I have the following equation -
$u_z = \frac{i}{2}u_{tt} + i|u|^2u$
where the subscripts denote partial derivatives with respect to the corresponding variable. I will be doing the stability analysis numerically, so for me, $u$ is a vector.
This equation has a solution $u(z,t) = sech(t)exp(iz/2)$.
My task is to determine the stability of the system for this solution.
I started by linearizing the equation as follows -
As $u$ is complex, let $u = v + iw$
Then - $$ \partial_{z}\left[\begin{array}{l} v \\ w \end{array}\right]=\frac{-\beta}{2} J \partial_{t}^{2}\left[\begin{array}{c} v \\ w \end{array}\right]+\gamma\left(v^{2}+w^{2}\right) J\left[\begin{array}{l} v \\ w \end{array}\right] $$ where $\mathrm{J}=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right] .$
Therefore -
$$ \partial_{z}\left[\begin{array}{c} v+\varepsilon \Delta v \\ w+\varepsilon \Delta w \end{array}\right]=\frac{-\beta}{2} J \partial_{t}^{2}\left[\begin{array}{c} v+\varepsilon \Delta v \\ w+\varepsilon \Delta w \end{array}\right]+\gamma\left((v+\varepsilon \Delta v)^{2}+(w+\varepsilon \Delta w)^{2}\right) J\left[\begin{array}{c} v+\varepsilon \Delta v \\ w+\varepsilon \Delta w \end{array}\right] $$ Keeping terms with $\mathcal{O}(\varepsilon)$ and simplifying, $$ \partial_{z}\left[\begin{array}{c} v+\varepsilon \Delta v \\ w+\varepsilon \Delta w \end{array}\right]=\partial_{z}\left[\begin{array}{c} v \\ w \end{array}\right]+\varepsilon\left(\frac{-\beta}{2} J \partial_{t}^{2}\left[\begin{array}{c} \Delta v \\ \Delta w \end{array}\right]+\gamma\left(v^{2}+w^{2}\right) J\left[\begin{array}{c} \Delta v \\ \Delta w \end{array}\right]+2 \gamma(w \Delta v+w \Delta w) \mathrm{J}\left[\begin{array}{c} v \\ w \end{array}\right]\right) $$
Thus, $$ \begin{aligned} \partial_{z}\left[\begin{array}{l} \Delta v \\ \Delta w \end{array}\right] &=\lim _{\varepsilon \rightarrow 0} \frac{\left[\begin{array}{c} v+\varepsilon \Delta v \\ w+\varepsilon \Delta w \end{array}\right]-\partial_{z}\left[\begin{array}{c} v \\ w \end{array}\right]}{\varepsilon} \\ &=\frac{-\beta}{2} J \partial_{t}^{2}\left[\begin{array}{c} \Delta v \\ \Delta w \end{array}\right]+\gamma\left(v^{2}+w^{2}\right) J\left[\begin{array}{c} \Delta v \\ \Delta w \end{array}\right]+2 \gamma \mathrm{J}\left[\begin{array}{c} v \\ w \end{array}\right]\left[\begin{array}{ll} v & w \end{array}\right]\left[\begin{array}{l} \Delta v \\ \Delta w \end{array}\right] . \end{aligned} $$ Therefore, the linearized equation is $$ \frac{\partial}{\partial z}\left[\begin{array}{c} \Delta v \\ \Delta w \end{array}\right]=\mathscr{L}_{S M F}(v, w)\left[\begin{array}{c} \Delta v \\ \Delta w \end{array}\right] $$
where
\begin{aligned} &\mathscr{L}_{SMF}(v, w)=\mathrm{L}+\mathrm{M}_{1}(v, w)+\mathrm{M}_{2}(v, w) \\ &\mathrm{L}=-\frac{\beta}{2}J \partial_{t}^{2} \\ &\mathrm{M}_{1}(v, w)=\gamma\left(v^{2}+w^{2}\right) \mathrm{J} \\ &\mathrm{M}_{2}(v, w)=2 \gamma \mathrm{J}\left[\begin{array}{l} v \\ w \end{array}\right]\left[\begin{array}{ll} v & w \end{array}\right] \end{aligned}
And now I am stuck here. I have never dealt with PDEs before and I have no idea how to proceed. Any help would be appreciated