Transverse Evolution Operator and Symmetry Operators for the Gross-Pitaevskii Equation in Semiclassical Approximation

Abstract:
The $n$-dimensional Gross-Pitaevskii equation (GPE) with an attractive self-action reads
\begin{equation}
\left(-i\hbar\partial_{t}+{\cal H}( \hat{\vec p},\vec x,t)-g^{2}|\Psi(\vec x,t,\hbar)|^{2}\right)\Psi(\vec x,t,\hbar)=0,
\end{equation}
where $\vec x \in {\mathbb R}^{n}$, $t\in{\mathbb R}^{1}$, $\hat{\vec p}=-i\hbar\nabla_{\vec x}$, $g$ is a real nonlinearity parameter, $\hbar$ is  a small asymptotical parameter, $\hbar\to 0$; ${\cal H}( \hat{\vec p},\vec x,t)$ is a linear operator quadratic in $\hat{\vec p}$ and Weyl ordered in $\hat{\vec p}$ and $\vec x$. The GPE (1) is one of the basic model equations in the theory of Bose-Einstein condensate (BEC). Localized solutions of the GPE describe the condensate in external electro-magnetic fields including fields of magnetic traps. In the context of the complex WKB-Maslov method  a class of one-soliton trajectory concentrated functions (OSTCF) is introduced. Functions of this class are soliton-like fast-oscillating wave packets concentrated in a neighborhood of a trajectory in an effective phase space. A solution of the Cauchy problem is presented for the Eq. (1) in the class of OSTCF in semiclassical approximation. The evolution operator acting on the variables transversal on a wave packet vector is derived to obtain the leading term of the asymptotic solution in the class of OSTCF. A class of symmetry operators for the GPE is constructed in semiclassical approximation using the evolution operator. The three-dimensional GPE with the oscillator external field is considered as an illustration and the collapse problem is discussed.

The work has been  supported in part by the Grant for Support of Russian Scientific Schools 1743.2003.2, and the Grant of the President of the Russian Federation YD-246.2003.02.

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