For $\epsilon=\epsilon(x,\;t)$, I have
$i\epsilon_{t}+\epsilon_{xx}+2(d_{1}^2+b_{2}^2)(\epsilon+\epsilon^\ast)+4d_1b_2\epsilon+4d_1b_2^2=0, \;\;\; (1)$
where $d_1,\;b_2$ are constants, and $\epsilon_{t}$ is the derivative w.r.t $t$. If the last term in equation $(1)$ is not present then for
$\epsilon=\epsilon_1e^{i(kx-wt)}+\epsilon_2e^{-i(kx-wt)}$
we can get dispersion relation between wave number $k$ and frquency $w$. But if the last term is present in equation $(1)$ then I am not able to find the dispersion relation for frquency $w$ and wave number $k$. Do you have any ideas how to do that?