%#! platex master_thesis-M2.tex
\chapter{$BGH?t6u4V$K$*$1$k7O$N@-<A(B}

$BK\>O$G$O(B, $B=`CO9U(B 2 $BAX%b%G%k$K$*$$$F@.N)$9$k%(%M%k%.!<$d%(%s%9%H%m%U%#!<(B
$B$NJ]B8B'$r@0M}$9$k(B. $B>ZL@$K$D$$$F$OIUO?(B \ref{chap:wave-eq} $B$r;2>H$N$3$H(B.

$B$^$:(B, $B=g05%b!<%I$H7905%b!<%I$K$*$1$k;YG[J}Dx<0(B (\ref{eq:troeq}) $B$H(B
(\ref{eq:clieq}) $B$NGH?t6u4VFb$G$NI=<0$O(B,
\begin{subequations} %17:31$B$N<072(B
 \begin{align}
  &\begin{aligned} \label{eq:gveq_wn-1}
  &\DP{}{t}|\bvec{k}|^2\hat{\psi}(\bvec{k},t) \\
 =&
 \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},t)\hat{\psi}^{\ast}(\bvec{m},t)+\hat{\tau}^{\ast}(\bvec{l},t)\hat{\tau}^{\ast}(\bvec{m},t)\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0} 
    \\ &\quad -ik_x U|\bvec{k}|^2\hat{\tau}(\bvec{k},t)
 -\frac{\kappa}{2}|\bvec{k}|^2\{\hat{\psi}(\bvec{k},t)-\hat{\tau}(\bvec{k},t)\}
    \\ &\qquad 
 -\nu|\bvec{k}|^{10}\hat{\psi}(\bvec{k},t)-\frac{1}{2}\Delta\nu|\bvec{k}|^8\{|\bvec{k}|^2\hat{\psi}(\bvec{k},t)-(|\bvec{k}|^2+k_d^2)\hat{\tau}(\bvec{k},t)\}   
   \end{aligned} \\ 
  &\begin{aligned} \label{eq:gveq_wn-2}
  &(|\bvec{k}|^2+k_d^2)\DP{}{t}\hat{\tau}(\bvec{k},t) \\ =& 
  \sum_{\bvec{l},\bvec{m}} \frac{1}{2}
    (\bvec{l}\times\bvec{m})_z\{|\bvec{m}|^2\hat{\psi}^{\ast}(\bvec{l},t)\hat{\tau}^{\ast}(\bvec{m},t)-|\bvec{l}|^2 
  \hat{\psi}^{\ast}(\bvec{m},t)\hat{\tau}^{\ast}(\bvec{l},t)\}\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},t)\hat{\tau}^{\ast}(\bvec{m},t)-|\bvec{l}|^2\hat{\psi}^{\ast}(\bvec{m},t)\hat{\tau}^{\ast}(\bvec{l},t)\}\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},t)\hat{\tau}^{\ast}(\bvec{m},t)-\hat{\psi}^{\ast}(\bvec{m},t)\hat{\tau}^{\ast}(\bvec{l},t)\}\delta_{\bvec{k}+\bvec{l}+\bvec{m},0}
  \\ 
  &\quad\qquad -ik_x U|\bvec{k}|^2\hat{\psi}(\bvec{k},t)+ik_x Uk_d^2\hat{\psi}(\bvec{k},t)-\frac{\kappa}{2}|\bvec{k}|^2\{\hat{\tau}(\bvec{k},t)-\hat{\psi}(\bvec{k},t)\}\\
  &\qquad\qquad
  -\nu|\bvec{k}|^8(|\bvec{k}|^2+k_d^2)\hat{\tau}(\bvec{k},t) \\
  &\qquad\qquad\quad -\frac{1}{2}\Delta\nu|\bvec{k}|^8\{(|\bvec{k}|^2+k_d^2)\hat{\tau}(\bvec{k},t)-|\bvec{k}|^2\hat{\psi}(\bvec{k},t)\}
   \end{aligned}
 \end{align}
\end{subequations}
$B$G$"$k(B. $B$?$@$7(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*}
$B$G$"$k(B. $B0J2<(B, $\hat{\psi}(\bvec{k},t)\equiv\hat{\psi}_{\bvec{k}},
\hat{\tau}(\bvec{k},t)\equiv\hat{\tau}_{\bvec{k}}$ $B$HI=5-$9$k(B.

\section{$BGH?t6u4VFb$K$*$1$k%(%M%k%.!<J]B8B'(B}

(\ref{eq:gveq_wn-1})$B$H(B(\ref{eq:gveq_wn-2})$B$+$iGH?t6u4VFb$G$N%(%M%k%.!<J}(B
$BDx<0$rF3$/$H(B,
\begin{subequations} %11:20$B$N<072(B
 \begin{align}
 &\begin{aligned} \label{eq:tro_ekeq-ws}
   &|\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}_{\bvec{k}}\hat{\tau}^{\ast}_{\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} \\
 &\begin{aligned} \label{eq:cli_ekeq-ws}
   &(|\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} 
  \\
  & + \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} \\
  &\quad + \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} \\
  &\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}}] \\
  &\quad\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}
\end{subequations}
$B$H$J$k(B. $B$3$3$G(B, $\bvec{k}=k_x\bvec{i}+k_y\bvec{j}$ $B$G$"$k(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$3$G(B,
\begin{align*}
 &T_{\mathrm{I}}^{\psi}(\bvec{k},\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}\}, \\
 &T_{\mathrm{II}}^{\psi}(\bvec{k},\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}\}, \\
 &T_{\mathrm{I}}^{\tau}(\bvec{k},\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},\\
 &T_{\mathrm{II}}^{\tau}(\bvec{k},\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},
 \\ 
 &T_{\mathrm{III}}^{\tau}(\bvec{k},\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{align*}
$B$G$"$k(B. $B$3$N$H$-(B, $B0J2<$N%(%M%k%.!<>\:YJ]B8B'$,@.N)$9$k(B:
\begin{screen}
 \begin{subequations} %11:36$B$N<072(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}) = 0, \label{eq:ek-conservation1}\\
   &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,\label{eq:ek-conservation2} \\
   &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, \label{eq:ek-conservation3}\\
   \begin{split} \label{eq:ek-conservation4}
   &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}) \\  
   &\quad +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.
   \end{split}
   \end{align}
 \end{subequations}
 \end{screen}

\section{$BGH?t6u4V$K$*$1$k%(%s%9%H%m%U%#!<J]B8B'(B}

$BF1MM$K(B, (\ref{eq:gveq_wn-1})$B$H(B(\ref{eq:gveq_wn-2})$B$+$iGH?t6u4VFb$G$N%(%s(B
$B%9%H%m%U%#!<J}Dx<0$rF3$/$H(B,
\begin{subequations} %11:51$B$N<072(B
 \begin{align}
  &\begin{aligned} \label{eq:tro_enseq-ws}
    &|\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 - k_x U|\bvec{k}|^4
    \mathrm{Im}\{\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\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} \label{eq:cli_enseq-ws}
    &(|\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} \\
    &+ \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}\\
    &\quad + \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} \\
    &\qquad
    + k_x U|\bvec{k}|^2(|\bvec{k}|^2+k_d^2)
    \mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}]
    - k_x Uk_d^2(|\bvec{k}|^2+k_d^2)
    \mathrm{Im}[\hat{\psi}_{\bvec{k}}\hat{\tau}^{\ast}_{\bvec{k}}] \\
    &\quad\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}
\end{subequations}
$B$H$J$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{screen}
 \begin{subequations} %12:06$B$N<072(B
  \begin{align}
   &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,\label{eq:ens-conservation1}\\
   &\begin{aligned} \label{eq:ens-conservation2}
     &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}) \\
     &\quad + 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, 
    \end{aligned}\\
     &\begin{aligned} \label{eq:ens-conservation3}
       &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}) \\
       &\quad +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. 
      \end{aligned}
  \end{align}
 \end{subequations}
\end{screen}

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