Kadomtsev–Petviashvili equation

In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili, the KP equation is usually written as

where ${\displaystyle \lambda =\pm 1}$. 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.

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.

${\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxt}u)+\lambda \partial _{yy}u=0}$

where ${\displaystyle \lambda =\pm 1}$. 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 ${\displaystyle \pm \infty }$. The BBM-KP equation can be viewed as a weak transverse perturbation of the Benjamin–Bona–Mahony equation. 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 ${\displaystyle L^{2}}$ -based Sobolev space ${\displaystyle H_{x}^{k}(\mathbb {R} )}$ for all ${\displaystyle k\geq 1}$, provided their corresponding initial data are close in ${\displaystyle H_{x}^{k}(\mathbb {R} )}$ as the transverse variable ${\displaystyle y\rightarrow \pm \infty }$.