Appendix
Coefficients in the Hamiltonian
(9) can be calculated plugging expressions for complex canonical variables
into the (2):
$$\begin{aligned} U_{k_1k_2k_3}= & {} \frac{1}{8}\frac{g^{\frac{1}{4}}}{\sqrt{\pi }} \left[ |\frac{k_1}{k_2k_3}|^{\frac{1}{4}}L_{k_2k_3} + |\frac{k_2}{k_1k_3}|^{\frac{1}{4}}L_{k_1k_3} + |\frac{k_3}{k_1k_2}|^{\frac{1}{4}}L_{k_1k_2} \right] ,\\V^{k_1}_{k_2k_3}= & {} \frac{1}{8}\frac{g^{\frac{1}{4}}}{\sqrt{\pi }} \left[ |\frac{k_1}{k_2k_3}|^{\frac{1}{4}}L_{k_2k_3} - |\frac{k_2}{k_1k_3}|^{\frac{1}{4}}L_{-k_1k_3} - |\frac{k_3}{k_1k_2}|^{\frac{1}{4}}L_{-k_1k_2} \right] . \end{aligned}$$
(18)
$$\begin{aligned} W_{k_1k_2}^{k_3k_4}= & {} \frac{-1}{32\pi } \bigg [ \left| \frac{k_1k_2}{k_3k_3} \right| ^\frac{1}{4}M_{-k_3-k_4}^{k_1k_2} + \left| \frac{k_3k_4}{k_1k_2} \right| ^\frac{1}{4}M_{k_1k_2}^{-k_3-k_4} - \left| \frac{k_1k_3}{k_2k_4} \right| ^\frac{1}{4}M_{k_2-k_4}^{k_1-k_3} - \left| \frac{k_2k_3}{k_1k_4} \right| ^\frac{1}{4}M_{k_1-k_4}^{k_2-k_3} -\\- & {} \left| \frac{k_1k_4}{k_2k_3} \right| ^\frac{1}{4}M_{k_2-k_3}^{k_1-k_4} - \left| \frac{k_2k_4}{k_1k_3} \right| ^\frac{1}{4}M_{k_1-k_3}^{k_2-k_4} \bigg ]\\G_{k_1k_2k_3}^{k_4}= & {} \frac{-1}{32\pi }\bigg [ \left| \frac{k_3k_4}{k_1k_2} \right| ^\frac{1}{4}M_{k_1k_2}^{k_3-k_4} + \left| \frac{k_2k_4}{k_1k_3} \right| ^\frac{1}{4}M_{k_1k_3}^{k_2-k_4} + \left| \frac{k_1k_4}{k_2k_3} \right| ^\frac{1}{4}M_{k_2k_3}^{k_1-k_4} - \left| \frac{k_1k_2}{k_3k_4} \right| ^\frac{1}{4}M_{k_3-k_4}^{k_1k_2} -\\- & {} \left| \frac{k_1k_3}{k_2k_4} \right| ^\frac{1}{4}M_{k_2-k_4}^{k_1k_3} - \left| \frac{k_2k_3}{k_1k_4} \right| ^\frac{1}{4}M_{k_1-k_4}^{k_2k_3} \bigg ]\\R_{k_1k_2k_3k_4}= & {} \frac{-1}{32\pi }\bigg [ \left| \frac{k_3k_4}{k_1k_2} \right| ^\frac{1}{4}M_{k_1k_2}^{k_3k_4} + \left| \frac{k_2k_4}{k_1k_3} \right| ^\frac{1}{4}M_{k_1k_3}^{k_2k_4} + \left| \frac{k_2k_3}{k_1k_4} \right| ^\frac{1}{4}M_{k_1k_4}^{k_2k_3} + \left| \frac{k_1k_4}{k_2k_3} \right| ^\frac{1}{4}M_{k_2k_3}^{k_1k_4} +\\+ & {} \left| \frac{k_1k_3}{k_2k_4} \right| ^\frac{1}{4}M_{k_2k_4}^{k_1k_3} + \left| \frac{k_1k_2}{k_3k_4} \right| ^\frac{1}{4}M_{k_3k_4}^{k_1k_2} \bigg ] \end{aligned}$$
(19)
Here
$$\begin{aligned} L_{k_1k_2}&= |k_1k_2| + k_1k_2\nonumber \\ M_{k_1k_2}^{k_3k_4}&= |k_1k_2|(|k_1+k_3|+|k_1+k_4|+|k_2+k_3|+|k_2+k_4|-2|k_1|-2|k_2|). \end{aligned}$$
(20)
To construct canonical transformation
of general form we follow the book (Zakharov et al. 1992) and use auxiliary Hamiltonian
:
$$\begin{aligned} \tilde{H}&=-i\int \tilde{V}^{k_1}_{k_2k_3}(b_{k_1}^*b_{k_2}b_{k_3}-b_{k_1}b_{k_2}^*b_{k_3}^*)\delta _{k_1-k_2-k_3} dk_1dk_2dk_3-\nonumber \\&-\frac{i}{3}\int \tilde{U}_{k_1k_2k_3}(b_{k_1}^*b_{k_2}^*b_{k_3}^*-b_{k_1}b_{k_2}b_{k_3})\delta _{k_1+k_2+k_3}dk_1dk_2dk_3,\\&+\frac{1}{2}\int (\tilde{\tilde{W}}_{k_1k_2}^{k_3k_4}+i\tilde{W}_{k_1k_2}^{k_3k_4})b_{k_1}^*b_{k_2}^*b_{k_3}b_{k_4}\delta _{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4-\nonumber \\&-\frac{i}{3}\int \tilde{G}_{k_1k_2k_3}^{k_4}(b_{k_1}^*b_{k_2}^*b_{k_3}^*b_{k_4}-b_{k_1}b_{k_2}b_{k_3}b_{k_4}^*)\delta _{k_1+k_2+k_3-k_4}dk_1dk_2dk_3dk_4-\nonumber \\&-\frac{i}{12}\int \tilde{R}_{k_1k_2k_3k_4}(b_{k_1}^*b_{k_2}^*b_{k_3}^*b_{k_4}^*-b_{k_2}b_{k_3}b_{k_4})\delta _{k_1+k_2+k_3+k_4}dk_1dk_2dk_3dk_4 \end{aligned}$$
