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In mathematics and physics, the Kadomtsev–Petviashvili equation – or KP equation, named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili – is a partial differential equation to describe nonlinear wave motion. The KP equation is usually written as:
where . The above form shows that the KP equation is a generalization to two spatial dimensions, x and y, of the one-dimensional Korteweg–de Vries (KdV) equation. To be physically meaningful, the wave propagation direction has to be not-too-far from the x direction, i.e. with only slow variations of solutions in the y direction.
The KP equation was first written in 1970 by Soviet physicists Boris B. Kadomtsev (1928–1998) and Vladimir I. Petviashvili (1936–1993); it came as a natural generalization of the KdV equation (derived by Korteweg and De Vries in 1895). Whereas in the KdV equation waves are strictly one-dimensional, in the KP equation this restriction is relaxed. Still, both in the KdV and the KP equation, waves have to travel in the positive x-direction.
Connections to physics
The KP equation can be used to model water waves of long wavelength with weakly non-linear restoring forces and frequency dispersion. If surface tension is weak compared to gravitational forces, is used; if surface tension is strong, then . Because of the asymmetry in the way x- and y-terms enter the equation, the waves described by the KP equation behave differently in the direction of propagation (x-direction) and transverse (y) direction; oscillations in the y-direction tend to be smoother (be of small-deviation).
For , typical x-dependent oscillations have a wavelength of giving a singular limiting regime as . The limit is called the dispersionless limit.
If we also assume that the solutions are independent of y as , then they also satisfy Burgers' equation:
Suppose the amplitude of oscillations of a solution is asymptotically small — — in the dispersionless limit. Then the amplitude satisfies a mean-field equation of Davey–Stewartson type.
- Kadomtsev, B. B.; Petviashvili, V. I. (1970). "On the stability of solitary waves in weakly dispersive media". Sov. Phys. Dokl. 15: 539–541. Bibcode:1970SPhD...15..539K.. Translation of "Об устойчивости уединенных волн в слабо диспергирующих средах". Doklady Akademii Nauk SSSR 192: 753–756.
- Previato, Emma (2001), "K/k120110", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4