Unified WKB description

It is interesting that taking the limit of zero condensate amplitude in the waveaction () results in the waveaction $\frac{1}{2}\vert\hat\Psi\vert^2$ of the Ehrenfest equation () which corresponds to the regime without condensate,

$$\lim\limits_{\rho\to 0}n({\bf k},x,t)\to \frac{1}{2}\rho \left\vert \widehat{\phi}\right\vert^{2} = \frac{1}{2}\vert\hat\Psi\vert^2.$$

On the other hand, $\lim\limits_{\rho\to 0} \omega \to k^2$ which is different from the Ehrenfest expression $\omega = k^2 +U$. Thus, one cannot recover the non-condensate (Ehrenfest) description by just taking the limit of zero condensate amplitude in (), () and (). However, one can easily write a unified WKB description which will be valid with or without condensate by simply adding $U+\rho$ to the frequency (). Indeed, for strong condensate $U+\rho$=const and, therefore, it does not alter the ray equations (which contain only derivatives of $\omega$). On the other hand, such an addition allows us to obtain the correct expression

$$\omega = k^2 +U,$$

in the limit $\rho\to 0$. Summarizing, we write the following equations of the linear WKB theory which are valid with or without the presence of a condensate,

$$D_{t}n(\vec{x},\vec{k},t) = 0,$$ (17)

where

$$n({\bf k},x,t) = \frac{1}{2} \frac{\omega\rho}{k^{2}} \left\ver... ...{ \Re \phi} - \frac{ i k^{2} }{\omega} \widehat{ \Im \phi} \right\vert^{2},$$ (18)

is the waveaction and

$$D_{t}\equiv \partial_{t} + \dot{\vec{x}}\cdot\nabla +\dot{\vec{k}}\cdot\partial_{k},$$ (19)

is the full time derivative along trajectories and

$$\dot{\vec{x}} = \partial_{k} \omega, \hspace{1cm} \dot{\vec{k}} = -\nabla \omega,$$ (20)

are the ray equations with

$$\omega = k \sqrt{k^{2}+2\varrho} + U + \rho.$$ (21)

Formula () is an important and nontrivial result which can be obtained neither from existing general facts about the WBK formalism nor from the linear theory of homogeneous systems.