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 $${\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0}$$ 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.^{[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.^[7]

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

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 ${\displaystyle y\rightarrow \pm \infty }$.^[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, ${\displaystyle \lambda =+1}$ is used; if surface tension is strong, then ${\displaystyle \lambda =-1}$. 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,^[9] as well as two-dimensional matter–wave pulses in Bose–Einstein condensates.

For ${\displaystyle \epsilon \ll 1}$, typical x-dependent oscillations have a wavelength of ${\displaystyle O(1/\epsilon )}$ giving a singular limiting regime as ${\displaystyle \epsilon \rightarrow 0}$. The limit ${\displaystyle \epsilon \rightarrow 0}$ is called the dispersionless limit.^[10][11][12]

If we also assume that the solutions are independent of y as ${\displaystyle \epsilon \rightarrow 0}$, then they also satisfy the inviscid Burgers' equation:

${\displaystyle \displaystyle \partial _{t}u+u\partial _{x}u=0.}$

Suppose the amplitude of oscillations of a solution is asymptotically small — ${\displaystyle O(\epsilon )}$ — in the dispersionless limit. Then the amplitude satisfies a mean-field equation of Davey–Stewartson type.

• Novikov–Veselov equation
• Schottky problem
• Dispersionless KP equation

• 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.
• Kodama, Y. (2017). KP Solitons and the Grassmannians: combinatorics and geometry of two-dimensional wave patterns. Springer. ISBN 978-981-10-4093-1.
• Lou, S. Y.; Hu, X. B. (1997). "Infinitely many Lax pairs and symmetry constraints of the KP equation". Journal of Mathematical Physics. 38 (12): 6401–6427. Bibcode:1997JMP....38.6401L. doi:10.1063/1.532219.
• Minzoni, A. A.; Smyth, N. F. (1996). "Evolution of lump solutions for the KP equation". Wave Motion. 24 (3): 291–305. Bibcode:1996WaMot..24..291M. CiteSeerX 10.1.1.585.6470. doi:10.1016/S0165-2125(96)00023-6.
• Nakamura, A. (1989). "A bilinear N-soliton formula for the KP equation". Journal of the Physical Society of Japan. 58 (2): 412–422. Bibcode:1989JPSJ...58..412N. doi:10.1143/JPSJ.58.412.
• Previato, Emma (2001) [1994], "KP-equation", Encyclopedia of Mathematics, EMS Press
• Xiao, T.; Zeng, Y. (2004). "Generalized Darboux transformations for the KP equation with self-consistent sources". Journal of Physics A: Mathematical and General. 37 (28): 7143. arXiv:nlin/0412070. Bibcode:2004JPhA...37.7143X. doi:10.1088/0305-4470/37/28/006. S2CID 18500877.

• Weisstein, Eric W. "Kadomtsev–Petviashvili equation". MathWorld.
• Gioni Biondini and Dmitri Pelinovsky (ed.). "Kadomtsev–Petviashvili equation". Scholarpedia.
• Bernard Deconinck. "The KP page". University of Washington, Department of Applied Mathematics. Archived from the original on 2006-02-06. Retrieved 2006-02-27.