(21)
with standard symmetry conditions for coefficients. Just mention that for \(\tilde{W}_{k_1k_2}^{k_3k_4}\) this condition is the following:
$$\begin{aligned} \tilde{W}_{k_1k_2}^{k_3k_4} = \tilde{W}_{k_2k_1}^{k_3k_4} = \tilde{W}_{k_1k_2}^{k_4k_3} = -\tilde{W}_{k_3k_4}^{k_1k_2}. \end{aligned}$$
(22)
Again, following Zakharov et al. (1992) general canonical transformation
from \(b_k\) to \(a_k\) can be written as the series:
$$\begin{aligned}&a_k = b_k + \int \left[ 2\tilde{V}^{k_1}_{kk_2}b_{k_1}b_{k_2}^*\delta _{k_1-k-k_2} - \tilde{V}^k_{k_1k_2}b_{k_1}b_{k_2}\delta _{k-k_1-k_2} - \tilde{U}_{kk_1k_2}b_{k_1}^*b_{k_2}^*\delta _{k+k_1+k_2} \right] dk_1dk_2 \nonumber \\&+ \int \left[ A^{k}_{k_1k_2k_3}b_{k_1}b_{k_2}b_{k_3} + A^{kk_1}_{k_2k_3}b_{k_1}^*b_{k_2}b_{k_3} + A^{kk_1k_2}_{k_3}b_{k_1}^*b_{k_2}^*b_{k_3} + A^{kk_1k_2k_3}b_{k_1}^*b_{k_2}^*b_{k_3}^* \right] dk_1dk_2dk_3 \end{aligned}$$
(23)
Coefficients A with upper and lower indices are equal to:
$$\begin{aligned} A^{k}_{k_1k_2k_3}= & {} \left[ \frac{1}{3}\tilde{G}_{k_1k_2k_3}^{k}+ \tilde{V}_{k_1k-k_1}^{k}\tilde{V}_{k_2k_3}^{k_2+k_3} - \tilde{V}_{kk_1-k}^{k_1}\tilde{U}_{-k_2-k_3k_2k_3}\right] \delta _{k-k_1-k_2-k_3},\\A^{kk_1}_{k_2k_3}= & {} \left[ -i\tilde{\tilde{W}}_{kk_1}^{k_2k_3} + \tilde{W}_{kk_1}^{k_2k_3} - 2\tilde{V}_{k_2k-k_2}^{k}\tilde{V}_{k_1k_3-k_1}^{k_3} - \tilde{V}_{kk_1}^{k+k_1}\tilde{V}_{k_2k_3}^{k_2+k_3} + 2\tilde{V}_{kk_3-k}^{k_3}\tilde{V}_{k_2k_1-k_2}^{k_1} +\right. \\+ & {} \left. \tilde{U}_{-k-k_1kk_1}\tilde{U}_{-k_2-k_3k_2k_3} \right] \delta _{k+k_1-k_2-k_3},\\A^{kk_1k_2}_{k_3}= & {} \left[ -\tilde{G}_{kk_1k_2}^{k_3} +\tilde{V}_{k_3k-k_3}^{k}\tilde{U}_{-k_2-k_1k_2k_1} - \tilde{V}_{kk_3-k}^{k_3}\tilde{V}_{k_1k_2}^{k_1+k_2} +2\tilde{V}_{kk_1}^{k+k_1}\tilde{V}_{k_2k_3-k_2}^{k_3} - \right. \\- & {} \left. 2\tilde{U}_{-k-k_1kk_1}\tilde{V}_{k_3k_2-k_3}^{k_2} \right] \delta _{k+k_1+k_2-k_3},\\A^{kk_1k_2k_3}= & {} \left[ -\frac{1}{3}\tilde{R}_{kk_1k_2k_3} -\tilde{V}_{kk_1}^{k+k_1}\tilde{U}_{-k_2-k_3k_2k_3} + \tilde{V}_{k_2k_3}^{k_2+k_3}\tilde{U}_{-k-k_1kk_1} \right] \delta _{k+k_1+k_2+k_3}. \end{aligned}$$
(24)
Let us now substitute transformation (23) into the Hamiltonian
(9) and calculate second, third and fourth order terms.
Collecting all cubic terms after substitution and making symmetrization one can get:
$$\begin{aligned}&\qquad H_3 =\int [V^{k_1}_{k_2k_3}-(\omega _{k_1}-\omega _{k_3}-\omega _{k_3})\tilde{V}^{k_1}_{k_2k_3}]b_{k_1}^*b_{k_2}b_{k_3} \delta _{k_1-k_2-k_3}dk_1dk_2dk_3 +\nonumber \\&+\frac{1}{3}\int [U_{k_1k_2k_3}-(\omega _{k_1}+\omega _{k_3}+\omega _{k_3})\tilde{U}_{k_1k_2k_3}]b_{k_1}^*b_{k_2}^*b_{k_3}^*\delta _{k_1+k_2+k_3}dk_1dk_2dk_3 + c.c. \end{aligned}$$
(25)
it is possible to cancel nonresonant both cubic and fourth order terms. If
$$\begin{aligned} \tilde{V}_{k_1k_2}^{k} = \frac{V_{k_1k_2}^{k}}{\omega _k-\omega _{k_1}-\omega _{k_2}},\qquad \tilde{U}_{kk_1k_2} = \frac{U_{kk_1k_2}}{\omega _k+\omega _{k_1}+\omega _{k_2}}. \end{aligned}$$
(26)
than \(H_3\) vanishes.
