Kadomtsev–Petviashvili equation

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: 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 λ = ± 1 {displaystyle lambda =pm 1} . 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. 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. 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, λ = + 1 {displaystyle lambda =+1} is used; if surface tension is strong, then λ = − 1 {displaystyle lambda =-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. For ϵ ≪ 1 {displaystyle epsilon ll 1} , typical x-dependent oscillations have a wavelength of O ( 1 / ϵ ) {displaystyle O(1/epsilon )} giving a singular limiting regime as ϵ → 0 {displaystyle epsilon ightarrow 0} . The limit ϵ → 0 {displaystyle epsilon ightarrow 0} is called the dispersionless limit. If we also assume that the solutions are independent of y as ϵ → 0 {displaystyle epsilon ightarrow 0} , then they also satisfy the inviscid Burgers' equation: Suppose the amplitude of oscillations of a solution is asymptotically small — O ( ϵ ) {displaystyle O(epsilon )} — in the dispersionless limit. Then the amplitude satisfies a mean-field equation of Davey–Stewartson type.