The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling it, but most websites discussing it use the heat equation as an example. Here we can replace the usual
t variable with
Xi, and the usual
Rho. Now unlike the heat equation, this equation has an additional first derivative of
Xi term (the first term gives both first and second derivative of
Xi). Does it matter what finite difference scheme I use to approximate this term?
Another thing is that the equation has nonlinear terms involving
|A|^4. How can I incorporate this to the matrix equation I am going to solve? (PS:
A is complex)
I am pretty new to numerical methods so forgive me if I have neglected anything significant.