Counting all fourth terms, making symmetrization and calculating new \(H_4\) one can get
$$\begin{aligned} H_4&= \frac{1}{2}\int [W_{k_1k_2}^{k_3k_4} + D_{k_1k_2}^{k_3k_4}+ (\omega _{k_1} + \omega _{k_2} -\omega _{k_3} - \omega _{k_4}) (\tilde{W}_{k_1k_2}^{k_3k_4}-i\tilde{\tilde{W}}_{k_1k_2}^{k_3k_4})]b_{k_1}^*b_{k_2}^*b_{k_3}b_{k_4} \delta _{k_1+}\nonumber \\&\qquad {k_2{-}k_3{-}k_4}dk_1dk_2dk_3dk_4 +\\&+\frac{1}{3}\int \left[ (G_{k_1k_2k_3}^{k_4} + D_{k_1k_2k_3}^{k_4} -(\omega _{k_1} + \omega _{k_2} +\omega _{k_3} - \omega _{k_4})\tilde{G}_{k_1k_2k_3}^{k_4}) b_{k_1}^*b_{k_2}^*b_{k_3}^*b_{k_4} + c.c.\right] \delta _{k_1+}\nonumber \\&\qquad {k_2{+}k_3{-}k_4}dk_1dk_2dk_3dk_4 +\\&+\frac{1}{12} \int \left[ (R_{k_1k_2k_3k_4} +\!D_{k_1k_2k_3k_4} -(\omega _{k_1} +\!\omega _{k_2} +\!\omega _{k_3}+ \omega _{k_4}) \tilde{R}_{k_1k_2k_3k_4})b_{k_1}^*b_{k_2}^*b_{k_3}^*b_{k_4}^*+ c.c.\right] \delta _{k_1+}\nonumber \\&\qquad {k_2{+}k_3{+}k_4}dk_1dk_2dk_3dk_4. \end{aligned}$$
(27)
Here
$$\begin{aligned} D_{k_1k_2}^{k_3k_4}= & {} \tilde{V}^{k_1}_{k_3k_1-k_3}\tilde{V}^{k_4}_{k_2k_4-k_2} \left[ \omega _{k_1}-\omega _{k_3}-\omega _{k_1-k_3}+\omega _{k_4}-\omega _{k_2}-\omega _{k_4-k_2}\right] +\\+ & {} \tilde{V}^{k_2}_{k_3k_2-k_3}\tilde{V}^{k_4}_{k_1k_4-k_1} \left[ \omega _{k_2}-\omega _{k_3}-\omega _{k_2-k_3}+\omega _{k_4}-\omega _{k_1}-\omega _{k_4-k_1}\right] +\\+ & {} \tilde{V}^{k_1}_{k_4k_1-k_4}\tilde{V}^{k_3}_{k_2k_3-k_2} \left[ \omega _{k_1}-\omega _{k_4}-\omega _{k_1-k_4}+\omega _{k_3}-\omega _{k_2}-\omega _{k_3-k_2}\right] +\\+ & {} \tilde{V}^{k_2}_{k_4k_2-k_4}\tilde{V}^{k_3}_{k_1k_3-k_1} \left[ \omega _{k_2}-\omega _{k_4}-\omega _{k_2-k_4}+\omega _{k_3}-\omega _{k_1}-\omega _{k_3-k_1}\right] -\\- & {} \tilde{V}^{k_1+k_2}_{k_1k_2}\tilde{V}^{k_3+k_4}_{k_3k_4} \left[ \omega _{k_1+k_2}-\omega _{k_1}-\omega _{k_2}+\omega _{k_3+k_4}-\omega _{k_3}-\omega _{k_4}\right] -\\- & {} \tilde{U}_{-k_1-k_2k_1k_2}\tilde{U}_{-k_3-k_4k_3 k_4} \left[ \omega _{k_1+k_2}+\omega _{k_1}+\omega _{k_2}+\omega _{k_3+k_4}+\omega _{k_3}+\omega _{k_4}\right] ,\qquad \end{aligned}$$
(28)
$$\begin{aligned} D_{k_1k_2k_3}^{k_4}= & {} \tilde{V}^{k_1+k_2}_{k_1k_2}\tilde{V}^{k_4}_{k_3k_4-k_3} (\omega _{k_1+k_2}-\omega _{k_1}-\omega _{k_2} -\omega _{k_4}+\omega _{k_3}+\omega _{k_3-k_4})+\\+ & {} \tilde{V}^{k_1+k_3}_{k_1k_3}\tilde{V}^{k_4}_{k_2k_4-k_2} (\omega _{k_1+k_3}-\omega _{k_1}-\omega _{k_3} -\omega _{k_4}+\omega _{k_2}+\omega _{k_2-k_4})+\\+ & {} \tilde{V}^{k_2+k_3}_{k_2k_3}\tilde{V}^{k_4}_{k_1k_4-k_1} (\omega _{k_2+k_3}-\omega _{k_2}-\omega _{k_3} -\omega _{k_4}+\omega _{k_1}+\omega _{k_1-k_4})+\\+ & {} \tilde{U}_{-k_1-k_2k_1k_2}\tilde{V}^{k_3}_{k_4k_3-k_4} (\omega _{k_1+k_2}+\omega _{k_1}+\omega _{k_2} -\omega _{k_3}+\omega _{k_4}+\omega _{k_3-k_4})+\\+ & {} \tilde{U}_{-k_1-k_3k_1k_3}\tilde{V}^{k_2}_{k_4k_2-k_4} (\omega _{k_1+k_3}+\omega _{k_1}+\omega _{k_3} -\omega _{k_2}+\omega _{k_4}+\omega _{k_2-k_4})+\\+ & {} \tilde{U}_{-k_2-k_3k_2k_3}\tilde{V}^{k_1}_{k_4k_1-k_4} (\omega _{k_2+k_3}+\omega _{k_2}+\omega _{k_3} -\omega _{k_1}+\omega _{k_4}+\omega _{k_1-k_4}),\qquad \end{aligned}$$
(29)
