Kadomtsev–Petviashvili equation
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In mathematics, the Kadomtsev–Petviashvili equation (or KP equation) is a partial differential equation. 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.
Like the KdV equation, the KP equation is completely integrable. It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation.
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[edit] History
The KP equation was first written in 1970 by Kadomtsev and Petviashvili; 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.
[edit] 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, λ = + 1 is used; if surface tension is strong, then λ = − 1. 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).
The KP equation can also be used to model waves in ferromagnetic media, as well as two-dimensional matter–wave pulses in Bose–Einstein condensates.
[edit] Limiting behavior
For 
, typical x-dependent oscillations have a wavelength of O(1 / ε) 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 — O(ε) — in the dispersionless limit. Then the amplitude satisfies a mean-field equation of Davey–Stewartson type.
[edit] References
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This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (October 2008) |
- Kadomtsev, BB; Petviashvili, VI (1970), "On the Stability of Solitary Waves in Weakly Dispersive Media", Sov. Phys. Dokl. 15: 539–541.
- Previato, Emma (2001), "Kadomtsev–Petviashvili equation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104, http://eom.springer.de/K/k120110.htm
[edit] External links
- Weisstein, Eric W., "Kadomtsev–Petviashvili equation" from MathWorld.
- The KP page of Bernard Deconinck at the University of Washington, Department of Applied Mathematics.
- Virtual Math Labs: Dynamical Systems—a Java applet showing solutions of the KP equations, at the Technical University of Berlin.
