I am dealing with the Nonlinear Schrodinger Equation and my task is to analyse the stability of the equilibrium solution.
The equation in fiber optics takes the following form:
$u_z = \phi u + \frac{1}{2}u_{tt} + |u|^2u$.
where $u$ is a vector in the $t$ axis and $\phi$ is a constant.
For finding the equilibrium point, I set $u_z=0$ and calculated the solution numerically. That works well.
A dissertation I am reading said that I have to linearize the above equation about the fixed point as :
$\frac{\partial}{\partial z}\Delta u = \mathscr{L} (\Delta u)$ where $\mathscr{L}$ is the linearized operator. The dissertation then says that I have to evaluare $\mathscr{L}$ at the fixed point and find the eigenvalues. This works.
But my question is - why not calculate the Jacobian of the right side of the nonlinear equation and then calculate its eigenvalues instead? I tried that, verified the accuracy of the Jacobian by testing its second order convergence and calculated the eigenvalues. It wasn't the same as the eigenvalues of the linearized operator.
So what is the fundamental difference between the linearized operator and the Jacobian of the nonlinear equation?