Weakly nonlinear GP equation

The derivation for the description of the nonuniform turbulence found in a BEC system consists of a amalgamation of a WKB method, for the description of the linear dynamics, and a standard weak turbulence theory (see e.g. [13]), with the noted modification that Gabor transforms are used instead of Fourier ones. We will now demonstrate the general ideas of such a derivation for the simple case of system where no condensate is present.

Consider the Gabor transformation of ():

$$i \partial_{t} \hat \psi + \triangle \hat \psi - \widehat{\vert\psi\vert^{2}\psi} - U\hat\psi + i {\bf } (\nabla_x U)\nabla_k\hat{\psi} = 0.$$ (22)

To calculate the $\widehat{\vert\psi\vert^{2}\psi}$ term let us first separate the Gabor transform into its correspondingly fast and slow spatial parts,

$$\hat\psi(\vec{x},\vec{k},t) = \underbrace{a(\vec{x},\vec{k},t)}_{\mbox{slow}} \underbrace{e^{i{\vec k} \cdot {\vec x}}}_{\mbox{fast}}.$$ (23)

Now by using the inverse Gabor transform

$$g(x,t)= \int \hat{g}(\vec{x},\vec{k},t) \, d {\vec k},$$ (24)

we find

$$\widehat{\vert\psi\vert^{2}\psi} = e^{i {\vec k} \cdot {\vec x}} \int f (\vec x-\vec x_0) \, e^{i \vec x_0 \cdot (\vec k_3+ \vec k_2- \vec k_1- \vec k)}$$

$$\times a^*(\vec k_1, \vec x_0) a(\vec k_2, \vec x_0)a(\vec k_3, \vec x_0) \, d {\vec x_0} d {\vec k_1} d {\vec k_2} d {\vec k_3}.$$ (25)

Note that the slow amplitudes $a$ do not change much over the characteristic width of the function $f$ and hence their argument ${\vec{x_0}}$ can be replaced by ${\vec{x}}$. Therefore, we can approximate () by

$$\widehat{\vert\psi\vert^{2}\psi} \simeq \frac{e^{i {\vec k} \cdot {\vec x}}}{(2 \pi)^{3 d/2}} \int F (\vec k_3+\vec k_2-\vec k_1-\vec k)$$

$$\times a^*(\vec k_1,\vec x) a(\vec k_2,\vec x)a(\vec k_3,\vec x) \, d {\vec k_1} d {\vec k_2} d {\vec k_3}.$$ (26)

Here $F(\vec k)$ is the Fourier transform of $f(\vec x)$. Note that for the spatially homogeneous systems, $\epsilon^*\to 0$, $F(\vec k)$ is just a delta function,

$$\lim\limits_{ \epsilon^*\to 0}F(\vec k)\to \delta(\vec k).$$

After dropping terms proportional to $\triangle a$, equation () then becomes

$$\partial_{t} a(\vec k,\vec x) = - 2 \vec k\cdot \nabla a(\vec k,\vec x)$$

$$- i(k^2 +\vec k\cdot(\nabla_x )) a(\vec k,\vec x) - (\nabla_x U) (\nabla_k a(\vec k,\vec x))$$

$$- \int F(\vec k_3 + \vec k_2 - \vec k_1 - \vec k) \,a^*(\vec k_1,\vec x) a(\vec k_2,\vec x) a(\vec k_3,\vec x) \,d {\vec k_1} d {\vec k_2} d {\vec k_3}.$$ (27)

