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Next: Appendix B: Hamiltonian formalism Up: Appendix A: derivation WKB Previous: The order -

The $ \epsilon^1$ order -

Let us split the wave amplitudes into fastly and slowly varying parts,

  $\displaystyle \lambda(\vec{x},\vec{k},t) = 
 \Lambda(\vec{x},\vec{k},t)\E^{i \vec{k}\cdot\vec{x}+i \omega t},$ $\displaystyle \mu (\vec{x},\vec{k},t) = 
 M(\vec{x},\vec{k},t)\E^{i \vec{k}\cdot\vec{x}-i \omega t},$ (47)

or, in shorthand notation,

  $\displaystyle \lambda = \Lambda\E^{+},$ $\displaystyle \mu = M\E^{-},$ (48)

where

  $\displaystyle \E^{+} \equiv \E^{i \vec{k}\cdot\vec{x}+i \omega t},$ $\displaystyle \E^{-}$ $\displaystyle \equiv \E^{i \vec{k}\cdot\vec{x}-i \omega t}.$    

The $ \E^{+}$ and $ \E^{-}$ represent the fastly oscillating parts of the Gabor transforms. From ([*]) it follows that

$\displaystyle \Lambda^*(\vec{x},-\vec{k},t) = M(\vec{x},\vec{k},t).$ (49)

Obviously,

  $\displaystyle \partial_{t}\lambda = i \omega\lambda + \E^{+}\partial_{t}\Lambda,$ $\displaystyle \partial_{t}\mu = -i \omega\mu + \E^{-}\partial_{t}M,$ (50)

  $\displaystyle \nabla\lambda = i \vec{k}\lambda + \E^{+}\nabla\Lambda 
 + i t\lambda\nabla\omega,$ $\displaystyle \nabla\mu = i \vec{k}\mu + \E^{-}\nabla M -i t\mu\nabla\omega,$ (51)

$\displaystyle \triangle \lambda$ $\displaystyle = -k^{2}\lambda 
 + 2i \E^{+}\vec{k}\cdot\nabla\Lambda 
 + \E^{+}\triangle\Lambda 
 - 2t\lambda\vec{k}\cdot\nabla\omega,$ (52)
$\displaystyle \triangle \mu$ $\displaystyle = -k^{2}\mu 
 + 2i \E^{+}\vec{k}\cdot\nabla M 
 + \E^{+}\triangle M 
 + 2t\mu\vec{k}\cdot\nabla\omega,$    

  $\displaystyle \partial_{k}\lambda = \E^{+}\partial_{k}\Lambda 
 + i \vec{x}\E^{+}\Lambda + i t\E^{+}\Lambda\partial_{k}\omega,$ (53)
  $\displaystyle \partial_{k}\mu = \E^{-}\partial_{k}M 
 + i \vec{x}\E^{-}M - i t\E^{-}M\partial_{k}\omega.$    

Our aim now is to derive equations for $ \partial_{t}\lambda$ and $ \partial_{t}\mu$. However, due to the relationship ([*]) it is sufficient to derive an equation for only one of the two, for example $ \lambda$. From ([*]) we find

$\displaystyle \partial_{t}\lambda = \partial_{t}\left(\frac{\hat{a}}{2} 
 - \frac{i k^{2}}{2\omega}\hat{b}\right).$    

After substituting our equations for $ \partial_{t}\hat{a}$ and $ \partial_{t}\hat{b}$, ([*]) and ([*]), and making use of the relationships ([*]) the equation for $ \lambda$ acquires the following form:

$\displaystyle \partial_{t}\lambda$ $\displaystyle = \lambda \left[-\frac{i \nabla\varrho\cdot\nabla\omega}{2k^{2}\v...
...k^{2}\varrho}
 - 2\vec{v}
 - \frac{i k^{2}\nabla\varrho}{2\omega\varrho}\right]$    
  $\displaystyle + \triangle\lambda \left[-\frac{i \omega}{2k^{2}}
 -\frac{i k^{2}}{2\omega}\right]
 -\frac{k^{2}}{\omega} \nabla\varrho\cdot\partial_{k}\lambda$    
  $\displaystyle + \mu \left[\frac{i \nabla\varrho\cdot\nabla\omega}{2k^{2}\varrho...
...la\varrho}{2k^{2}\varrho}
 - \frac{i k^{2}\nabla\varrho}{2\omega\varrho}\right]$    
  $\displaystyle + \triangle\mu \left[+\frac{i \omega}{2k^{2}}
 -\frac{i k^{2}}{2\...
...a}\right]
 -\frac{k^{2}}{\omega}\nabla\varrho\cdot\partial_{k}\mu - {\cal{NL}}.$    

Here the nonlinear term $ {\cal{NL}}$ is given by

$\displaystyle {\cal{NL}}={\cal G} \left[\rho(2 a b + b(a^2+b^2))\right] -\frac{i
k^2}{2\omega_k} {\cal G}\left[( \varrho(3 a^2+b^2+a(a^2+b^2)))\right].$

Note that we have neglected $ \dot \omega$ in the above expressions because, according to the dispersion relationship ([*]), it is of the order of $ \dot \rho$ which is $ O(\epsilon^2)$ by virtue of ([*]). We will also drop the nonlinear term in the subsequent calculation.

Our next step is to eliminate the fast oscillations associated with the Gabor transforms and derive an equation for $ \vert\Lambda\vert^{2}$. This in turn will lead to a natural waveaction quantity which can be used to describe the behavior of our wavepackets in phase space. Using ([*]-[*]) we obtain

$\displaystyle \partial_{t}\Lambda$ $\displaystyle = \Lambda \left[-\frac{i \nabla\varrho\cdot\nabla\omega}{2k^{2}\varrho}
 + \frac{i k^{2}\varrho}{\omega} -i \omega \right]$    
  $\displaystyle + \left[i \vec{k}\Lambda+\nabla\Lambda+i t\Lambda\nabla\omega\rig...
...k^{2}\varrho}
 - 2\vec{v}
 - \frac{i k^{2}\nabla\varrho}{2\omega\varrho}\right]$    
  $\displaystyle + \left[-k^{2}\Lambda+2i \vec{k}\cdot\nabla\Lambda-2t\Lambda\vec{...
...ega\right]
 \left[-\frac{i \omega}{2k^{2}}-\frac{i \vec{k}^{2}}{2\omega}\right]$    
  $\displaystyle -\frac{k^{2}}{\omega}\nabla\varrho\cdot\partial_{k}\Lambda
 -\fra...
...varrho
 -\frac{i tk^{2}\Lambda}{\omega}\nabla \varrho
 \cdot\partial_{k}\omega.$    
  (54)

Please note that all the terms involving $ M$ drop out. This stems from the fact that, in deriving an equation for $ \Lambda$, we have had to divide through by $ \E^{+}$. Therefore, any terms involving $ M$ will result in a factor

$\displaystyle \E^{-}/\E^{+} = \E^{-2i \omega t}.$    

Thus, after time averaging over a few wave periods, all the $ M$ terms drop out.

Expanding out equation ([*]) we find the $ O(1)$ terms cancel out and using the dispersion relationship ([*]) we find

$\displaystyle \partial_{t}\Lambda = \partial_{k}\omega\cdot\nabla\Lambda +
 \fr...
...k^{2}}\vec{k}\cdot\nabla\varrho -
 \nabla\omega\cdot\partial_{k}\Lambda + i J ,$ (55)

where

$\displaystyle J = \frac{t\Lambda\omega}{k^{2}}\vec{k}\cdot\nabla\omega
 + \frac...
...bla\varrho
 - \frac{tk^{2}\Lambda}{\omega}\nabla\varrho\cdot\partial_{k}\omega,$    

