We consider the Schrödinger equation with a cubic nonlinearity and periodic boundary conditions

\[ \mathrm{i} u_t + u_{xx} = \lambda |u|^2\,u,\quad u(-\pi,t)=u(\pi,t),\quad u(x,0)=f(x). \]

The solution \(u=u(x,t)\) is a complex function of two real variebales \(x\) and \(t\) , og \(\mathrm{i}=\sqrt{-1}\). The constant \(\lambda\) kan be both positive and negative.

Google: nonlinear Schrödinger equation, cubic Schrödinger equation, solitons, nonlinear wave equations, completely integrable systems, Lax pair, optical fibers, modulational instability, dispersive waves