Integrability of the Gross–Pitaevskii equation with Feshbach resonance management

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 Schrödinger equation. By this transformation, each exact solution of the standard nonlinear Schrödinger 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.

Compression of Bright Bound Solitons in the Bose-Einstein Condensates with Exponentially Time-Dependent Atomic Scattering Length by the Feshbach Resonance

“…In fact, the complete integrability requires that 4V(t) = G(t)d 2 [1/G(t)]/dt 2 [9,29,30], and such constraint is derived and applied in the sense of the Painlevé integrability in Refs. [31][32][33]. If the Feshbach-resonancemanaged coefficient G(t) varies exponentially with time, i.e., G(t) = g 0 exp(λt), V(t) is confirmed as V(t) = λ 2 /4, which accords with the situations in Refs.…”

Solitons of two-component Bose–Einstein condensates modulated in space and time

“…Other related investigations are as follows: the dispersion and nonlinear management for femtosecond optical solitons was considered in [23], in Ref. [24] one deals with the integrability of the Gross-Pitaevskii equation with Feshbach resonance management, and in Ref. [25] one treats with the eigenvalues of the Zakharov-Shabat scattering problem for two separated sech-shaped pulses.…”

Enhanced soliton spectral tunneling effect of self-compressing nonautonomous colored femtosecond solitons

“…for the complex amplitude of the laser pulse envelope q(z,τ) coupled with the so-called oscillator model of stimulated Raman scattering effect 10,11,[15][16][17][18] …”