Trial solution for modulational instability

Question

I have a simple linear differential equation of the following form:

$\frac{\partial a}{\partial z} + i\beta_2\frac{\partial^2 a}{\partial t^2} = i\gamma P(a+a^*)$

I seek the solution to $a(z,t)$, and $a^*$ is the complex conjugate. $\beta_2$, $\gamma$ and $P$ are all constants, and $i$ is the imaginary unit.

Apparently, the following trial solution is valid:

$a = c_1\exp[i(kz-\omega t)] + c_1^*\exp[-i(kz-\omega t)]$

if the following dispersion relation is satisfied:

$k = \pm \sqrt{\beta_2^2\omega^4 + 2\gamma P\beta_2\omega^2}$

However, I am unable to validate this. When I plug the trial solution into the differential equation, there is no way that a $k^2$ pops out, which would eventually let me get a dispersion relation of the form shown (in a square root).

This problem is related to modulation instabilities (MI) in nonlinear optical systems. The wiki page on MI makes the claim that this solution is valid, but does not provide a proof, and the book referenced by that page does the same thing, without proof. Any ideas?

Answer 1

Substiuting trial solution in the differential equation. Compare +ve and -ve exponential part. I got 2 equation(1 & 2), multiply them and find k^2. To match with dispersion relation, subtract these two equations. After that, you can figure out by yourself.