Nonlinear Schrödinger equation

Summary

The nonlinear Schrödinger equation is the Schrödinger equation for a complex wave $u$, with an added term that depends on the size $\abs{u}$. Its most studied case is the cubic equation $i \partial_t u + \Delta u = \pm \abs{u}^2 u$. It describes slowly varying wave packets in optics, water waves and Bose–Einstein condensates. It is a model problem for dispersive equations, and in one space dimension the cubic case is completely integrable.