Consider the following initial value problem:
$i\psi_t = -\psi_{xx} - 2|\psi|^2\psi$ with $x\in[0,2\pi)$ and $t\geq 0$, and $\psi(x,0) = \frac{3}{2}\left(1 - \frac{1}{10}\cos(x-\pi)\right)$.
The above in its more general form is called the Nonlinear Schrödinger's Equation. What I would like to know if the above can be solved analytically. I know that solutions to this problem exist, but it is not clear to me whether methods exist to produce solutions, given the boundary conditions.
Context of the question: I have to solve this thing numerically and I'd like to know more about the equation.