$$\begin{aligned} D_{k_1k_2k_3k_4} =- & {} \tilde{U}_{-k_1-k_2k_1k_2}\tilde{V}^{k_3+k_4}_{k_3k_4} (\omega _{k_1+k_2}+\omega _{k_1}+\omega _{k_2} +\omega _{k_3+k_4}-\omega _{k_3}-\omega _{k_4})-\\- & {} \tilde{U}_{-k_1-k_3k_1k_3}\tilde{V}^{k_2+k_4}_{k_2k_4} (\omega _{k_1+k_3}+\omega _{k_1}+\omega _{k_3} +\omega _{k_2+k_4}-\omega _{k_2}-\omega _{k_4})-\\- & {} \tilde{U}_{-k_1-k_4k_1k_4}\tilde{V}^{k_3+k_2}_{k_3k_2} (\omega _{k_1+k_4}+\omega _{k_1}+\omega _{k_4} +\omega _{k_3+k_2}-\omega _{k_3}-\omega _{k_2})-\\- & {} \tilde{U}_{-k_2-k_3k_2k_3}\tilde{V}^{k_1+k_4}_{k_1k_4} (\omega _{k_2+k_3}+\omega _{k_2}+\omega _{k_3} +\omega _{k_1+k_4}-\omega _{k_1}-\omega _{k_4})-\\- & {} \tilde{U}_{-k_2-k_4k_2k_4}\tilde{V}^{k_1+k_3}_{k_1k_3} (\omega _{k_2+k_4}+\omega _{k_2}+\omega _{k_4} +\omega _{k_1+k_3}-\omega _{k_1}-\omega _{k_3})-\\- & {} \tilde{U}_{-k_3-k_4k_3k_4}\tilde{V}^{k_1+k_2}_{k_1k_2} (\omega _{k_3+k_4}+\omega _{k_3}+\omega _{k_4} +\omega _{k_1+k_2}-\omega _{k_1}-\omega _{k_2}).\qquad \end{aligned}$$
(30)
To cancel nonresonant fourth order terms in (27) relations given below must be valid:
$$\begin{aligned} \tilde{G}_{k_1k_2k_3}^{k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}+\omega _{k_3}-\omega _{k_4}} (G_{k_1k_2k_3}^{k_4}+D_{k_1k_2k_3}^{k_4}),\\\tilde{R}_{k_1k_2k_3k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}+\omega _{k_3}+\omega _{k_4}} (R_{k_1k_2k_3k_4}+D_{k_1k_2k_3k_4}). \end{aligned}$$
(31)
Now the Hamiltonian
has only resonant four-wave interaction term (\(2\Leftrightarrow 2\)):
$$\begin{aligned} H= & {} \int \omega _k|b_k|^2 dk+\\+ & {} \frac{1}{2}\int [W_{k_1k_2}^{k_3k_4} +D_{k_1k_2}^{k_3k_4}+ (\omega _{k_1} + \omega _{k_2} -\omega _{k_3} - \omega _{k_4}) (\tilde{W}_{k_1k_2}^{k_3k_4}\nonumber \\&-i\tilde{\tilde{W}}_{k_1k_2}^{k_3k_4})]b_{k_1}^*b_{k_2}^*b_{k_3}b_{k_4} \delta _{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4 \end{aligned}$$
(32)
If we put
$$\begin{aligned} \tilde{W}_{k_1k_2}^{k_3k_4}-i\tilde{\tilde{W}}_{k_1k_2}^{k_3k_4} = 0, \end{aligned}$$
(33)
we obtain so-called Zakharov equation
with the following Hamiltonian
:
$$\begin{aligned} H= & {} \int \omega _k|b_k|^2 dk+\frac{1}{2}\int T_{k_1k_2}^{k_3k_4}b_{k_1}^*b_{k_2}^*b_{k_3}b_{k_4} \delta _{k_1+k_2-k_3-k_4}dk_1dk_2dk_3dk_4 \\T_{k_1k_2}^{k_3k_4}= & {} W_{k_1k_2}^{k_3k_4} +D_{k_1k_2}^{k_3k_4} \end{aligned}$$
(34)
At this moment the key point of the transformation takes place: we explicitly use property of vanishing of \(T_{k_1k_2}^{k_3k_4}\) on the resonant manifold and consider waves propagating in the same direction. Then we chose instead of (33) the following expression:
$$\begin{aligned} \tilde{W}_{k_1k_2}^{k_3k_4}-i\tilde{\tilde{W}}_{k_1k_2}^{k_3k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}-\omega _{k_3}-\omega _{k_4}}(\tilde{T}_{k_1k_2}^{k_3k_4} - W_{k_1k_2}^{k_3k_4} - D_{k_1k_2}^{k_3k_4}), \end{aligned}$$
(35)
here
$$\begin{aligned} \tilde{T}_{k_2k_3}^{kk_1}= \frac{\theta (k)\theta (k_1)\theta (k_2)\theta (k_3)}{8\pi }\left[ (kk_1(k+k_1) + k_2k_3(k_2+k_3))\right. - \\\left. -(kk_2|k-k_2| + kk_3|k-k_3| + k_1k_2|k_1-k_2| + k_1k_3|k_1-k_3) \right] , \end{aligned}$$
(36)
This coefficient \(\tilde{T}_{k_2k_3}^{kk_1}\) gives us simple Hamiltonian
(11).
Now we can calculate symmetrized coefficients A of the cubic part of the transformation:
$$\begin{aligned}&A^{k_1k_2}_{k_3k_4} = \frac{1}{\omega _{k_1} +\omega _{k_2}-\!\omega _{k_3}-\!\omega _{k_4}} \left[ \tilde{T}_{k_1k_2}^{k_3k_4} - W_{k_1k_2}^{k_3k_4} + 2(U_{-k_1-k_2k_1k_2}\tilde{U}_{-k_3-k_4k_3 k_4} +\! V^{k_1+k_2}_{k_1k_2}\tilde{V}^{k_3+k_4}_{k_3k_4} \right. \nonumber \\&- \left. V^{k_1}_{k_3k_1-k_3}\tilde{V}^{k_4}_{k_2k_4-k_2} - \tilde{V}^{k_2}_{k_3k_2-k_3} V^{k_4}_{k_1k_4-k_1}- V^{k_1}_{k_4k_1-k_4}\tilde{V}^{k_3}_{k_2k_3-k_2} - \tilde{V}^{k_2}_{k_4k_2-k_4}V^{k_3}_{k_1k_3-k_1})\right] \end{aligned}$$
(37)
