Perturbation expansion for the Hamiltonian

One can introduce new variable $\tilde x$ as

$$x=u+\tilde x,\hspace{0.3cm} x_u = 1+\tilde x_u, \hspace{0.3cm} \tilde x = -\hat Hz$$

Then

$$\Psi = \frac{\mbox{$\cal P$}+ \mbox{$\cal P$}\tilde x_u + \hat H(\mbox{$\cal P$}z_u)}{(1+\tilde x_u)^2+z_u^2}$$

Now one can expand $\Psi$ in powers of $\tilde x_u$ and $z_u$

$$\Psi = \Psi^{(1)} + \Psi^{(2)} + \Psi^{(3)} + \Psi^{(4)} + \ldots \cr \Psi^{(1)}$$

The Hamiltonian of the system is

$$\mbox{$\cal H$}= \frac{1}{2} \int \{g z^2 (1+\tilde x_u) - \Psi \hat H\Psi_u\} du$$

Now $\mbox{$\cal H$}$ can be expanded as follow

$$\mbox{$\cal H$}= \mbox{$\cal H$}_2 + \mbox{$\cal H$}_3 + \mbox{$\cal H$}_4 + \mbox{$\cal H$}_5 + \ldots$$ (3.1)

here

$$\mbox{$\cal H$}_2 = {1\over2}\int(g z^2 - \Psi^{(1)} \hat H\Psi^{(1)}_u) du \cr \mbox{$\cal H$}_3$$

Let's introduce the Fourier transform:

$$f_k = {1\over \sqrt{2\pi}} \int f(u) e^{-i k u} du,\hspace{1cm} f(u) = {1\over \sqrt{2\pi}} \int f_k e^{i k u} dk$$

After simple, but a little bit tedious calculations one can find

$$\mbox{$\cal H$}_2 = \frac{1}{2}\int (g \vert z_k\vert^2 + \vert k\vert\vert\mbox{$\cal P$}_k\vert^2) dk, \cr \mbox{$\cal H$}_3$$

Here $S_{k_1 k_2 k_3}$, $F^{k_1}_{k_2 k_3}$, $M^{k_1 k_2}_{k_3 k_4}$ and $N^{k_1 k_2 k_3}_{k_4 k_5}$ are the functions symmetric inside of upper and lower groups of indices. Namely

$$S_{k_1 k_2 k_3} = {g\over 3}(\vert k_1\vert+\vert k_2\vert+\vert k_3\vert)\cr L_{k_1 k_2}$$ (3.2)

The expression for $M^{k_1 k_2}_{k_3 k_4}$ and $N^{k_1 k_2 k_3}_{k_4 k_5}$ are given in the Appendix A.

It is convenient to introduce a normal complex variable $a_k$

$$\nonumber y_k = \sqrt{\frac{\omega_k}{2g}}(a_k+a^*_{-k}) \hs... ...\mbox{$\cal P$}_k = -i\sqrt{\frac{2g}{\omega_k}}(a_k-a^*_{-k})$$

which satisfies the equation of motion

$$\nonumber \frac{\partial a_k}{\partial t} + i\frac{\delta \mbox{$\cal H$}}{\delta a_k^*}=0.$$

here $\omega_k = \sqrt{g\vert k\vert}$ -is the dispersion law for the gravity waves.

In the normal variable $a_k$ second order term in the Hamiltonian acquires the form:

$$\mbox{$\cal H$}_2 = \int \omega_k a_k a_k^* dk$$

The third order term is:

$$\mbox{$\cal H$}_3 = \frac{1}{2}\int\!V^{k_1}_{k_2 k_3} \{a_{k_1}^* a_{k_2} a_{k_3}+a_{k_1} a_{k_2}^* a_{k_3}^*\} \delta_{k_1-k_2-k_3}\!dk_1dk_2dk_3+\cr$$

$$V^{k_1}_{k_2 k_3} = \frac{g^{\frac{1}{4}}}{2\sqrt{4\pi}} (\vert k_1\vert+\vert k_2\vert+\vert k_3\vert)\times\cr$$

Fourth order term in the Hamiltonian consists of three terms:

$$\mbox{$\cal H$}_4 = \mbox{$\cal H$}_4^{4\Leftrightarrow 0} + \mbox{$\cal H$}_4^{3\Leftrightarrow 1} + \mbox{$\cal H$}_4^{2\Leftrightarrow 2}$$

describing different types of the wave-wave interactions. The term corresponding to $4\Leftrightarrow 0$ interaction has a form:

$$\mbox{$\cal H$}_4^{4\Leftrightarrow 0} = \frac{1}{24}\int R_{k_1 k_2 k_3 k_4} (a_{k_1}a_{k_2}a_{k_3}a_{k_4}+a_{k_1}^*a_{k_2}^*a_{k_3}^*a_{k_4}^*)\times\cr$$

where $R_{k_1 k_2 k_3 k_4}$ is:

$$R_{k_1 k_2 k_3 k_4} = \frac{-1}{4\pi} ( {\left\vert\frac{k_1 k_2}{k_3 k_4}\right\vert}^\frac{1}{4}M^{k_... ...ft\vert\frac{k_1 k_4}{k_2 k_3}\right\vert}^\frac{1}{4}M^{k_1 k_4}_{k_2 k_3}+\cr$$

Term corresponding to $3\Leftrightarrow 1$ interaction has a form:

$$\mbox{$\cal H$}_4^{3\Leftrightarrow 1} = \frac{1}{6}\int G^{k_1}_{k_2 k_3 k_4} (a_{k_1}^*a_{k_2}a_{k_3}a_{k_4}+a_{k_1}a_{k_2}^*a_{k_3}^*a_{k_4}^*)\times\cr$$

where $G^{k_1}_{k_2 k_3 k_4}$:

$$G^{k_1}_{k_2 k_3 k_4} = \frac{-1}{4\pi} ( {\left\vert\frac{k_1 k_2}{k_3 k_4}\right\vert}^\frac{1}{4}M^{-k... ...t\vert\frac{k_1 k_4}{k_2 k_3}\right\vert}^\frac{1}{4}M^{-k_1 k_4}_{k_2 k_3}-\cr$$

Term corresponding to $2\Leftrightarrow 2$ interaction has a form:

$$\mbox{$\cal H$}_4^{2\Leftrightarrow 2} = \frac{1}{4}\int W^{k_1 k... ... k_4} a_{k_1}^*a_{k_2}^*a_{k_3}a_{k_4} \delta_{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4$$

where $W^{k_1 k_2}_{k_3 k_4}$:

$$W^{k_1 k_2}_{k_3 k_4} = \frac{-1}{4\pi} ( {\left\vert\frac{k_1 k_2}{k_3 k_4}\right\vert}^\frac{1}{4}M^{-k... ...\vert\frac{k_1 k_4}{k_2 k_3}\right\vert}^\frac{1}{4}M^{-k_1 k_4}_{k_3 -k_2}-\cr$$

Among the different terms of the fifth order we consider only the term, corresponding to the process (1.6):

$$\mbox{$\cal H$}_5 = \!\frac{1}{12}\int Q^{k_1 k_2 k_3}_{k_4 k_5} ... ...}^* a_{k_4} a_{k_5}+\!c.c.\} \delta_{k_1+k_2+k_3-k_4-k_5}\!dk_1dk_2dk_3dk_4dk_5$$

where

$$Q^{k_1 k_2 k_3}_{k_4 k_5} = \frac{-3}{(4\pi)^{\frac{3}{2}}g^{\frac{1}{4}}}\times\{\cr$$