I got stuck in my research. Briefly speaking, the following is a system of 6 variables ($u,v,p,h_{11},h_{12},h_{22}$) I need to analyze: \begin{equation} g^2\frac{\partial u}{\partial X}+\frac{\partial v}{\partial Y}=\frac{h_{11}+h_{22}}{2} \end{equation}
\begin{equation} \frac{dh_{11}}{dt}=2\gamma h_{11}+\beta(2\frac{\partial u}{\partial X}-2g^2\frac{\partial v}{\partial Y}-g^2h_{11}+g^2h_{22}) \end{equation}
\begin{equation} \frac{dh_{12}}{dt}=2\gamma h_{12}+\beta[2\frac{\partial u}{\partial Y}+2g^2\frac{\partial v}{\partial X}-(g^2+\frac{1}{g^2})h_{12}] \end{equation}
\begin{equation} \frac{dh_{22}}{dt}=2\gamma h_{22}+\beta(2g^2\frac{\partial v}{\partial Y}-2\frac{\partial u}{\partial X}+\frac{1}{g^2}h_{11}-\frac{1}{g^2}h_{22}) \end{equation}
\begin{equation} -\frac{1}{2}\frac{\partial h_{11}}{\partial X}+\frac{1}{2}\frac{\partial h_{22}}{\partial X}-\frac{\partial h_{12}}{\partial Y}+g^2(\frac{\partial^2u}{\partial X^2}+\frac{\partial^2u}{\partial Y^2})=g^4\frac{\partial p}{\partial X} \end{equation}
\begin{equation} \frac{1}{2}\frac{\partial h_{11}}{\partial Y}-\frac{1}{2}\frac{\partial h_{22}}{\partial Y}-\frac{\partial h_{12}}{\partial X}+(\frac{\partial^2v}{\partial X^2}+\frac{\partial^2v}{\partial Y^2})=\frac{\partial p}{\partial Y} \end{equation}
I used the ansatz that all 6 variables have the form $e^{\omega t}f_i(Y)\sin(kX)$ or $e^{\omega t}f_i(Y)\cos(kX)$, and got some results. Most coincide with previous work but some suggested the $e^{\omega t}$ part is too strong (they don't necessarily share the same growth rate $\omega$), and the $t,Y$ part in real solutions are likely to be coupled together.
I tried several different approaches: Take $t$-derivatives of equations 1,5,6, then apply 2D Fourier Transform in $X,Y$; Assume the solutions are of the form $f_i(t,Y)\sin(kX)$ or $f_i(t,Y)\cos(kX)$ then apply 1D Fourier Transform in $Y$, etc., but none seems promising.
Previous work dealt with the simpler system of only 3 variables (setting $h_{11}=h_{12}=h_{22}=\beta=0$ in my new system):
\begin{equation} g^2\frac{\partial u}{\partial X}+\frac{\partial v}{\partial Y}=0 \end{equation}
\begin{equation} \frac{\partial^2u}{\partial X^2}+\frac{\partial^2u}{\partial Y^2}=g^2\frac{\partial p}{\partial X} \end{equation}
\begin{equation} \frac{\partial^2v}{\partial X^2}+\frac{\partial^2v}{\partial Y^2}=\frac{\partial p}{\partial Y} \end{equation}
And the solutions $u,v$ are of the following form to satisfy certain boundary conditions:
\begin{equation} u=f_u(t)\sin(kX)[\cosh(kY)-\cosh(g^2kY)+\tau(\sinh(kY)-g^2\sinh(g^2kY))] \end{equation}
\begin{equation} v=f_v(t)\cos(kX)[\tau(\cosh(kY)-\cosh(g^2kY))+\sinh(kY)-\frac{\sinh(g^2kY)}{g^2}] \end{equation}
where $\tau$ has the expression
\begin{equation} \tau=\frac{2g^2\sinh(g^2k)-(1+g^4)\sinh(k)}{(1+g^4)\cosh(k)-2g^4\cosh(g^2k)} \end{equation}
I'm thinking of using the difference of the R.H.S. of the first equations in the old and new system, somehow modify the $Y$ terms in old solutions and use it to solve my new system, but haven't got the right idea so far.
I'd appreciate a lot if someone can give me some insights or thoughts what might work. Thanks in advance!