$$\begin{aligned}&A^{k_1k_2k_3k_4} = \frac{1}{3(\omega _{k_1} +\omega _{k_2}+\omega _{k_3}+\omega _{k_4})}\left[ -R_{k_1k_2k_3k_4}+ 2(U_{-k_1-k_2k_1k_2}\tilde{V}^{k_3+k_4}_{k_3k_4} + U_{-k_1-k_3k_1k_3}\tilde{V}^{k_2+k_4}_{k_2k_4}\right. \nonumber \\&+ \left. U_{-k_1-k_4k_1k_4}\tilde{V}^{k_2+k_3}_{k_2k_3} + \tilde{U}_{-k_2-k_3k_2k_3}V^{k_1+k_4}_{k_1k_4}+ \tilde{U}_{-k_2-k_4k_2k_4}V^{k_1+k_3}_{k_1k_3} + \tilde{U}_{-k_3-k_4k_3k_4}V^{k_1+k_2}_{k_1k_2})\right] \end{aligned}$$
(38)
$$\begin{aligned}&A^{k_1k_2k_3}_{k_4}=\frac{-1}{\omega _{k_1} +\omega _{k_2}+\omega _{k_3}-\omega _{k_4}}\left[ G_{k_1k_2k_3}^{k_4} +2(V^{k_1+k_2}_{k_1k_2}\tilde{V}^{k_4}_{k_3k_4-k_3} + V^{k_1+k_3}_{k_1k_3}\tilde{V}^{k_4}_{k_2k_4-k_2}\right. \nonumber \\ +&\left. U_{-k_1-k_2k_1k_2}\tilde{V}^{k_3}_{k_4k_3-k_4} +\! U_{-k_1-k_3k_1k_3}\tilde{V}^{k_2}_{k_4k_2-k_4} \right. \left. -\!\tilde{V}^{k_2+k_3}_{k_2k_3}V^{k_4}_{k_1k_4-k_1} -\! \tilde{U}_{-k_2-k_3k_2k_3}V^{k_1}_{k_4k_1-k_4})\right] \nonumber \\ \end{aligned}$$
(39)
$$\begin{aligned}&A^{k_1}_{k_2k_3k_4}=\frac{-1}{3(\omega _{k_1} -\omega _{k_2}-\omega _{k_3}-\omega _{k_4})}\left[ G_{k_2k_3k_4}^{k_1} - 2(\tilde{V}^{k_2+k_3}_{k_2k_3} V^{k_1}_{k_4k_1-k_4} + \tilde{V}^{k_2+k_4}_{k_2k_4} V^{k_1}_{k_3k_1-k_3} \right. \nonumber \\&\left. +\tilde{V}^{k_3+k_4}_{k_3k_4} V^{k_1}_{k_2k_1-k_2} +\tilde{U}_{-k_2-k_3k_2k_3} V^{k_4}_{k_1k_4-k_1}+ \tilde{U}_{-k_2-k_4k_2k_4} V^{k_3}_{k_1k_3-k_1}+ \tilde{U}_{-k_3-k_4k_3 k_4} V^{k_2}_{k_1k_2-k_1})\right] .\nonumber \\ \end{aligned}$$
(40)
Below we calculate \(A^{k_1}_{k_2k_3k_4}\), \(A^{k_1k_2k_3k_4}\), \(A^{k_1k_2k_3}_{k_4}\) and \(A_{k_1k_2}^{k_3k_4}\) for the case when canonical variable \(b_k\) has harmonics with positive k only.
Let us start with \(A^{k_1}_{k_2k_3k_4}\), expression (40). According to \(\delta \)-function in (24) \(k_1\) is also positive. One can finally get:
$$\begin{aligned} A^{k_1}_{k_2k_3k_4} = \frac{\omega _{k_1} +\omega _{k_2}+\omega _{k_3}+\omega _{k_4}}{48\pi g}k_1(k_1k_2k_3k_4)^{\frac{1}{4}}. \end{aligned}$$
(41)
Coefficient \(A^{k_1k_2k_3k_4}\) has to be calculated for negative \(k_1\) (according to \(\delta \)-function in (24), so we will calculate it as \(A^{-k_1k_2k_3k_4}\).
$$\begin{aligned} A^{-k_1k_2k_3k_4} = \frac{\omega _{k_1} -\omega _{k_2}-\omega _{k_3}-\omega _{k_4}}{48\pi g}k_1(k_1k_2k_3k_4)^{\frac{1}{4}}. \end{aligned}$$
(42)
Coefficient \(A^{k_1k_2k_3}_{k_4}\) has to be calculated both for positive and negative \(k_1\). For \(k_i>0\) the following is valid:
$$\begin{aligned} A^{k_1k_2k_3}_{k_4} = \frac{\omega _{k_1} +\omega _{k_2}+\omega _{k_3}+\omega _{k_4}}{16\pi g}k_1(k_1k_2k_3k_4)^{\frac{1}{4}}. \end{aligned}$$
(43)
For \(k_1<0\) we will calculate it as \(A^{-k_1k_2k_3}_{k_4}\). Let us start with the case \(k_4>k_2,k_3>k_1\):
$$\begin{aligned} A^{-k_1k_2k_3}_{k_4}= & {} \frac{\omega _{k_4} +\omega _{k_3}+\omega _{k_2}-\omega _{k_1}}{16 \pi g}(k_1k_2k_3k_4)^{\frac{1}{4}}k_1\frac{3 \sqrt{k_1k_4}-\sqrt{k_2k_3} }{\sqrt{k_1k_4}+\sqrt{k_2k_3}} \end{aligned}$$
(44)
In the case \(k_2>k_1,k_4>k_3\):
$$\begin{aligned} A^{-k_1k_2k_3}_{k_4} = \frac{\omega _{k_4}+\omega _{k_3}+\omega _{k_2}-\omega _{k_1}}{16 \pi g}(k_1k_2k_3k_4)^{\frac{1}{4}}k_1\frac{\sqrt{k_1k_4}(2k_3+k_1)-\sqrt{k_2k_3}(2k_3-k_1)}{\sqrt{k_1k_4}+\sqrt{k_2k_3}}\nonumber \\ \end{aligned}$$
(45)
In the case \(k_3>k_1,k_4>k_2\):
$$\begin{aligned} A^{-k_1k_2k_3}_{k_4}=\frac{\omega _{k_4}+\!\omega _{k_3}+\!\omega _{k_2}-\!\omega _{k_1}}{16 \pi g}(k_1k_2k_3k_4)^{\frac{1}{4}}k_1\frac{\sqrt{k_1k_4}(2k_2+k_1)-\!\sqrt{k_2k_3}(2k_2-\!k_1)}{\sqrt{k_1k_4}+\sqrt{k_2k_3}}\nonumber \\ \end{aligned}$$
(46)
In the case \(k_1>k_2,k_3>k_4\):
