii and nonlinear Schrödinger equations.

Enables systematic construction of solutions for Bose-Einstein condensates.

Abstract

In this paper we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V. N. Serkin et al., Phys. Rev. Lett. 98, 074102 (2007)]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schr\"odinger equation. By this transformation, each exact solution of the standard nonlinear Schr\"odinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitions and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique.