---
tag: GEJHRXC176
kind: concept
name: Nonlinear Schrödinger equation
msc: [35Q55, 35Q51]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Nonlinear_Schrödinger_equation
same_as:
  wikidata: Q2579914
  wikipedia: Nonlinear Schrödinger equation
---

## Summary

The nonlinear Schrödinger equation is the [[Schrödinger equation|EPZR6H2WJN]] 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.

## Explanations

### L1 human

A wave packet spreads out as it moves, because waves of different frequencies travel at different speeds.
In some media the speed also depends on how strong the wave is, and this adds a nonlinear term.
If the nonlinearity pulls the packet together while dispersion spreads it out, the two effects can balance.
Then a solitary wave forms that keeps its shape, such as $u(x,t) = \sqrt{2}\,\mathrm{sech}(x)\, e^{it}$ for the focusing cubic equation on the line.
Light pulses in optical fibres behave like this.