$$\begin{aligned} A^{-k_1k_2k_3}_{k_4}=\!\frac{\omega _{k_4}+\!\omega _{k_3}+\!\omega _{k_2}-\!\omega _{k_1}}{16 \pi g}(k_1k_2k_3k_4)^{\frac{1}{4}}k_1\frac{\sqrt{k_1k_4}(2k_4 + k_1)-\sqrt{k_2k_3}(2k_4 -k_1)}{\sqrt{k_1k_4}+\sqrt{k_2k_3}}\nonumber \\ \end{aligned}$$
(47)
Coefficient \(A^{k_1k_2}_{k_3k_4}\) has to be calculated both for positive and negative \(k_1\). Below we calculate \(A^{k_1k_2}_{k_3k_4}\) for the case \(k_1,k_2,k_3,k_4>0\). Let us start with the case \(k_2>k_3,k_4>k_1\):
$$\begin{aligned} A^{k_1k_2}_{k_3k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}-\omega _{k_3}-\omega _{k_4}}\left[ \tilde{T}_{k_1k_2}^{k_3k_4}-\!\frac{(k_1k_2k_3k_4)^{\frac{1}{4}}}{8\pi }k_1\left( 3\sqrt{k_1k_2}+\!\sqrt{k_3k_4} \right) \right] \nonumber \\ \end{aligned}$$
(48)
In the case \(k_1>k_3,k_4>k_2\):
$$\begin{aligned} A^{k_1k_2}_{k_3k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}-\omega _{k_3}-\omega _{k_4}}\times \\\times & {} \left[ \tilde{T}_{k_1k_2}^{k_3k_4}-\frac{(k_1k_2k_3k_4)^{\frac{1}{4}}}{8\pi }\left( \sqrt{k_1k_2}(2k_2+k_1)+\sqrt{k_3k_4}(2k_2-k_1) \right) \right] \quad \end{aligned}$$
(49)
In the case \(k_4>k_1,k_2>k_3\):
$$\begin{aligned} A^{k_1k_2}_{k_3k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}-\omega _{k_3}-\omega _{k_4}}\times \\\times & {} \left[ \tilde{T}_{k_1k_2}^{k_3k_4}-\frac{(k_1k_2k_3k_4)^{\frac{1}{4}}}{8\pi }\left( \sqrt{k_1k_2}(2k_3+k_1)+\sqrt{k_3k_4}(2k_3-k_1) \right) \right] \quad \end{aligned}$$
(50)
In the case \(k_3>k_1,k_2>k_4\):
$$\begin{aligned} A^{k_1k_2}_{k_3k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}-\omega _{k_3}-\omega _{k_4}}\times \\\times & {} \left[ \tilde{T}_{k_1k_2}^{k_3k_4}-\frac{(k_1k_2k_3k_4)^{\frac{1}{4}}}{8\pi }\left( \sqrt{k_1k_2}(2k_4+k_1)+\sqrt{k_3k_4}(2k_4-k_1) \right) \right] \quad \end{aligned}$$
(51)
For \(k_1<0\) we will calculate it as \(A^{-k_1k_2}_{k_3k_4}\) and \(k_1,k_2,k_3,k_4>0\):
$$\begin{aligned} A^{-k_1k_2}_{k_3k_4}= & {} \frac{1}{\omega _{k_1} +\omega _{k_2}-\omega _{k_3}-\omega _{k_4}}\left[ \frac{(k_1k_2k_3k_4)^{\frac{1}{4}}}{8\pi }k_1\left( \sqrt{k_1k_4}+\sqrt{k_1k_3}-\sqrt{k_3k_4} \right) \right] = \\= & {} \frac{(k_1k_2k_3k_4)^{\frac{1}{4}}}{16\pi g}k_1\left( \omega _{k_2}+\omega _{k_3}+\omega _{k_4} - \omega _{k_1}\right) \end{aligned}$$
(52)
It appears that if spectrum of b(x) consists of harmonics with positive k only, transformation from \(b_k\) to \(\eta _k\) and \(\psi _k\) can be considerably simplified. To prove that, let us calculate \(\eta _k\) and \(\psi _k\) for positive k using transformations (23) and (7). To recover \(\eta _k\) and \(\psi _k\) for negative k one can use the following relations:
$$\begin{aligned} \eta _{-k} = \eta _k^*, \qquad \psi _{-k} = \psi _k^*. \end{aligned}$$
(53)
But first let us write \(\eta _k\) and \(\psi _k\) as a power series of \(b_k\) up to the third order:
$$\begin{aligned} \eta _k = \eta _k^{(1)} + \eta _k^{(2)} + \eta _k^{(3)}, \qquad \psi _k = \psi _k^{(1)} + \psi _k^ {(2)} + \psi _k^{(3)}. \end{aligned}$$
(54)
Obviously
$$\begin{aligned} \eta _k^{(1)} = \sqrt{\frac{\omega _k}{2g}}[b_k+b_{-k}^*], \qquad \psi _k^{(1)} = -i\sqrt{\frac{g}{2\omega _k}}[b_k-b_{-k}^*]. \end{aligned}$$
(55)
Or
$$\begin{aligned} \eta ^{(1)}(x) = \frac{1}{\sqrt{2}g^{\frac{1}{4}}}(\hat{k}^{\frac{1}{4}}b(x)+\hat{k}^{\frac{1}{4}}b(x)^*), \quad \qquad \psi ^{(1)}(x) = -i\frac{g^{\frac{1}{4}}}{\sqrt{2}}(\hat{k}^{-\frac{1}{4}}b(x)-\hat{k}^{-\frac{1}{4}}b(x)^*). \end{aligned}$$
(56)
Operators \(\hat{k}^{\alpha }\) act in Fourier space as multiplication by \(|k|^\alpha \).
