How to linearize this in equilibrium? Consider the pde system $$ u_t=-uu_x-vu_y+fv-g\cdot (H+c)_x\\\ v_t=-uv_x-vv_y-fu-g\cdot (H+c)_y\\\ H_t=-(uH)_x-(vH)_y $$ where $u=u(x,y,t), v=v(x,y,t), H=H(x,y,t)$ and $f\in\mathbb{R}$ is a parameter, $c$ is a constant. $g$ is the gravitational acceleration. A trivial equilibrium is given by $E=(0,0,H_0)$ with $H_0$ being a constant. My question is: How do I linearize in this equilibrium? I really have no idea.

We linearize by dividing $u$, $v$, and $H$ into the following:

$$u(x,y,t)=u_0(x,y)+\epsilon u_1(x,y,t)$$ $$v(x,y,t)=v_0(x,y)+\epsilon v_1(x,y,t)$$ $$H(x,y,t)=H_0(x,y)+\epsilon H_1(x,y,t)$$

where $\epsilon$ is small. Substituting into the above, with $u_0=v_0=0$ and $H_0$ constant, and discarding terms of order $\epsilon^2$ or higher:

$$u_{1,t}=fv_1-g\cdot H_{1,x}$$ $$v_{1,t}=-fu_1-g\cdot H_{1,y}$$ $$H_{1,t}=-H_0(u_{1,x}+v_{1,y})$$

Hopefully that's a simple enough form.