%#! platex master_thesis-M2.tex
\chapter{$BGH?t6u4V$K$*$1$kJ}Dx<07O(B} \label{chap:wave-eq}

\section{$BGH?t6u4V$K$*$1$kJ}Dx<07O$NF3=P(B}

$B=g05%b!<%I$N;YG[J}Dx<0(B
\begin{align} \tag{\ref{eq:troeq}}
 &\begin{aligned}
   &\DP{\bnabla^2\psi}{t}+J(\psi,\bnabla^2\psi)+J(\tau,\bnabla^2\tau)+U\DP{\bnabla^2\tau}{x}
   \\ &\quad
   =-\kappa\bnabla^2\frac{\psi-\tau}{2}-\nu\bnabla^8(\bnabla^2\psi)
   -\frac{1}{2}\Delta\nu\bnabla^8\{\bnabla^2\psi
   - (\bnabla^2-k_d^2)\tau\}
  \end{aligned}
\end{align}
$B$rGH?t6u4V$K$*$1$k<0$KD>$9$?$a$K(B, $B=g05%b!<%I$NN.@~H!?t$H7905%b!<%I$NN.@~(B
$BH!?t$r$=$l$>$l(B
\begin{align}
 \psi=\sum_{\bvec{k}}\hat{\psi}(\bvec{k},t)\ee^{i\bvec{k}\cdot\bvec{r}} \\
 \tau=\sum_{\bvec{k}}\hat{\tau}(\bvec{k},t)\ee^{i\bvec{k}\cdot\bvec{r}}
\end{align}
$B$H(B Fourier $B5i?tE83+$9$k(B. $B$3$3$G(B, $\bvec{k}=(k,l), \bvec{r}=(x,y)$ $B$G$"$k(B.
$B0J2<(B,
\[
 \hat{\psi}(\bvec{k},t)\equiv\hat{\psi}_{\bvec{k}}, \, \hat{\tau}(\bvec{k},t)\equiv\hat{\tau}_{\bvec{k}}
\]
$B$H=q$/(B. $B$3$l$i$rBeF~$9$k(B;
\begin{align*}
 &\DP{\bnabla^2\psi}{t}=-\DP{}{t}\sum_{\bvec{k}}|\bvec{k}|^2 \hat{\psi}_{\bvec{k}}\ee^{i\bvec{k}\cdot\bvec{r}}, \\
 &U\DP{\bnabla^2\tau}{x}=-U\sum_{\bvec{k}}ik|\bvec{k}|^2 \hat{\tau}_{\bvec{k}}\ee^{i\bvec{k}\cdot\bvec{r}}, \\
 &\kappa\frac{\bnabla^2(\psi-\tau)}{2}=-\frac{\kappa}{2}\sum_{\bvec{k}}|\bvec{k}|^2 (\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}})\ee^{i\bvec{k}\cdot\bvec{r}}, \\
 &\nu\bnabla^8(\bnabla^2\psi) =
 -\nu\sum_{\bvec{k}}(|\bvec{k}|^2)^{5}\hat{\psi}_{\bvec{k}}, \\
 &\frac{1}{2}\Delta\nu\bnabla^8\{\bnabla^2\psi - (\bnabla^2-k_d^2)\tau\}
 =
 -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{|\bvec{k}|^2\hat{\psi}_{\bvec{k}}-(|\bvec{k}|^2+k_d^2)\hat{\tau}_{\bvec{k}}\}, \\ 
 &J(\psi,\bnabla^2\psi)=\DP{\psi}{x}\DP{\bnabla^2\psi}{y}-\DP{\psi}{y}\DP{\bnabla^2\psi}{x} \\
 =& \left(\sum_{\bvec{k}}ik\hat{\psi}_{\bvec{k}}\ee^{i\bvec{k}\cdot\bvec{r}}\right)\left(\sum_{\bvec{l}}-in|\bvec{l}|^2 \hat{\psi}_{\bvec{l}}\ee^{i\bvec{l}\cdot\bvec{r}}\right)
 \\ &\quad -
 \left(\sum_{\bvec{k}}il\hat{\psi}_{\bvec{k}}\ee^{i\bvec{k}\cdot\bvec{r}}\right)\left(\sum_{\bvec{l}}-im|\bvec{l}|^2 \hat{\psi}_{\bvec{l}}\ee^{i\bvec{l}\cdot\bvec{r}}\right) \\
 =&
 \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(kn-ml)|\bvec{l}|^2 \hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}} \\
 &\quad +
 \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(ml-nk)|\bvec{k}|^2 \hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}} \\
 =&
 \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(\bvec{k}\times\bvec{m})_{z}\{|\bvec{l}|^2 -|\bvec{k}|^2 \}\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}, \\
 &J(\tau,\bnabla^2\tau)=\sum_{\bvec{k},\bvec{l}}\frac{1}{2}(\bvec{k}\times\bvec{m})_{z}\{|\bvec{l}|^2 -|\bvec{k}|^2 \}\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}. 
\end{align*}
$B$3$3$G(B, $\bvec{l}=(m,n)$ $B$H$7$?(B. $B$3$l$i$K(B $\ee^{-i\bvec{m}\cdot\bvec{r}}\,($B$?$@(B
$B$7(B, \bvec{m}=(k_x ,k_y ))$ $B$r3](B
$B$1(B, $[0,L_x]\times[0,L_y]$$B$NNN0h$G@QJ,$9$k$H(B,
\begin{align*}
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd y\,
 \ee^{-i\bvec{m}\cdot\bvec{r}}\DP{\bnabla^2\psi}{t} \\=&
 -\DP{}{t}\left\{L_xL_y\sum_{\bvec{k}}|\bvec{k}|^2
 \hat{\psi}_{\bvec{k}}\delta_{\bvec{k},\bvec{m}}\right\} \\ 
 =&-\DP{}{t}\left\{L_xL_y|\bvec{m}|^2 \hat{\psi}_{\bvec{m}}\right\}, \\
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd
 y\,\ee^{-i\bvec{m}\cdot\bvec{r}} U\DP{\bnabla^2\tau}{x} \\ =&
 -UL_xL_y\sum_{\bvec{k}}ik|\bvec{k}|^2
 \hat{\tau}_{\bvec{k}}\delta_{\bvec{k},\bvec{m}} \\ 
 =&-UL_xL_yik_x |\bvec{m}|^2 \hat{\tau}_{\bvec{m}}, \\
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd y\,\ee^{-i\bvec{m}\cdot\bvec{r}}
 \kappa\frac{\bnabla^2(\psi-\tau)}{2} \\
 =&-\frac{\kappa}{2}L_xL_y\sum_{\bvec{k}}|\bvec{k}|^2
 (\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}})\delta_{\bvec{k},\bvec{m}}
 \\ 
 =&-\frac{\kappa}{2}L_xL_y|\bvec{m}|^2 (\hat{\psi}_{\bvec{m}}-\hat{\tau}_{\bvec{m}}),  \\
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd
 y\,\ee^{-i\bvec{m}\cdot\bvec{r}}\nu\bnabla^8(\bnabla^2\psi) \\ =&-\nu
 L_xL_y\sum_{\bvec{k}}(|\bvec{k}|^2)^{5}\hat{\psi}_{\bvec{k}}\delta_{\bvec{k},\bvec{m}}\\ 
 =&-\nu L_xL_y(|\bvec{m}|^2)^{5}\hat{\psi}_{\bvec{m}}, \\
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd
 y\,\ee^{-i\bvec{m}\cdot\bvec{r}}\frac{1}{2}\Delta\nu\bnabla^8\{\bnabla^2\psi
   - (\bnabla^2-k_d^2)\tau\} \\ =&-\frac{1}{2}\Delta\nu 
 L_xL_y\sum_{\bvec{k}}|\bvec{k}|^8\{|\bvec{k}|^2\hat{\psi}_{\bvec{k}}-(|\bvec{k}|^2+k_d^2)\hat{\tau}_{\bvec{k}}\}\delta_{\bvec{k},\bvec{m}}\\ 
 =&-\frac{1}{2}\Delta\nu L_xL_y|\bvec{m}|^8\{|\bvec{m}|^2\hat{\psi}_{\bvec{m}}-(\bvec{m}^2-k_d^2)\hat{\tau}_{\bvec{m}}\}.
\end{align*}
$BHs@~7?9`$K4X$7$F$O(B, $B:G8e$K(B, $\bvec{k}\rightarrow -\bvec{k},\bvec{l}\rightarrow-\bvec{l}$ $B$H$$$&CV$-49$($r$9$k(B
$B$H(B, 
\begin{align*}
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd
 y\,\ee^{-i\bvec{m}\cdot\bvec{r}} J(\psi,\bnabla^2\psi) \\
 =&
 \frac{1}{2}L_xL_y\sum_{\bvec{k},\bvec{l}}(\bvec{k\times\bvec{l}})_z\{|\bvec{l}|^2 -|\bvec{k}|^2 \}\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\delta_{(\bvec{k}+\bvec{l}),\bvec{m}}\\
 =&
 \frac{1}{2}L_xL_y\sum_{\bvec{k},\bvec{l}}(\bvec{k}\times\bvec{l})_{z}(|\bvec{l}|^2-|\bvec{k}|^2)\hat{\psi}^{\ast}_{\bvec{k}}\hat{\psi}^{\ast}_{\bvec{l}}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}, \\
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd
 y\,\ee^{-i\bvec{m}\cdot\bvec{r}} J(\tau,\bnabla^2\tau) \\
 =& \frac{1}{2}L_xL_y\sum_{\bvec{k},\bvec{l}}(\bvec{k}\times\bvec{l})_{z}(|\bvec{l}|^2-|\bvec{k}|^2)\hat{\tau}^{\ast}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{l}}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
\end{align*}
$B$H=q$1$k(B. $B7k6I(B, (\ref{eq:troeq}) $B$NGH?t6u4V$K$*$1$kI=8=$O(B
($\bvec{m}\rightarrow\bvec{k},\bvec{m}\rightarrow\bvec{l},\bvec{k}\rightarrow\bvec{m}$
$B$HCV$-49$($k$H(B), 
\begin{align}
 &\DP{}{t}|\bvec{k}|^2\hat{\psi}_{\bvec{k}} \nonumber\\
 =&
 \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z(|\bvec{m}|^2-|\bvec{l}|^2)\{\hat{\psi}^{\ast}_{\bvec{l}}\hat{\psi}^{\ast}_{\bvec{m}}-\hat{\tau}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
 \nonumber \\ &\quad -ik_x U|\bvec{k}|^2\hat{\tau}_{\bvec{k}}
 -\frac{\kappa}{2}|\bvec{k}|^2\{\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}}\}
 \nonumber  
 \\ &\qquad
 -\nu|\bvec{k}|^{10}\hat{\psi}_{\bvec{k}}-\frac{1}{2}\Delta\nu|\bvec{k}|^8\{|\bvec{k}|^2\hat{\psi}_{\bvec{k}}-(|\bvec{k}|^2+k_d^2)\hat{\tau}_{\bvec{k}}\} 
 \label{eq:barotro4}
\end{align}
$B$H$J$k(B.

$BF1MM$K(B, $B7905%b!<%I$N;YG[J}Dx<0(B
\begin{align} \tag{eq:clieq}
 \begin{aligned}
  &\DP{(\bnabla^2\tau-k_d^2\tau)}{t}+J(\psi,\bnabla^2\tau-k_d^2\tau)+J(\tau,\bnabla^2\psi)+U\DP{\bnabla^2\psi}{x}+k_d^2U\DP{\psi}{x}
  \\ &\quad
  =-\kappa\bnabla^2\frac{\tau-\psi}{2}-\nu\bnabla^8(\bnabla^2\tau-k_d^2\tau)
  -\frac{1}{2}\Delta\nu\bnabla^8\{(\bnabla^2-k_d^2)\tau - \bnabla^2\psi \}
 \end{aligned}  
\end{align}
$B$rGH?t6u4V$K$*$1$k<0$KD>$9(B: $B;~4VJQ2=9`$H@~7?9`$K4X$7$F$O@hDx$HF1MM$J$N$G(B, $BHs@~(B
$B7?9`$N$_7W;;2aDx$r5-$9(B. $B$^$:(B, Fourier $B5i?tE83+$9$k$H(B,
\begin{align*}
 J(\psi,\bnabla^2\tau-k_d^2\tau)=&\DP{\psi}{x}\DP{(\bnabla^2\tau-k_d^2\tau)}{y}-\DP{\psi}{y}\DP{(\bnabla^2\tau-k_d^2\tau)}{x} \\
 =&\left(\sum_{\bvec{k}}ik\hat{\psi}_{\bvec{k}}\ee^{i\bvec{k}\cdot\bvec{r}}\right)\left(\sum_{\bvec{l}}-in\{|\bvec{l}|^2-k_d^2\}\hat{\tau}_{\bvec{l}}\ee^{i\bvec{l}\cdot\bvec{r}}\right)
 \\ &\quad -
 \left(\sum_{\bvec{k}}il\hat{\psi}_{\bvec{k}}\ee^{i\bvec{k}\cdot\bvec{r}}\right)\left(\sum_{\bvec{l}}-im\{|\bvec{l}|^2 -k_d^2\}\hat{\tau}_{\bvec{l}}\ee^{i\bvec{l}\cdot\bvec{r}}\right) \\
 =&
 \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(kn-lm)\{|\bvec{l}|^2 +k_d^2\}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}} \\
 &\quad +
 \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(ml-nk)\{|\bvec{k}|^2 +k_d^2\}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}} \\
 =& \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(kn-lm)\{|\bvec{l}|^2 \hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-|\bvec{k}|^2 \hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{k}}\}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}} \\
 &\quad
 +\sum_{\bvec{k},\bvec{l}}\frac{1}{2}(kn-lm)k_d^2\{\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{k}}\}
 \ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}\\
 =&\sum_{\bvec{k},\bvec{l}}\frac{1}{2}(\bvec{k}\times\bvec{l})_z\{|\bvec{l}|^2\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-|\bvec{k}|^2\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{k}}\}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}
 \\
 &\quad +\sum_{\bvec{k},\bvec{l}}\frac{1}{2}(\bvec{k}\times\bvec{l})_z k_d^2\{\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{k}}\}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}} \\
 J(\tau,\bnabla^2\psi)=&\sum_{\bvec{k},\bvec{l}}\frac{1}{2}(kn-lm)|\bvec{l}|^2 \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}
 \\ &\quad +
 \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(ml-nk)|\bvec{k}|^2 \hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}, \\
 =&
 \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(kn-lm)\{|\bvec{l}|^2 \hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{k}}-|\bvec{k}|^2 \hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}, \\
 =& \sum_{\bvec{k},\bvec{l}}\frac{1}{2}(\bvec{k}\times\bvec{l})_z\{|\bvec{l}|^2\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{k}}-|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\}\ee^{i(\bvec{k}+\bvec{l})\cdot\bvec{r}}
\end{align*}
$B$H$J$k(B. $B$3$l$i$K(B $\ee^{-i\bvec{m}\cdot\bvec{r}}$ $B$r3]$1(B, $[0,L_x]\times[0,L_y]$ $B$G@QJ,$7(B, $B@h$[$I$HF1$8CV$-49$($r$9$k$H(B,
\begin{align*}
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd y\,\ee^{-i\bvec{m}\cdot\bvec{x}}
 J(\psi,\bnabla^2\tau-k_d^2\tau)\\
 =& \frac{1}{2}L_xL_y
 \sum_{\bvec{k},\bvec{l}}(\bvec{k}\times\bvec{l})_z\{|\bvec{l}|^2\hat{\psi}(-\bvec{k})\hat{\tau}(-\bvec{l})-|\bvec{k}|^2\hat{\psi}(-\bvec{l})\hat{\tau}(-\bvec{k})\}\delta_{(\bvec{k}+\bvec{l}),\bvec{m}}
 \\
 &\quad +\frac{1}{2}L_xL_y\sum_{\bvec{k},\bvec{l}}(\bvec{k}\times\bvec{l})_z
 k_d^2\{\hat{\psi}(-\bvec{k})\hat{\tau}(-\bvec{l})-\hat{\psi}(-\bvec{l})\hat{\tau}(-\bvec{k})\}\delta_{(\bvec{k}+\bvec{l}),\bvec{m}}\\
 =& \frac{1}{2}L_xL_y
 \sum_{\bvec{k},\bvec{l}}(\bvec{k}\times\bvec{l})_z\{|\bvec{l}|^2\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{l}}-|\bvec{k}|^2\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{k}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
 \\
 &\quad +\frac{1}{2}L_xL_y\sum_{\bvec{k},\bvec{l}}(\bvec{k}\times\bvec{l})_z k_d^2\{\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{l}}-\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{k}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
 &\int_{0}^{L_x}\dd x\int_{0}^{L_y}\dd y\,\ee^{-i\bvec{m}\cdot\bvec{x}}
 J(\tau,\bnabla^2\psi) \\
 =&
 \frac{1}{2}L_xL_y\sum_{\bvec{k},\bvec{l}}(\bvec{k}\times\bvec{l})_z\{|\bvec{l}|^2\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{k}}-|\bvec{k}|^2\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{l}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
\end{align*}
$B$H$J$k(B. $B$h$C$F(B, (\ref{eq:clieq}) $B$NGH?t6u4V$K$*$1$kI=8=$O(B
($\bvec{m}\rightarrow\bvec{k},\bvec{m}\rightarrow\bvec{l},\bvec{k}\rightarrow\bvec{m}$
$B$HCV$-49$($k$H(B),
\begin{align}
 \begin{aligned}
  (|\bvec{k}|^2+k_d^2)\DP{}{t}\hat{\tau}_{\bvec{k}} =& 
  \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z\{|\bvec{m}|^2\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}-|\bvec{l}|^2
  \hat{\psi}^{\ast}_{\bvec{m}}\hat{\tau}^{\ast}_{\bvec{l}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
  \\
  &\quad
  +\sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z\{|\bvec{m}|^2\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}-|\bvec{l}|^2\hat{\psi}^{\ast}_{\bvec{m}}\hat{\tau}^{\ast}_{\bvec{l}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
  \\ 
  &\qquad +\sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  k_d^2\{\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}-\hat{\tau}^{\ast}_{\bvec{l}}\hat{\psi}^{\ast}_{\bvec{m}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
  \\ 
  &\quad\qquad -ik_x U|\bvec{k}|^2\hat{\psi}_{\bvec{k}}+ik_x Uk_d^2\hat{\psi}_{\bvec{k}}-\frac{\kappa}{2}|\bvec{k}|^2\{\hat{\tau}_{\bvec{k}}-\hat{\psi}_{\bvec{k}}\}\\
  &\qquad\qquad
  -\nu|\bvec{k}|^8(|\bvec{k}|^2+k_d^2)\hat{\tau}_{\bvec{k}} \\
  &\qquad\qquad\quad -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{(|\bvec{k}|^2+k_d^2)\hat{\tau}_{\bvec{k}}-|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\}
  \label{eq:barocli4}
 \end{aligned}
\end{align}
$B$H$J$k(B.

\section{$BGH?t6u4V$K$*$1$k%(%M%k%.!<J}Dx<0(B}

(\ref{eq:barotro4}) $B$NN>JU$K(B$\hat{\psi}^{\ast}_{\bvec{k}}$ $B$r3]$1$k$H(B,
\begin{align}
 &|\bvec{k}|^2\hat{\psi}^{\ast}_{\bvec{k}}\DP{\hat{\psi}_{\bvec{k}}}{t} \nonumber\\
 =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z(|\bvec{m}|^2-|\bvec{l}|^2)\{\hat{\psi}^{\ast}_{\bvec{k}}\hat{\psi}^{\ast}_{\bvec{l}}\hat{\psi}^{\ast}_{\bvec{m}}-\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
 \nonumber \\ &\quad -ik_x U|\bvec{k}|^2\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}
 -\frac{\kappa}{2}|\bvec{k}|^2\{\hat{\psi}^{\ast}_{\bvec{k}}(\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}})\}-\nu|\bvec{k}|^{10}\hat{\psi}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}
 \nonumber \\
&\qquad
 -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{|\bvec{k}|^2\hat{\psi}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}-(|\bvec{k}|^2+k_d^2)\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}\}  
 \label{eq:barotro5}
\end{align}
$B$H$J$k(B. $B%(%M%k%.!<J}Dx<0$rF3$/$?$a$K(B,
$\{\mathrm{(\ref{eq:barotro5})+(\ref{eq:barotro5})^{\ast}}\}\div 2$ $B$r7W;;$9$k$H(B,
\begin{align*}
 (l.h.s.)=&\frac{1}{2}\left\{|\bvec{k}|^2\hat{\psi}^{\ast}_{\bvec{k}}\DP{\hat{\psi}_{\bvec{k}}}{t}+|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\DP{\hat{\psi}^{\ast}_{\bvec{k}}}{t}\right\} \\
 =&|\bvec{k}|^2\DP{}{t}\frac{1}{2}|\hat{\psi}_{\bvec{k}}|^2, \\
 (r.h.s.)=&\sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{m}|^2-|\bvec{l}|^2)
 \mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}+\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
 &\quad
 -k_x U|\bvec{k}|^2\mathrm{Im}[\hat{\tau}_{\bvec{k}}\hat{\psi}^{\ast}_{\bvec{k}}]
 -\frac{1}{2}\kappa|\bvec{k}|^2\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\{\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}}\}]-\nu|\bvec{k}|^{10}|\hat{\psi}_{\bvec{k}}|^2 \\
 &\qquad -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{|\bvec{k}|^2
 |\hat{\psi}_{\bvec{k}}|^2 -
 (|\bvec{k}|^2+k_d^2)\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}]
 \} 
\end{align*}
$B$H$J$k(B. $B$3$3$G$O(B,
\begin{equation}
 \frac{f+f^{\ast}}{2}=\mathrm{Re}(f), \, \frac{f-f^{\ast}}{2}=\mathrm{Im}(f)
\end{equation}
$B$rMQ$$$?(B. $B$h$C$F(B, $B=g05%b!<%I$N%(%M%k%.!<J}Dx<0$O(B,
\begin{align}
 \begin{aligned}
  |\bvec{k}|^2\DP{}{t}\frac{1}{2}|\hat{\psi}_{\bvec{k}}|^2
  =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{m}|^2-|\bvec{l}|^2)
  \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\} \\
  &+ \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{m}|^2-|\bvec{l}|^2)
  \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\} \\
  &\quad - k_x U|\bvec{k}|^2
  \mathrm{Im}\{\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}\} \\
  &\qquad - \frac{1}{2}\kappa|\bvec{k}|^2\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\{\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}}\}]
  - \nu|\bvec{k}|^{10}|\hat{\psi}_{\bvec{k}}|^2 \\
  &\quad\qquad -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{|\bvec{k}|^2
 |\hat{\psi}_{\bvec{k}}|^2-(|\bvec{k}|^2+k_d^2)\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}]\}
 \end{aligned}
\end{align}
$B$H$J$k(B.

$B<!$K(B, $B7905%b!<%I$N%(%M%k%.!<J}Dx<0$rF3$/(B: (\ref{eq:barocli4}) $B$N(B
$BN>JU$K(B $\hat{\tau}^{\ast}_{\bvec{k}}$ $B$r3]$1$k$H(B,
\begin{align}
 \begin{aligned}
  &(|\bvec{k}|^2 -
  k_d^2)\hat{\tau}^{\ast}_{\bvec{k}}\DP{}{t}\hat{\tau}_{\bvec{k}} \\
  =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z\{|\bvec{m}|^2\hat{\tau}^{\ast}_{\bvec{k}}
  \hat{\tau}^{\ast}_{\bvec{l}}\hat{\psi}^{\ast}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} 
  \\
  &\quad +\sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z\{|\bvec{m}|^2\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}^{\ast}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{l}}\hat{\psi}^{\ast}_{\bvec{m}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
  &\qquad +\sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  k_d^2\{\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}^{\ast}_{\bvec{l}}\hat{\tau}^{\ast}_{\bvec{m}}-\hat{\tau}^{\ast}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{l}}\hat{\psi}^{\ast}_{\bvec{m}}\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  &\quad\qquad +ik_x U|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}+ik_x Uk_d^2\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}\\
  &\qquad\qquad
  -\frac{\kappa}{2}|\bvec{k}|^2\{\hat{\tau}_{\bvec{k}}-\hat{\psi}_{\bvec{k}}\}\hat{\tau}^{\ast}_{\bvec{k}}-\nu|\bvec{k}|^8(|\bvec{k}|^2+k_d^2)\hat{\tau}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}} \\
  &\qquad\qquad\quad
  -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{(|\bvec{k}|^2+k_d^2)\hat{\tau}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}-|\bvec{k}|^2\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}\}
  \label{eq:barocli5}
 \end{aligned}
\end{align}
$B$H$J$k(B. $B@hDx$HF1MM$K(B
$\{\mathrm{(\ref{eq:barocli5})+(\ref{eq:barocli5})^{\ast}}\}\div 2$ $B$r7W;;$9$k$H(B,
\begin{align*}
 (l.h.s) =& (|\bvec{k}|^2+k_d^2)\DP{}{t}\frac{1}{2}|\hat{\tau}_{\bvec{k}}|^2, \\
 (r.h.s) =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z \mathrm{Re}[\{|\bvec{m}|^2
 \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2 \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} 
  \\
  &\quad +\sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
 \mathrm{Re}[\{|\bvec{m}|^2
 \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
  &\qquad +\sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  k_d^2 \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  &\quad\qquad +k_x U|\bvec{k}|^2\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}]-k_x Uk_d^2\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}]\\
  &\qquad\qquad
  -\frac{\kappa}{2}|\bvec{k}|^2
 \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}-\hat{\psi}_{\bvec{k}}\}\hat{\tau}^{\ast}_{\bvec{k}}]-\nu|\bvec{k}|^8
 (|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2 \\
 &\qquad\qquad\quad -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{(|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2-|\bvec{k}|^2\mathrm{Re}[\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}]\}
\end{align*}
$B$H$J$k(B. $B$h$C$F(B, $B7905%b!<%I$N%(%M%k%.!<J}Dx<0$O(B
\begin{align}
 \begin{aligned}
  &(|\bvec{k}|^2+k_d^2)\DP{}{t}\frac{1}{2}|\hat{\tau}_{\bvec{k}}|^2 \\
  =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  \mathrm{Re}[\{|\bvec{m}|^2
  \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2
  \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}   
  \\
  &\quad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  \mathrm{Re}[\{|\bvec{m}|^2
  \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  &\qquad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  k_d^2 \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  &\quad\qquad
  + k_x U|\bvec{k}|^2\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}]
  - k_x
  Uk_d^2\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}] \\
  &\qquad\qquad - \frac{\kappa}{2}|\bvec{k}|^2
  \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}-\hat{\psi}_{\bvec{k}}\}\hat{\tau}^{\ast}_{\bvec{k}}]
  - \nu|\bvec{k}|^8 (|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2 \\
  &\qquad\qquad\quad
  -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{(|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2-|\bvec{k}|^2\mathrm{Re}[\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}]\} 
 \end{aligned}
\end{align}

\section{$B%(%M%k%.!<>\:YJ]B8(B}

$BJ?6QN.$dK`;$(B, $BG4@-$,$J$$$H$-(B, 
\begin{subequations} %11:32$B$N<072(B
 \begin{align}
 &\begin{aligned}
   &|\bvec{k}|^2\DP{}{t}\frac{1}{2}|\hat{\psi}_{\bvec{k}}|^2 \\
  =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{m}|^2-|\bvec{l}|^2)
  \mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  &+ \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{m}|^2-|\bvec{l}|^2)
  \mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  \end{aligned}\\
 &\begin{aligned}
   &(|\bvec{k}|^2+k_d^2)\DP{}{t}\frac{1}{2}|\hat{\tau}_{\bvec{k}}|^2 \\
   =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z \mathrm{Re}[\{|\bvec{m}|^2
   \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2 \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
   \\
   &\quad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
   \mathrm{Re}[\{|\bvec{m}|^2
   \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
   &\qquad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
   k_d^2 \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
  \end{aligned}
 \end{align}
\end{subequations}
$B$NN><0$rB-$7$F(B $\bvec{k}$ $B$K$D$$$FOB$r<h$k$HJ]B8$9$k(B. $B$3$l$i$r(B
\begin{subequations} %11:34$B$N<072(B
 \begin{align} 
  \DP{}{t}\mathcal{E}_{\psi}(\bvec{k}) =&
  \sum_{\bvec{l},\bvec{m}} \{T_{\mathrm{I}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) +
  T_{\mathrm{II}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) \} \\
  \DP{}{t}\mathcal{E}_{\tau}(\bvec{k}) =&
  \sum_{\bvec{l},\bvec{m}} \{T_{\mathrm{I}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) +
  T_{\mathrm{II}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) + T_{\mathrm{III}}^{\tau}(\bvec{k},\bvec{l},\bvec{m})\}
 \end{align}
\end{subequations}
$B$H=q$/$3$H$K$9$k(B. 

$B$3$N$H$-(B, $B0J2<$N$h$&$J%(%M%k%.!<>\:YJ]B8B'$,@.N)$9$k(B.

