---
tag: AYMJ2NFR4W
kind: concept
name: Dispersive partial differential equation
msc: [35Q55, 35Q53, 35B40]
sources:
  - url: https://en.wikipedia.org/wiki/Dispersive_partial_differential_equation
same_as:
  wikidata: Q5282591
  wikipedia: Dispersive partial differential equation
---

## Summary

A dispersive partial differential equation is an evolution equation whose plane waves travel at speeds that depend on their frequency.
A localized solution, made of many frequencies, therefore spreads out in space and decays in amplitude while its energy is conserved.
The main examples are the [[Schrödinger equation|EPZR6H2WJN]], the [[Korteweg–De Vries equation|KTT75QC72Z]] and the [[Klein–Gordon equation|AYCCDQWCJF]].
Dispersion competes with nonlinear effects, and the balance between the two produces phenomena such as [[solitons|63D4AXYCT4]].

## Explanations

### L1 human

Throw a stone into a pond: the ring of waves spreads into a train of ripples, because long and short ripples travel at different speeds.
This separation by frequency is called dispersion.
In a dispersive equation, a bump does not keep its shape; it spreads out and gets lower.
For the Schrödinger equation $i u_t + u_{xx} = 0$, a plane wave $e^{i(kx - k^2 t)}$ has frequency $k^2$, and a packet of such waves near $k$ moves with speed $2k$, so fast and slow parts separate.