This is the master equation formulating the nonlinear dynamics in terms of the Gabor amplitudes. This can serve as a starting point for the statistical averaging which in turn leads to the weak turbulence formalism. Note that this equation can be written in Hamiltonian form,

$$i \frac{\partial}{\partial t} a_{{\bf k},{\bf x}}= \frac{\delta H} {\delta a_{{\bf x},{\bf k}}^*},$$ (28)

with a Hamiltonian function

$$H = \int ( \omega_{k,x} - {\bf x} \cdot \nabla_x \omega_{k,x} ) \vert a_{k,x}\vert^2$$

$$+ \frac{i}{2} (\nabla_x \omega_{k,x}) ( a_{k,x}^{*}\nabla_{k}a_{k,x} - a_{k,x}\nabla_{k}a^{*}_{k,x} )$$

$$+ \frac{i}{2} (\nabla_k \omega_{k,x}) ( a_{k,x}\nabla_{x}a_{k,x}^{*} - a_{k,x}^{*}\nabla_{x}a_{k,x} ) \, d {\bf k} d {\bf x}$$

$$+ \int F({\bf k_3} + {\bf k_2} - {\bf k_1} - {\bf k})$$

$$a^{*}({\bf k_1} ,{\bf x}) a({\bf k_2},{\bf x}) a({\bf k_3},{\bf x}) a({\bf k_4},{\bf x}) \, d {\vec k_1} d {\vec k_2} d {\vec k_3} d {\vec k_4},$$ (29)

where $\omega_{k,x}= k^2 + U(x)$. In fact, such a Hamiltonian description can be derived directly, in terms of the Gabor amplitudes, from the Hamiltonian formulation of the original GP equation (see Appendix B).

If a condensate is present in the system, one can also re-write the equations in a Hamiltonian form with an identical quadratic part. That is, with $a$ being replaced by the normal amplitude, and $\omega$ by the frequency of waves, found in the presence of the condensate. It appears that the quadratic part of the Hamiltonian () is generic in the WKB context. Indeed, let us consider a typical Hamiltonian for linear waves in weakly inhomogeneous media [32] expressed in terms of Fourier amplitudes $a_{\bf q_1}$ and $a^*_{\bf q_1}$

$${\cal H}=\int\Omega({\bf q_1}, {\bf q}) \, a_{\bf q_1} a^*_{\bf q_1} \, d {\bf q} d {\bf q_1},$$ (30)

with a hermitian kernel $\Omega({\bf q_1}, {\bf q})= \Omega({\bf q}, {\bf q_1})$ which is strongly peaked at ${\bf q}- {\bf q_1} =0$. As we will show in a separate paper [26], this Hamiltonian can be represented in terms of the Gabor transforms as

$$H = \int ( \omega_{\vec k,\vec x} - {\bf x} \cdot \nabla_x \omega_{\vec k,\vec x} ) \vert a_{k,x}\vert^2$$

$$+ \frac{i}{2} (\nabla_x \omega_{\vec k,\vec x} ) ( a_{k,x}^*\nabla_{k}a_{k,x} - a_{k,x}\nabla_{k}a^*_{k,x} )$$

$$+ \frac{i}{2} ( \nabla_k \omega_{\vec k,\vec x} ) ( a_{k,x}\nabla_{x}a_{k,x}^* - a_{k,x}^*\nabla_{x}a_{k,x} ) \, d {\bf k} d {\bf x},$$ (31)

where $a_{k x}$ are the Gabor coefficients, and $\omega_{\vec k\vec x}$ is the position dependent frequency, related to $\Omega({\bf q}, {\bf q_1})$ via

$$\omega_{\vec k,\vec x}=\int e^{- 2 i \vec q\cdot\vec x} \, \Omega(\vec k,\vec k+2\vec q) \, d \vec q.$$ (32)

Actually, such an expression is a canonical form, even for a much broader class of Hamiltonians that correspond to a significant class of linear equations with coordinate dependent coefficients [26]. That is,

$${\cal H}=\int [A({\bf q_1}, {\bf q}) \, a_{\bf q_1} a^*_{\bf q} \... ...\bf q_1}, {\bf q}) \, a_{\bf q_1} a_{\bf - q} + c.c.] \, d {\bf q} d {\bf q_1},$$ (33)

where functions $A$ and $B$ peaked at ${\bf q}- {\bf q_1} =0$.