I'm trying to solve the following coupled system of linear PDE's
$$ u=(\lambda+2 \mu)\frac{\partial^2}{\partial x^2}u+\mu \frac{\partial^2}{\partial y^2}u +(\lambda+\mu)\frac{\partial}{\partial x}\frac{\partial}{\partial y} v $$ $$ v=(\lambda+2 \mu)\frac{\partial^2}{\partial x^2}v+\mu \frac{\partial^2}{\partial y^2}v +(\lambda+\mu)\frac{\partial}{\partial x}\frac{\partial}{\partial y}u $$
Subject to the initial conditions: $$ u(x=0,y)=0\\ u(x=L,y)=0\\ \partial_y v(x,y=0)=0\\ \partial_y v(x,y=0)=0 $$
Where $u$ and $v$ are functions of $x,y$ and are at least $C^2$ continuous.
I have tried several methods to solve this problem, but none seem to work. My first instinct is a 2D Fourier Transform which yields:
$$ \tilde{u}=-(2\pi)^2((\lambda+2 \mu)(k_x)^2 \tilde{u}+\mu (k_y)^2 \tilde{u} +(\lambda+\mu)k_xk_y \tilde{v}) $$ $$ \tilde{v}=-(2\pi)^2((\lambda+2 \mu)(\xi_x)^2 \tilde{v}+\mu (\xi_y)^2 \tilde{u} +(\lambda+\mu)\xi_x \xi_y \tilde{u}) $$
Where I have used $F\{u\}=\tilde{u}$ and my Fourier Transform variables as $k$'s and $\xi$'s respectively. However, a really quick inspection shows the above equations can be easily substituted. Solving for $v$ in the second equation and plugging it into the first yields:
$$ \tilde{u}+(\lambda+2\mu)k_x^2 \tilde{u}+\mu k_y^2 \tilde{u} = -(\lambda+\mu)k_xk_y\frac{-(\lambda+\mu)\xi_x\xi_y\tilde{u}}{1+(\lambda+2\mu)k_x^2+\mu k_y^2} $$
which of course has a $\tilde{u}$ in every term, and assuming $\tilde{u} \neq 0$, and dividing through by $\tilde{u}$ gives no information as to the solution to this PDE. I would like to have an analytic solution, but I'm not sure these equations admit one. However, these are linear PDE's and they don't seem like they should be all that difficult. Do you all have any suggestions? Also, I've tried separation of variables and it leads to a similar conundrum. Ultimately we wish to extract the spectrum from these equations, but I can't seem to figure out how to do that.
Hint.
We have
$$ (u+v) = (\lambda+2\mu)(u+v)_{xx}+\mu(u+v)_{yy}+(\lambda+\mu)(u+v)_{xy}\\ (u-v) = (\lambda+2\mu)(u-v)_{xx}+\mu(u-v)_{yy}-(\lambda+\mu)(u-v)_{xy} $$
and then
$$ p = (\lambda+2\mu)p_{xx}+\mu p_{yy}+(\lambda+\mu)p_{xy}\\ q = (\lambda+2\mu)q_{xx}+\mu q_{yy}-(\lambda+\mu)q_{xy} $$
NOTE
The $p$ equation can be stated as
$$ \left(\frac{\left(-\sqrt{\lambda ^2-2 \lambda \mu -7 \mu ^2}+\lambda +\mu \right)}{2 \mu }\partial_x+\partial_y\right) \left(\frac{1}{2} \left(\sqrt{\lambda ^2-2 \lambda \mu -7 \mu ^2}+\lambda +\mu \right)\partial_x+\mu \partial_y\right)p = p $$
The $q$ equation can be stated as
$$ \left(\frac{\left(-\sqrt{\lambda ^2-2 \lambda \mu -7 \mu ^2}-\lambda -\mu \right)}{2 \mu }\partial_x+\partial_y\right) \left(\frac{1}{2} \left(\sqrt{\lambda ^2-2 \lambda \mu -7 \mu ^2}-\lambda -\mu \right)\partial_x+\mu \partial_y\right)q = q $$