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Integrability of the Gross-Pitaevskii Equation with Feshbach Resonance management
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.Comment: 13 page