Theoretical Physics Department,

Tomsk State University,

36 Lenin ave.,

Tomsk,

634050, RUSSIA

and

Mathematical Physics Laboratory of the High Mathematics

and Mathematical Physics Department,

Tomsk Polytechnic University,

30 Lenin ave.,

Tomsk,

634050, RUSSIA

E-mail: shpv@phys.tsu.ru, trifonov@mph.phtd.tpu.edu.ru

**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.

See abstract in PDF.