---
tag: P93R13XP0P
kind: concept
name: Kadomtsev–Petviashvili equation
msc: [35Q53, 35Q51]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Kadomtsev–Petviashvili_equation
same_as:
  wikidata: Q6345653
  wikipedia: Kadomtsev–Petviashvili equation
---

## Summary

The Kadomtsev–Petviashvili equation, or KP equation, is a nonlinear wave equation in two space variables $x$ and $y$.
It extends the [[Korteweg–De Vries equation|KTT75QC72Z]] to waves that move mainly in the $x$ direction but vary slowly in the $y$ direction.
It comes in two versions, KP-I and KP-II, which differ by the sign of the $u_{yy}$ term.
Like the KdV equation it is completely integrable.

## Explanations

### L1 human

The KdV equation describes waves in a narrow canal, where the water moves along one line.
On a wide sea surface a wave crest is a long line, and it can bend a little.
The KP equation adds a small term that accounts for this sideways bending.
Every KdV wave that does not depend on $y$, such as a soliton with a straight crest, is also a KP wave.
The two sign choices model water with weak surface tension (KP-II) and with strong surface tension (KP-I).