Consider the following PDE system:
$$ \frac{\partial u}{\partial t}=-u\frac{\partial u}{\partial x}-v\frac{\partial u}{\partial y}+fv-g(\frac{\partial (H+h)}{\partial x}),\\ \frac{\partial v}{\partial t}=-u\frac{\partial v}{\partial x}-v\frac{\partial v}{\partial y}-fu-g(\frac{\partial (H+h)}{\partial y}),\\ \frac{\partial H}{\partial t}=-\frac{\partial uH}{\partial x}-\frac{\partial vH}{\partial y}, $$ with functions
$u(x,y,t), v(x,y,t)$ (=velocity in the $x$-, $y$-direction)
$H(x,y,t)$ (=depth of fluid)
$h(x,y)$ (=surface of the bottom)
and parameter $f\in\mathbb{R}$, gravitational acceleration $g$.
Assume $h$ and $H_0$ to be constant. Then $E=(0,0,H_0)^T$ is an equilibrium.
The question is whether this equilibrium is stable with respect to wave perturbations. To this end linearize the system in $E$ and insert solutions of the form $$ (u,v,H)^T=e^{i(kx+ly)+\lambda t}\cdot (u_0,v_0,z_0)^T. $$
This yields $$ \lambda \eta = A(k,l)\eta,~~~\text{ where }A(k,l)=\begin{pmatrix}0 & f & -gik\\-f & 0 & -gil\\-H_0ik & -H_0il & 0 \end{pmatrix}, \eta=(u,v,H). $$
Hence, $\lambda$ is eigenvalue of $A(k,l)$. This gives the dispersion relation $\lambda=\lambda(k,l)$ to be $$ \lambda (\lambda^2+gH_0(k^2+l^2)+f^2)=0 $$ having for $k,l\in\mathbb{R}$ solutions $$ \lambda=0,~~\lambda=\pm i\sqrt{gH_0(k^2+l^2)+f^2} $$
I have some questions to this.
(1) This is the computation of the point spectrum of the linear operator $A(k,l)$, isn't it?
(2) It is said that $E$ is stable if $Re(\lambda(k,l))\leq 0$. Up to this, it would be stable for $f\in\mathbb{R}, g,H_0\in\mathbb{R}^+$ since the computed eigenvalues have zero real part then. The condition $Re(\lambda(k,l))\leq 0$ for stability is strange to me: I always thought that the eigenvalues have to live in the left half-side(i.e. Re <0) to have stability with respect to the point spectrum?
(3) What is with the essential spectrum here? I guess $A(k,l)$ is a linear operator from $\mathbb{C}^3$ to itself and hence a linear operator on a finite dim. space, hence its spectrum only consists of the point spectrum?
(4) Why are solutions of the form $(u,v,H)^T=e^{i(kx+ly)+\lambda t}\cdot (u_0,v_0,z_0)^T$ called linear wave-solutions: What is linear about them? - Or are they only called linear waves since we insert them in a linear system?
(5) Is there a special reason why we insert solutions of the form $(u,v,H)^T=e^{i(kx+ly)+\lambda t}\cdot (u_0,v_0,z_0)^T$ and not solutions of the form
$$ (u,v,H)^T=e^{i(kx+ly-\lambda t)}\cdot (u_0,v_0,z_0)^T $$ which would be the typical 2D-plane wave ansatz?