Kadomtsev–Petviashvili equation
Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili, the KP equation is usually written as
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.
[1][2][3][4][5] It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation.
[6] In 2002, the regularized version of the KP equation, naturally referred to as the Benjamin–Bona–Mahony–Kadomtsev–Petviashvili equation (or simply the BBM-KP equation), was introduced as an alternative model for small amplitude long waves in shallow water moving mainly in the x direction in 2+1 space.
The BBM-KP equation provides an alternative to the usual KP equation, in a similar way that the Benjamin–Bona–Mahony equation is related to the classical Korteweg–de Vries equation, as the linearized dispersion relation of the BBM-KP is a good approximation to that of the KP but does not exhibit the unwanted limiting behavior as the Fourier variable dual to x approaches
As a result, the solutions of their corresponding Cauchy problems share an intriguing and complex mathematical relationship.
Aguilar et al. proved that the solution of the Cauchy problem for the BBM-KP model equation converges to the solution of the Cauchy problem associated to the Benjamin–Bona–Mahony equation in the
, provided their corresponding initial data are close in
[8] 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.
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,
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,[9] as well as two-dimensional matter–wave pulses in Bose–Einstein condensates.
, typical x-dependent oscillations have a wavelength of
giving a singular limiting regime as
, then they also satisfy the inviscid Burgers' equation: Suppose the amplitude of oscillations of a solution is asymptotically small —
Then the amplitude satisfies a mean-field equation of Davey–Stewartson type.