Quadratic terms in (54) are the following:
$$\begin{aligned} \eta _k^{(2)}= & {} \sqrt{\frac{\omega _k}{2g}}\left[ 2\int (\tilde{V}^{k_2}_{kk_1} +\tilde{V} ^{k_1}_{-kk_2})b_{k_1}^*b_{k_2}\delta _{k+k_1-k_2}dk_1dk_2 \right. - \\- & {} \left. \int (\tilde{V}^{k}_{k_1k_2} + \tilde{U}_{-kk_1k_2})b_{k_1}b_{k_2}\delta _{k-k_1-k_2}dk_1dk_2\right] ,\\\psi _k^{(2)}= & {} -i\sqrt{\frac{g}{2\omega _k}}\left[ 2\int (\tilde{V}^{k_2}_{kk_1}-\!\tilde{V}^{k_1}_{-kk_2})b_{k_1}^*b_{k_2}\delta _{k+k_1-k_2}dk_1dk_2- \right. \\- & {} \left. \int (\tilde{V}^{k}_{k_1k_2}-\!\tilde{U}_{-kk_1k_2})b_{k_1}b_{k_2}\delta _{k-k_1-k_2}dk_1dk_2\right] . \end{aligned}$$
(57)
All coefficients in (57) can be easily calculated using expressions (18), (26), properties (53) and little algebra. The following formulae are valid for both positive and negative k:
$$\begin{aligned} \eta _k^{(2)}= & {} \frac{|k|}{4\sqrt{2g\pi }} \left[ \int k_1^{\frac{1}{4}}b_{k_1}k_2^{\frac{1}{4}}b_{k_2}\delta _{k-k_1-k_2}dk_1dk_2 + \int k_1^{\frac{1}{4}}b_{k_1}^*k_2^{\frac{1}{4}}b_{k_2}^*\delta _{k+k_1+k_2}dk_1dk_2 \right. \\- & {} \left. 2\int k_1^{\frac{1}{4}}b_{k_1}^*k_2^{\frac{1}{4}}b_{k_2}\delta _{k+k_1-k_2}dk_1dk_2 \right] , \\\psi _k^{(2)}= & {} -\frac{i}{4\sqrt{2\pi }} \left[ \int (\sqrt{k_1}+\sqrt{k_2})k_1^{\frac{1}{4}}b_{k_1}k_2^{\frac{1}{4}}b_{k_2}\delta _{k-k_1-k_2}dk_1dk_2 - \right. \\- & {} \left. \int (\sqrt{k_1}+\sqrt{k_2})k_1^{\frac{1}{4}}b_{k_1}^*k_2^{\frac{1}{4}}b_{k_2}^*\delta _{k+k_1+k_2}dk_1dk_2 -\right. \\&\left. -2\mathbf{{sign}}(k)\int (\sqrt{k_1}+\sqrt{k_2})k_1^{\frac{1}{4}}b_{k_1}^*k_2^{\frac{1}{4}}b_{k_2}\delta _{k+k_1-k_2}dk_1dk_2 \right] . \end{aligned}$$
(58)
Applying Fourier transformation to (58) one can get
$$\begin{aligned} \eta ^{(2)}(x)= & {} \frac{\hat{k}}{4\sqrt{g}}[\hat{k}^{\frac{1}{4}}b(x) - \hat{k}^{\frac{1}{4}}b^*(x)]^2,\\\psi ^{(2)}(x)= & {} \frac{i}{2}[\hat{k}^{\frac{1}{4}}b^*(x)\hat{k}^{\frac{3}{4}}b^*(x) - \hat{k}^{\frac{1}{4}}b(x)\hat{k}^{\frac{3}{4}}b(x)]+ \\+ & {} \frac{1}{2}\hat{H}[\hat{k}^{\frac{1}{4}}b(x)\hat{k}^{\frac{3}{4}}b^*(x) + \hat{k}^{\frac{1}{4}}b^*(x)\hat{k}^{\frac{3}{4}}b(x)]. \end{aligned}$$
(59)
Here \(\hat{H}\)—is Hilbert transformation
with eigenvalue \(i\mathbf{{sign}}(k)\).
Cubic terms in (54) are the following (k, \(k_1\), \(k_2\) and \(k_3\) are positive):
$$\begin{aligned} \eta _k^{(3)}= & {} \sqrt{\frac{\omega _k}{2g}}\left[ \int (A^{k}_{k_1k_2k_3} + A_{-kk_1k_2k_3})b_{k_1}b_{k_2}b_{k_3}\delta _{k-k_1-k_2-k_3}dk_1dk_2dk_3 \right. +\\+ & {} \int (A^{kk_1}_{k_2k_3} + A^{-kk_2k_3}_{k_1})b_{k_1}^*b_{k_2}b_{k_3}\delta _{k+k_1-k_2-k_3}dk_1dk_2dk_3 +\\+ & {} \left. \int (A^{kk_1k_2}_{k_3} + A^{-kk_3}_{k_1k_2})b_{k_1}^*b_{k_2}^*b_{k_3}\delta _{k+k_1+k_2-k_3}dk_1dk_2dk_3 \right] ,\\\psi _k^{(3)}= & {} -i\sqrt{\frac{g}{2\omega _k}}\left[ \int (A^{k}_{k_1k_2k_3} - A_{-kk_1k_2k_3})b_{k_1}b_{k_2}b_{k_3}\delta _{k-k_1-k_2-k_3}dk_1dk_2dk_3 \right. +\\+ & {} \int (A^{kk_1}_{k_2k_3} - A^{-kk_2k_3}_{k_1})b_{k_1}^*b_{k_2}b_{k_3}\delta _{k+k_1-k_2-k_3}dk_1dk_2dk_3 +\\+ & {} \left. \int (A^{kk_1k_2}_{k_3} - A^{-kk_3}_{k_1k_2})b_{k_1}^*b_{k_2}^*b_{k_3}\delta _{k+k_1+k_2-k_3}dk_1dk_2dk_3 \right] \end{aligned}$$
(60)
Some of coefficients in (60) can be easily calculated using expressions for A and little algebra :
$$\begin{aligned} A^{k}_{k_1k_2k_3} + A_{-kk_1k_2k_3}= & {} \frac{\omega _{k}}{24\pi g}k(kk_1k_2k_3)^{\frac{1}{4}}\\A^{k}_{k_1k_2k_3} - A_{-kk_1k_2k_3}= & {} \frac{\omega _{k_1}+\omega _{k_2}+\omega _{k_3}}{24\pi g}k(kk_1k_2k_3)^{\frac{1}{4}} \end{aligned}$$
(61)
$$\begin{aligned} A^{kk_1k_2}_{k_3} + A^{-kk_3}_{k_1k_2}= & {} \frac{\omega _{k_1}+\omega _{k_2}+\omega _{k_3}}{8\pi g}k(kk_1k_2k_3)^{\frac{1}{4}}\\A^{kk_1k_2}_{k_3} - A^{-kk_3}_{k_1k_2}= & {} \frac{\omega _{k}}{8\pi g}k(kk_1k_2k_3)^{\frac{1}{4}} \end{aligned}$$
