Specifically, I have this equation
$\frac{\partial^{2}u}{\partial t^{2}}(x, t) = \frac{\partial^{2}u}{\partial x^{2}}(x, t) -sen(u(x, t)); \quad L_{0} \le x \le L_{1}; \quad t \geq t_{0}$
and them I applied the finite difference method and found this
$U^{i + 1}_{j} = v^2U^{i}_{j + 1} + (2 - 2v^2)U^{i}_{j} + v^2U^{i}_{j - 1} - U^{i - 1}_{j} -k^2\sin{\left(U^{i}_{j}\right)}$
$k$ and $h$ are my discretizations in time and space respectialy, $v = \frac{k}{h}$ is the courant number, $U^{i}_{j} \approx u(x_j, t_i)$
I want to know the contitions this equation is stable. To do that, I want to linearize it and apply the Von Neumann stability analysis. Can someone help me to do that ?