\begin{enumerate}
 \item \fbox{$\displaystyle
       T_{\mathrm{I}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) +
       T_{\mathrm{I}}^{\psi}(\bvec{l},\bvec{m},\bvec{k}) +
       T_{\mathrm{I}}^{\psi}(\bvec{m},\bvec{k},\bvec{l}) = 0 $} \\
       \textbf{$B!Z>ZL@![(B}
       \begin{align*}
        &T_{\mathrm{I}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) +
        T_{\mathrm{I}}^{\psi}(\bvec{l},\bvec{m},\bvec{k}) +
        T_{\mathrm{I}}^{\psi}(\bvec{m},\bvec{k},\bvec{l}) \\
        =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z
        \mathrm{Re}[\{|\bvec{m}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad + \frac{1}{2}(\bvec{m}\times\bvec{k})_z
        \mathrm{Re}[\{|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{m}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\qquad + \frac{1}{2}(\bvec{k}\times\bvec{l})_z
        \mathrm{Re}[\{|\bvec{l}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}  
       \end{align*}
       $B$3$3$G(B, $\bvec{k}=-\bvec{l}-\bvec{m}$ $B$h$j(B
       \begin{align*}
        (\bvec{m}\times\bvec{k})_z = \{\bvec{m}\times(-\bvec{l}-\bvec{m})\}_z =
        -(\bvec{m}\times\bvec{l})_z = (\bvec{l}\times\bvec{m})_z \\
        (\bvec{k}\times\bvec{l})_z = \{(-\bvec{l}-\bvec{m})\times\bvec{l}\}_z =
        -(\bvec{m}\times\bvec{l})_z = (\bvec{l}\times\bvec{m})_z
       \end{align*}
       $B$H$J$k$N$G(B,
       \begin{align*}
        &T_{\mathrm{I}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) +
        T_{\mathrm{I}}^{\psi}(\bvec{l},\bvec{m},\bvec{k}) +
        T_{\mathrm{I}}^{\psi}(\bvec{m},\bvec{k},\bvec{l}) \\
        =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z
        \mathrm{Re}[\{|\bvec{m}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad + \frac{1}{2}(\bvec{l}\times\bvec{m})_z
        \mathrm{Re}[\{|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{m}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\qquad + \frac{1}{2}(\bvec{l}\times\bvec{m})_z
        \mathrm{Re}[\{|\bvec{l}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{k}|^2\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        =& 0
       \end{align*}
       $B$H$J$k(B. $\Box$
 \item \fbox{$\displaystyle
       T_{\mathrm{I}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) +
       T_{\mathrm{I}}^{\tau}(\bvec{l},\bvec{m},\bvec{k}) +
       T_{\mathrm{I}}^{\tau}(\bvec{m},\bvec{k},\bvec{l}) = 0 $} \\
       \textbf{$B!Z>ZL@![(B}
       \begin{align*}
        &T_{\mathrm{I}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) +
        T_{\mathrm{I}}^{\tau}(\bvec{l},\bvec{m},\bvec{k}) +
        T_{\mathrm{I}}^{\tau}(\bvec{m},\bvec{k},\bvec{l}) \\
        =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z
        \mathrm{Re}[\{|\bvec{m}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad + \frac{1}{2}(\bvec{m}\times\bvec{k})_z
        \mathrm{Re}[\{|\bvec{k}|^2\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}-|\bvec{m}|^2\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\qquad +
        \frac{1}{2}(\bvec{k}\times\bvec{l})_z\mathrm{Re}[\{|\bvec{l}|^2\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}-|\bvec{k}|^2\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z
        \mathrm{Re}[\{|\bvec{m}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\} \\
        &\qquad\qquad\qquad +
        \{|\bvec{k}|^2\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}-|\bvec{m}|^2\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\} \\
        & \qquad\qquad\qquad\quad
        +\{|\bvec{l}|^2\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}-|\bvec{k}|^2\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\}]\\
        =& 0 \hspace{3pt} \Box 
       \end{align*}
 \item \fbox{$\displaystyle
       T_{\mathrm{III}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) +
       T_{\mathrm{III}}^{\tau}(\bvec{l},\bvec{m},\bvec{k}) +
       T_{\mathrm{III}}^{\tau}(\bvec{m},\bvec{k},\bvec{l}) = 0 $} \\
       \textbf{$B!Z>ZL@![(B}
       \begin{align*}
        &T_{\mathrm{III}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) +
        T_{\mathrm{III}}^{\tau}(\bvec{l},\bvec{m},\bvec{k}) +
        T_{\mathrm{III}}^{\tau}(\bvec{m},\bvec{k},\bvec{l}) \\
        =&
        \frac{1}{2}(\bvec{l}\times\bvec{m})_z k_d^2
        \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad +\frac{1}{2}(\bvec{m}\times\bvec{k})_z k_d^2
        \mathrm{Re}[\{\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\qquad +\frac{1}{2}(\bvec{k}\times\bvec{l})_z k_d^2
        \mathrm{Re}[\{\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z k_d^2
        \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\} \\
        &\qquad\qquad\qquad\quad +
        \{\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\} \\
        &\qquad\qquad\qquad\qquad +
        \{\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}] \\
        =& 0 \hspace{3pt} \Box
       \end{align*}
 \item \fbox{$\displaystyle
       T_{\mathrm{II}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) +
       T_{\mathrm{II}}^{\psi}(\bvec{l},\bvec{m},\bvec{k}) +
       T_{\mathrm{II}}^{\psi}(\bvec{m},\bvec{k},\bvec{l}) +
       T_{\mathrm{II}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) +
       T_{\mathrm{II}}^{\tau}(\bvec{l},\bvec{m},\bvec{k}) +
       T_{\mathrm{II}}^{\tau}(\bvec{m},\bvec{k},\bvec{l}) = 0 $} \\
       \textbf{$B!Z>ZL@![(B}
       \begin{align*}