(62)
For \(k,k_1,k_2,k_3>0\)
$$\begin{aligned}&A^{kk_1}_{k_2k_3} + A^{-kk_2k_3}_{k_1} = \frac{\tilde{T}_{kk_1}^{k_2k_3}}{\omega _{k} +\omega _{k_1}-\omega _{k_2}-\omega _{k_3}} - \frac{\omega _{k}}{8\pi g}(kk_1k_2k_3)^{\frac{1}{4}}k-\\&-\frac{(kk_1k_2k_3)^{\frac{1}{4}}}{8\pi g}min(k,k_1,k_2,k_3)\times \\&\times \left[ \frac{\sqrt{kk_1}+\!\sqrt{k_2k_3}}{\sqrt{kk_1}-\sqrt{k_2k_3}}\left( \omega _{k}+\!\omega _{k_1}+\!\omega _{k_2}+\!\omega _{k_3}\right) +\! \frac{\sqrt{kk_1}-\!\sqrt{k_2k_3}}{\sqrt{kk_1}+\!\sqrt{k_2k_3}}\left( \omega _{k} -\omega _{k_1}-\omega _{k_2}-\!\omega _{k_3}\right) \right] \\&A^{kk_1}_{k_2k_3} - A^{-kk_2k_3}_{k_1} = \frac{\tilde{T}_{kk_1}^{k_2k_3}}{\omega _{k} +\omega _{k_1}-\omega _{k_2}-\omega _{k_3}} - \frac{\omega _{k_1}+\omega _{k_2}+\omega _{k_3}}{8\pi g}(kk_1k_2k_3)^{\frac{1}{4}}k-\\&-\frac{(kk_1k_2k_3)^{\frac{1}{4}}}{8\pi g}min(k,k_1,k_2,k_3)\times \\&\times \left[ \frac{\sqrt{kk_1}+\sqrt{k_2k_3}}{\sqrt{kk_1}-\!\sqrt{k_2k_3}}\left( \omega _{k}+\!\omega _{k_1}+\omega _{k_2}+\omega _{k_3}\right) - \frac{\sqrt{kk_1}-\!\sqrt{k_2k_3}}{\sqrt{kk_1}+\sqrt{k_2k_3}}\left( \omega _{k} -\omega _{k_1}-\!\omega _{k_2}-\!\omega _{k_3}\right) \right] \nonumber \\ \end{aligned}$$
(63)
Using properties (53) expressions for \(\eta _k^{(3)}\) and \(\psi _k^{(3)}\) can be extended for negative k, so that the following formulae are valid for both positive and negative k:
$$\begin{aligned}&\eta _k^{(3)} = \frac{k^2}{24\pi g^{\frac{3}{4}} \sqrt{2}}\int k_1^{\frac{1}{4}}b_{k_1} k_2^{\frac{1}{4}}b_{k_2} k_3^{\frac{1}{4}}b_{k_3}\delta _{k-k_1-k_2-k_3}dk_1dk_2dk_3 +\\&+\frac{k^2}{24\pi g^{\frac{3}{4}} \sqrt{2}}\int k_1^{\frac{1}{4}}b_{k_1}^* k_2^{\frac{1}{4}}b_{k_2}^* k_3^{\frac{1}{4}}b_{k_3}^*\delta _{k+k_1+k_2+k_3}dk_1dk_2dk_3+\\&+\int \left[ \sqrt{\frac{\omega _k}{2g}}(A^{kk_1}_{k_2k_3}+\!A^{-kk_2k_3}_{k_1})+ \frac{k^{\frac{3}{2}}(k_1k_2k_3)^{\frac{1}{4}}\left( k_1^{\frac{1}{2}}+k_2^{\frac{1}{2}}+k_3^{\frac{1}{2}}\right) }{8\pi g^{\frac{3}{4}}\sqrt{2}}\right] \times \\&\times \; b_{k_1}^*b_{k_2}b_{k_3}\delta _{k+\!k_1-\!k_2-\!k_3}dk_1dk_2dk_3+\\&+\int \left[ \sqrt{\frac{\omega _k}{2g}}(A^{-kk_3}_{k_2k_1}+\!A^{kk_2k_1}_{k_3})+ \frac{k^{\frac{3}{2}}(k_1k_2k_3)^{\frac{1}{4}}\left( k_1^{\frac{1}{2}}+k_2^{\frac{1}{2}}+k_3^{\frac{1}{2}}\right) }{8\pi g^{\frac{3}{4}}\sqrt{2}}\right] \times \\&\times \; b_{k_1}^*b_{k_2}^*b_{k_3}\delta _{k+k_1+k_2-\!k_3}dk_1dk_2dk_3 \end{aligned}$$
(64)
$$\begin{aligned} \psi _k^{(3)}&=-i\frac{|k|}{24\pi g^{\frac{1}{4}}\sqrt{2}}\int \left( k_1^{\frac{3}{4}}k_2^{\frac{1}{4}}k_3^{\frac{1}{4}}+\!k_1^{\frac{1}{4}}k_2^{\frac{3}{4}}k_3^{\frac{1}{4}}+\!k_1^{\frac{1}{4}}k_2^{\frac{1}{4}}k_3^{\frac{3}{4}}\right) b_{k_1}b_{k_2}b_{k_3}\delta _{k-k_1-k_2-k_3}dk_1dk_2dk_3 +\\&+i\frac{|k|}{24\pi g^{\frac{1}{4}}\sqrt{2}}\int \left( k_1^{\frac{3}{4}}k_2^{\frac{1}{4}}k_3^{\frac{1}{4}}+k_1^{\frac{1}{4}}k_2^{\frac{3}{4}}k_3^{\frac{1}{4}}+k_1^{\frac{1}{4}}k_2^{\frac{1}{4}}k_3^{\frac{3}{4}}\right) b_{k_1}^*b_{k_2}^*b_{k_3}^*\delta _{k+k_1+k_2+k_3}dk_1dk_2dk_3+\\&+\int \left[ -i\sqrt{\frac{g}{2\omega _k}}(A^{kk_1}_{k_2k_3} - A^{-kk_2k_3}_{k_1})+i\frac{k^{\frac{3}{2}}\left( k_1k_2k_3\right) ^{\frac{1}{4}}}{8\pi g^{\frac{1}{4}}\sqrt{2}}\right] b_{k_1}^*b_{k_2}b_{k_3}\delta _{k+k_1-k_2-k_3}dk_1dk_2dk_3+\\&+\int \left[ i\sqrt{\frac{g}{2\omega _k}}(A^{-kk_3}_{k_2k_1} - A^{kk_2k_1}_{k_3})-i\frac{k^{\frac{3}{2}}\left( k_1k_2k_3\right) ^{\frac{1}{4}}}{8\pi g^{\frac{1}{4}}\sqrt{2}}\right] b_{k_1}^*b_{k_2}^*b_{k_3}\delta _{k+k_1+k_2-k_3}dk_1dk_2dk_3 \end{aligned}$$
(65)
