# Band structure of nonlinear Schrodinger equation with one dimensional potential

I have a nonlinear Schrodinger equation which reads:

$$\frac{1}{2} \frac{d^2u}{dx^2}+ |u|^2u + V(x)u = -i \frac{du}{dz},$$

where $V(x)=\cos(wx)+ i a \sin(wx)$ and $w$, $a$ are numbers.

How to solve its band structure, meaning its dispersion relation?

The plane wave expansion method is used for a real potential. Can it be applied for this complex potential also?

What are other methods, if any, in Matlab or Mathematica or other software?

• welcome to scicomp! please use mathjax to format your equations. – GoHokies Dec 21 '17 at 10:48
• I think that you should not have a source term to compute the dispersion relation – nicoguaro Dec 21 '17 at 13:59
• @ Kirill : Yes you're right, it was just a typo. – foi Dec 22 '17 at 11:42