        &T_{\mathrm{II}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) +
        T_{\mathrm{II}}^{\psi}(\bvec{l},\bvec{m},\bvec{k}) +
        T_{\mathrm{II}}^{\psi}(\bvec{m},\bvec{k},\bvec{l}) +
        T_{\mathrm{II}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) +
        T_{\mathrm{II}}^{\tau}(\bvec{l},\bvec{m},\bvec{k}) +
        T_{\mathrm{II}}^{\tau}(\bvec{m},\bvec{k},\bvec{l}) \\
        =&\hspace{1zw} \frac{1}{2}(\bvec{l}\times\bvec{m})_z
  \mathrm{Re}[\{|\bvec{m}|^2
        \hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad + \frac{1}{2}(\bvec{m}\times\bvec{k})_z
        \mathrm{Re}[\{|\bvec{k}|^2
        \hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-|\bvec{m}|^2\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
        &\qquad + \frac{1}{2}(\bvec{k}\times\bvec{l})_z
        \mathrm{Re}[\{|\bvec{l}|^2
        \hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-|\bvec{k}|^2\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &+ \frac{1}{2}(\bvec{l}\times\bvec{m})_z
        \mathrm{Re}[\{|\bvec{m}|^2
        \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{m}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad + \frac{1}{2}(\bvec{m}\times\bvec{k})_z
        \mathrm{Re}[\{|\bvec{k}|^2
        \hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-|\bvec{m}|^2\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\qquad +\frac{1}{2}(\bvec{k}\times\bvec{l})_z
        \mathrm{Re}[\{|\bvec{l}|^2
        \hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-|\bvec{k}|^2\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        =& \hspace{1zw} \frac{1}{2}(\bvec{l}\times\bvec{m})_z
  \mathrm{Re}[\{|\bvec{m}|^2
        \hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\} \\
        &\qquad\qquad\qquad\quad + \{|\bvec{k}|^2
        \hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-|\bvec{m}|^2\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\} \\
        &\qquad\qquad\qquad\qquad + \{|\bvec{l}|^2
        \hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-|\bvec{k}|^2\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\} \\
        &\qquad\qquad\qquad\quad + \{|\bvec{m}|^2
        \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{m}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\} \\
        &\qquad\qquad\qquad\qquad + \{|\bvec{k}|^2
        \hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-|\bvec{m}|^2\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\} \\
        &\qquad\qquad\qquad\qquad\quad + \{|\bvec{l}|^2
        \hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-|\bvec{k}|^2\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\}]  \\
        =& 0 \hspace{3pt} \Box
       \end{align*}
\end{enumerate}

\section{$B%(%s%9%H%m%U%#!<>\:YJ]B8(B}

(\ref{eq:barotro4}) $B$NN>JU$K(B$|\bvec{k}|^2\hat{\psi}^{\ast}_{\bvec{k}}$ $B$r3]$1(B
$B$?<0$H$=$NJ#AG6&Lr$rB-$7$F(B 2 $B$G3d$k$H(B, $B=g05%b!<%I$N%(%s%9%H%m%U%#!<(B
$BJ}Dx<0$,F3$+$l(B, (\ref{eq:barocli4}) $B$NN>JU$K(B$(|\bvec{k}|^2+k_d^2)\hat{\tau}^{\ast}_{\bvec{k}}$ $B$r3]$1(B
$B$?<0$H$=$NJ#AG6&Lr$rB-$7$F(B 2 $B$G3d$k$H(B, $B7905%b!<%I$N%(%s%9%H%m%U%#!<(B
$BJ}Dx<0$,F3$+$l$k(B:
\begin{align}
 &\begin{aligned}
  &|\bvec{k}|^4\DP{}{t}\frac{1}{2}|\hat{\psi}_{\bvec{k}}|^2 \\
  =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
   |\bvec{k}|^2 (|\bvec{m}|^2-|\bvec{l}|^2)
  \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\} \\
  &+ \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
   |\bvec{k}|^2 (|\bvec{m}|^2-|\bvec{l}|^2)
  \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\} \\
  &\quad - ik_x U|\bvec{k}|^4
  \mathrm{Im}\{\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}\} \\
  &\qquad - \frac{1}{2}\kappa|\bvec{k}|^4\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\{\hat{\psi}_{\bvec{k}}-\hat{\tau}_{\bvec{k}}\}]
   - \nu|\bvec{k}|^{12}|\hat{\psi}_{\bvec{k}}|^2 \\
   &\quad\qquad -\frac{1}{2}\Delta\nu|\bvec{k}|^{10}\{|\bvec{k}|^2
 |\hat{\psi}_{\bvec{k}}|^2-(|\bvec{k}|^2+k_d^2)\mathrm{Re}[\hat{\psi}^{\ast}_{\bvec{k}}\hat{\tau}_{\bvec{k}}]\},
 \end{aligned} \\
 &\begin{aligned}
  &(|\bvec{k}|^2+k_d^2)^2\DP{}{t}\frac{1}{2}|\hat{\tau}_{\bvec{k}}|^2 \\ =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{k}|^2+k_d^2)\mathrm{Re}[\{|\bvec{m}|^2
  \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2 \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
  \\
  &\quad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  (|\bvec{k}|^2+k_d^2) \mathrm{Re}[\{|\bvec{m}|^2
  \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  &\qquad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
  k_d^2
   (|\bvec{k}|^2+k_d^2)\mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
  &\quad\qquad
  + ik_x
   U|\bvec{k}|^2(|\bvec{k}|^2+k_d^2)\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}] 
  + ik_x
   Uk_d^2(|\bvec{k}|^2+k_d^2)\mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}] \\
  &\qquad\qquad
  - \frac{\kappa}{2}|\bvec{k}|^2 (|\bvec{k}|^2+k_d^2)
  \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}-\hat{\psi}_{\bvec{k}}\}\hat{\tau}^{\ast}_{\bvec{k}}] 
   - \nu|\bvec{k}|^8 (|\bvec{k}|^2+k_d^2)^2|\hat{\tau}_{\bvec{k}}|^2 \\
   &\qquad\qquad\quad -\frac{1}{2}\Delta\nu|\bvec{k}|^8 (|\bvec{k}|^2+k_d^2)\{(|\bvec{k}|^2+k_d^2)|\hat{\tau}_{\bvec{k}}|^2-|\bvec{k}|^2\mathrm{Re}[\hat{\tau}^{\ast}_{\bvec{k}}\hat{\psi}_{\bvec{k}}]\}.
 \end{aligned}
\end{align}