At this point let us drop the nonlinear term and concentrate on the linear dynamics. Multiplying ([*]) by $ \Lambda^{*}$ and combining it with the complex conjugate equation the $ J$ terms cancel, leading to

$\displaystyle \partial_{t}\vert\Lambda\vert^{2} - \partial_{k}\omega\cdot\nabla...
... \frac{2\vert\Lambda\vert^{2}\omega}
 {\varrho k^{2}}\vec{k}\cdot\nabla\varrho.$ (56)

A similar equation for $ \vert M\vert^{2}$ can be easily obtained by replacing $ {\bf k} \to {- \bf k}$ in ([*]) and using ([*]),

$\displaystyle \partial_{t}\vert M\vert^{2} + \partial_{k}\omega\cdot\nabla\vert...
...
 = - \frac{2\vert M\vert^{2}\omega}
 {\varrho k^{2}}\vec{k}\cdot\nabla\varrho.$ (57)

The LHS of this equation is the full time derivative of $ \vert M\vert^{2}$ along trajectories. If $ \vert M\vert^{2}$ were to be a correct phase-space waveaction, the right hand side of this equation would be zero, however, this is not the case. We find the correct waveaction $ n(\vec{x},\vec{k},t) $ by setting

$\displaystyle \vert M\vert^{2} = \alpha(\vec{x},\vec{k})n(\vec{x},\vec{k},t),$    

and finding such $ \alpha(\vec{x},\vec{k})$ that the the full time derivative of $ n(\vec{x},\vec{k},t) $ is zero. This leads to the following condition on $ \alpha$,

$\displaystyle \partial_{k}\omega\cdot\nabla\alpha 
 - \nabla\omega\cdot\partial_{k}\alpha 
 + \frac{2\alpha\omega}{\varrho k^{2}}\vec{k}\cdot\nabla\varrho = 0.$    

By choosing $ \alpha = k^{X}\varrho^{Y}$ and substituting it to ([*]) we find $ x=2$, $ y=-1$. Therefore the correct form of the waveaction is $ n = \frac{\varrho}{k^{2}}\vert M\vert^{2}.$ Summarizing, we have got the following transport equation for the waveaction $ n$ in the linear approximation,

$\displaystyle D_{t}n(\vec{x},\vec{k},t) = 0,$ (58)

where

$\displaystyle D_{t}\equiv \partial_{t} + \dot{\vec{x}}\cdot\nabla +\dot{\vec{k}}\cdot\partial_{k},$ (59)

is the full time derivative along trajectories and

$\displaystyle \dot{\vec{x}} = \partial_{k} \omega, \hspace{1cm}
 \dot{\vec{k}} = -\nabla \omega,$ (60)

are the ray equations with

$\displaystyle \omega = k \sqrt{k^{2}+2\varrho}.$ (61)

Obviously, the dynamics in this case cannot be reduced to the Ehrenfest theorem with any shape of potential $ U$. Therefore, approaches that model the condensate effect by introducing a renormalized potential are misleading.

Finally, it is useful to express the waveaction $ n$ in terms of the original variables,

$\displaystyle n({\bf k},x,t) = \frac{1}{2}
 \frac{\omega\rho}{k^{2}} \left\vert...
...hat{ \Re \phi}
 - \frac{ i k^{2} }{\omega} \widehat{ \Im \phi} \right\vert^{2}.$ (62)

It is interesting that such a waveaction is in agreement with that found in [13]. In fact in [13] the homogeneous case with non-zero nonlinearity ( $ \varepsilon = 0$, $ \sigma \neq 0 $) was considered. This is the opposite limit to the one we have considered above (where $ \varepsilon \neq 0$, $ \sigma = 0$).


next up previous
Next: Appendix B: Hamiltonian formalism Up: Appendix A: derivation WKB Previous: The order -
Dr Yuri V Lvov 2007-01-23