$B%(%M%k%.!<$N$H$-$HF1MM$K(B, $BJ?6QN.$dK`;$(B, $BG4@-$,L5$$>l9g(B,
\begin{subequations} %12:01$B$N<072(B
 \begin{align}
  &\begin{aligned}
    &|\bvec{k}|^4\DP{}{t}\frac{1}{2}|\hat{\psi}_{\bvec{k}}|^2 \\
    =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z |\bvec{k}|^2
    (|\bvec{m}|^2-|\bvec{l}|^2)
    \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}  \\
    &\quad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
    |\bvec{k}|^2 (|\bvec{m}|^2-|\bvec{l}|^2)
    \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}, \\
   \end{aligned}\\
  &\begin{aligned}
    &(|\bvec{k}|^2+k_d^2)^2\DP{}{t}\frac{1}{2}|\hat{\tau}_{\bvec{k}}|^2 \\
    =& \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
    (|\bvec{k}|^2+k_d^2)\mathrm{Re}[\{|\bvec{m}|^2
    \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2
    \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
    & \quad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
    (|\bvec{k}|^2+k_d^2) \mathrm{Re}[\{|\bvec{m}|^2
    \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}
    -|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]
    \delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
    &\qquad + \sum_{\bvec{l},\bvec{m}}\frac{1}{2}(\bvec{l}\times\bvec{m})_z
    k_d^2 (|\bvec{k}|^2+k_d^2)\mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}
    \hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}
    \hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0},
   \end{aligned} 
 \end{align}
\end{subequations}
$B$NN><0$rB-$7$F(B $\bvec{k}$ $B$K$D$$$FB-$9$HJ]B8$9$k(B. $B$3$l$i$r(B
\begin{subequations} %12:05$B$N<072(B
 \begin{align}
  &\DP{}{t}\mathcal{Q}_{\psi}(\bvec{k})=\sum_{\bvec{l},\bvec{m}}\{C_{\mathrm{I}}^{\psi}(\bvec{k},\bvec{l},\bvec{m})+C_{\mathrm{II}}^{\psi}(\bvec{k},\bvec{l},\bvec{m})\},\\
  &\DP{}{t}\mathcal{Q}_{\tau}(\bvec{k})=\sum_{\bvec{l},\bvec{m}}\{C_{\mathrm{I}}^{\tau}(\bvec{k},\bvec{l},\bvec{m})
  + C_{\mathrm{II}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) + C_{\mathrm{III}}^{\tau}(\bvec{k},\bvec{l},\bvec{m})\}
 \end{align}
\end{subequations}
$B$H=q$/(B. $B$3$3$G(B,
\begin{align*}
 &C_{\mathrm{I}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) =
 \frac{1}{2}(\bvec{l}\times\bvec{m})_z |\bvec{k}|^2
 (|\bvec{m}|^2-|\bvec{l}|^2)\{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\}, \\
 &C_{\mathrm{II}}^{\psi}(\bvec{k},\bvec{l},\bvec{m}) =
 \frac{1}{2}(\bvec{l}\times\bvec{m})_z |\bvec{k}|^2 (|\bvec{m}|^2-|\bvec{l}|^2)
 \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\}, \\
 &C_{\mathrm{I}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) \\ &\quad =
 \frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{k}|^2 - k_d^2)\mathrm{Re}[\{|\bvec{m}|^2
 \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2 \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0},\\
 &C_{\mathrm{II}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) \\&\quad =
 \frac{1}{2}(\bvec{l}\times\bvec{m})_z 
 (|\bvec{k}|^2 - k_d^2) \mathrm{Re}[\{|\bvec{m}|^2\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0},
 \\ 
 &C_{\mathrm{III}}^{\tau}(\bvec{k},\bvec{l},\bvec{m}) \\&\quad= \frac{1}{2}(\bvec{l}\times\bvec{m})_z
 k_d^2 (|\bvec{k}|^2 - k_d^2) \mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
\end{align*}
$B$G$"$k(B.

$B$3$N$H$-(B, $B<!$N%(%s%9%H%m%U%#!<>\:YJ]B8B'$,@.N)$9$k(B:
\begin{enumerate}
 \item \fbox{$C_{\mathrm{I}}^{\psi}(\bvec{k},\bvec{l},\bvec{m})+C_{\mathrm{I}}^{\psi}(\bvec{l},\bvec{m},\bvec{k})+C_{\mathrm{I}}^{\psi}(\bvec{m},\bvec{k},\bvec{l})=0$}\\
      \textbf{$B!Z>ZL@![(B}
      \begin{align*}
        (l.h.s) =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z |\bvec{k}|^2
        (|\bvec{m}|^2-|\bvec{l}|^2)
        \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad + \frac{1}{2}(\bvec{m}\times\bvec{k})_z |\bvec{l}|^2
        (|\bvec{k}|^2-|\bvec{m}|^2)
        \{\mathrm{Re}[\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\psi}_{\bvec{k}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\qquad + \frac{1}{2}(\bvec{k}\times\bvec{l})_z |\bvec{m}|^2
        (|\bvec{l}|^2-|\bvec{k}|^2)
        \{\mathrm{Re}[\hat{\psi}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z 
        \{\mathrm{Re}[|\bvec{k}|^2
        (|\bvec{m}|^2-|\bvec{l}|^2) \hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}
        \\ &\qquad\qquad\quad + |\bvec{l}|^2
        (|\bvec{k}|^2-|\bvec{m}|^2)\hat{\psi}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\psi}_{\bvec{k}} \\
        &\qquad\qquad\qquad + |\bvec{m}|^2
        (|\bvec{l}|^2-|\bvec{k}|^2)\hat{\psi}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\psi}_{\bvec{l}}] \\
        =& 0 \hspace{3pt} \Box
       \end{align*}
 \item \fbox{$C_{\mathrm{II}}^{\psi}(\bvec{k},\bvec{l},\bvec{m})+C_{\mathrm{II}}^{\psi}(\bvec{l},\bvec{m},\bvec{k})+C_{\mathrm{II}}^{\psi}(\bvec{m},\bvec{k},\bvec{l})+C_{\mathrm{I}}^{\tau}(\bvec{k},\bvec{l},\bvec{m})+C_{\mathrm{I}}^{\tau}(\bvec{l},\bvec{m},\bvec{k})+C_{\mathrm{I}}^{\tau}(\bvec{m},\bvec{k},\bvec{l})=0$} \\
      \textbf{$B!Z>ZL@![(B}
      \begin{align*}
       &(l.h.s.) \\ =&\frac{1}{2}(\bvec{l}\times\bvec{m})_z |\bvec{k}|^2
       (|\bvec{m}|^2-|\bvec{l}|^2)
       \{\mathrm{Re}[\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
       &\quad + \frac{1}{2}(\bvec{m}\times\bvec{k})_z |\bvec{l}|^2
       (|\bvec{k}|^2-|\bvec{m}|^2)
       \{\mathrm{Re}[\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
       &\qquad + \frac{1}{2}(\bvec{k}\times\bvec{l})_z |\bvec{m}|^2
       (|\bvec{l}|^2-|\bvec{k}|^2)
       \{\mathrm{Re}[\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
       &\qquad\quad + \frac{1}{2}(\bvec{l}\times\bvec{m})_z (|\bvec{k}|^2+k_d^2)\mathrm{Re}[\{|\bvec{m}|^2
  \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2
       \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
       &\qquad\qquad + \frac{1}{2}(\bvec{m}\times\bvec{k})_z (|\bvec{l}|^2-k_d^2)\mathrm{Re}[\{|\bvec{k}|^2
  \hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}-|\bvec{m}|^2
       \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
       &\quad\qquad\qquad + \frac{1}{2}(\bvec{k}\times\bvec{l})_z (|\bvec{m}|^2-k_d^2)\mathrm{Re}[\{|\bvec{l}|^2
  \hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}-|\bvec{k}|^2
 \hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
       =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z \{\mathrm{Re}[|\bvec{k}|^2(|\bvec{m}|^2-|\bvec{l}|^2)
       \hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}} \\
       &\qquad\qquad\qquad + |\bvec{l}|^2(|\bvec{k}|^2-|\bvec{m}|^2)
       \hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}} \\
       &\qquad\qquad\qquad\quad +
       |\bvec{m}|^2 (|\bvec{l}|^2-|\bvec{k}|^2)
       \hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}} \\
       &\qquad\qquad\quad + |\bvec{k}|^2\{|\bvec{m}|^2
       \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}-|\bvec{l}|^2
       \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\} \\
       & \qquad\qquad\qquad + |\bvec{l}|^2\{|\bvec{k}|^2
  \hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}-|\bvec{m}|^2
       \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\} \\
        & \qquad\qquad\qquad\quad + |\bvec{m}|^2 \{|\bvec{l}|^2
  \hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}-|\bvec{k}|^2
       \hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\} \\
       & \qquad\qquad\quad + k_d^2\{|\bvec{m}|^2
       \hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}} -
       |\bvec{l}|^2
       \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}} \\
       & \qquad\qquad\qquad + |\bvec{k}|^2
       \hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}} - |\bvec{m}|^2
       \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}} \\
       & \qquad\qquad\qquad\quad + |\bvec{l}|^2
  \hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}-|\bvec{k}|^2
       \hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\}] \\
       =& 0 \hspace{3pt} \Box
      \end{align*}
 \item \fbox{$C_{\mathrm{II}}^{\tau}(\bvec{k},\bvec{l},\bvec{m})+C_{\mathrm{II}}^{\tau}(\bvec{l},\bvec{m},\bvec{k})+C_{\mathrm{II}}^{\tau}(\bvec{m},\bvec{k},\bvec{l})+C_{\mathrm{III}}^{\tau}(\bvec{k},\bvec{l},\bvec{m})+C_{\mathrm{III}}^{\tau}(\bvec{l},\bvec{m},\bvec{k})+C_{\mathrm{III}}^{\tau}(\bvec{m},\bvec{k},\bvec{l})=0$} \\
       \textbf{$B!Z>ZL@![(B}
       \begin{align*}
        &(l.h.s.) \\ =& \frac{1}{2}(\bvec{l}\times\bvec{m})_z
  (|\bvec{k}|^2+k_d^2) \mathrm{Re}[\{|\bvec{m}|^2
 \hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-|\bvec{l}|^2\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
        &+ \frac{1}{2}(\bvec{m}\times\bvec{k})_z
  (|\bvec{l}|^2-k_d^2) \mathrm{Re}[\{|\bvec{k}|^2
 \hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-|\bvec{m}|^2\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
        &\quad + \frac{1}{2}(\bvec{k}\times\bvec{l})_z
  (|\bvec{m}|^2-k_d^2) \mathrm{Re}[\{|\bvec{l}|^2
 \hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-|\bvec{k}|^2\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
        +& \frac{1}{2}(\bvec{l}\times\bvec{m})_z k_d^2
        (|\bvec{k}|^2+k_d^2)\mathrm{Re}[\{\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\hat{\tau}_{\bvec{m}}-\hat{\tau}_{\bvec{k}}\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}\\
        &+ \frac{1}{2}(\bvec{m}\times\bvec{k})_z k_d^2
        (|\bvec{l}|^2-k_d^2)\mathrm{Re}[\{\hat{\tau}_{\bvec{l}}\hat{\psi}_{\bvec{m}}\hat{\tau}_{\bvec{k}}-\hat{\tau}_{\bvec{l}}\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        &\quad + \frac{1}{2}(\bvec{k}\times\bvec{l})_z
        k_d^2
        (|\bvec{m}|^2-k_d^2)\mathrm{Re}[\{\hat{\tau}_{\bvec{m}}\hat{\psi}_{\bvec{k}}\hat{\tau}_{\bvec{l}}-\hat{\tau}_{\bvec{m}}\hat{\tau}_{\bvec{k}}\hat{\psi}_{\bvec{l}}\}]\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} \\
        =& 0
       \end{align*}
       
\end{enumerate}




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