Kadomtsev–Petviashvili equation explained

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$$\backslash displaystyle\; \backslash partial\_x(\backslash partial\_t\; u+u\; \backslash partial\_x\; u+\backslash epsilon^2\backslash partial\_u)+\backslash lambda\backslash partial\_u=0$$where

λ= \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 \partial _{x(\partial} _{t} u + u \partial _{x} u + \epsilon ^{2 \partial} _{xxt} u) +λ \partial _{yy} u =0 where λ= \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 \pm inf ty . 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 L ^{2} -based Sobolev space H ^{k} _{x} (\R) for all k \ge 1 , provided their corresponding initial data are close in H ^{k} _{x} (\R) as the transverse variable y → \pm inf ty .^[8]

History

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.

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,^[9] as well as two-dimensional matter–wave pulses in Bose–Einstein condensates.

Limiting behavior

For

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

If we also assume that the solutions are independent of y as

\epsilon → 0 , then they also satisfy the inviscid Burgers' equation:\displaystyle \partial _{t} u + u\partial _{x} u =0 .

Suppose the amplitude of oscillations of a solution is asymptotically small —

O(\epsilon) — in the dispersionless limit. Then the amplitude satisfies a mean-field equation of Davey–Stewartson type.

See also

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

Further reading

• B. B.. Kadomtsev. V. I.. Petviashvili. On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl.. 15. 1970. 539–541. 1970SPhD...15..539K . . Translation of Об устойчивости уединенных волн в слабо диспергирующих средах . . 192 . 753–756.
• Book: Kodama, Y. . 2017 . KP Solitons and the Grassmannians: combinatorics and geometry of two-dimensional wave patterns . Springer . 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 . 10.1063/1.532219 . 1997JMP....38.6401L .
• Minzoni . A. A. . Smyth . N. F. . 1996 . Evolution of lump solutions for the KP equation . Wave Motion . 24 . 3 . 291–305 . 10.1016/S0165-2125(96)00023-6 . 1996WaMot..24..291M . 10.1.1.585.6470 .
• Nakamura . A. . 1989 . A bilinear N-soliton formula for the KP equation . Journal of the Physical Society of Japan . 58 . 2 . 412–422 . 10.1143/JPSJ.58.412 . 1989JPSJ...58..412N .
  • 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 . 10.1088/0305-4470/37/28/006 . nlin/0412070 . 2004JPhA...37.7143X . 18500877 .

External links

• • • Web site: The KP page . Bernard Deconinck . University of Washington, Department of Applied Mathematics . 2006-02-27 . https://web.archive.org/web/20060206111407/http://www.amath.washington.edu/~bernard/kp.html . 2006-02-06 . dead .

Notes and References

1. Wazwaz . A. M. . 2007 . Multiple-soliton solutions for the KP equation by Hirota's bilinear method and by the tanh–coth method . Applied Mathematics and Computation . 190 . 1 . 633–640 . 10.1016/j.amc.2007.01.056 .
2. Cheng . Y. . Li . Y. S. . 1991 . The constraint of the Kadomtsev-Petviashvili equation and its special solutions . Physics Letters A . 157 . 1 . 22–26 . 10.1016/0375-9601(91)90403-U . 1991PhLA..157...22C .
3. Ma . W. X. . 2015 . Lump solutions to the Kadomtsev–Petviashvili equation . Physics Letters A . 379 . 36 . 1975–1978 . 10.1016/j.physleta.2015.06.061 . 2015PhLA..379.1975M .
4. Kodama . Y. . 2004 . Young diagrams and N-soliton solutions of the KP equation . Journal of Physics A: Mathematical and General . 37 . 46 . 11169–11190 . 10.1088/0305-4470/37/46/006 . nlin/0406033 . 2004JPhA...3711169K . 2071043 .
5. Deng . S. F. . Chen . D. Y. . Zhang . D. J. . 2003 . The multisoliton solutions of the KP equation with self-consistent sources . Journal of the Physical Society of Japan . 72 . 9 . 2184–2192 . 10.1143/JPSJ.72.2184 . 2003JPSJ...72.2184D .
6. Book: Ablowitz . M. J. . Segur . H. . 1981 . Solitons and the inverse scattering transform . SIAM .
7. J. L.. Bona . Jerry L. Bona . Y.. Liu . M. M.. Tom. 2002 . The Cauchy problem and stability of solitary-wave solutions for RLW-KP-type equations . Journal of Differential Equations . 185 . 2 . 437–482 . 10.1006/jdeq.2002.4171. 2002JDE...185..437B . free .
8. J. B.. Aguilar . M.M.. Tom . 2024 . Convergence of solutions of the BBM and BBM-KP model equations . Differential and Integral Equations . 37 . 3/4 . 187–206. 10.57262/die037-0304-187. free . 2204.06016 .
9. Leblond . H. . 2002 . KP lumps in ferromagnets: a three-dimensional KdV–Burgers model . Journal of Physics A: Mathematical and General . 35 . 47 . 10149–10161 . 10.1088/0305-4470/35/47/313 . 2002JPhA...3510149L .
10. Book: Zakharov, V. E. . 1994 . Dispersionless limit of integrable systems in 2+1 dimensions . Singular limits of dispersive waves . 165–174 . Springer . Boston . 0-306-44628-6 .
11. Strachan . I. A. . 1995 . The Moyal bracket and the dispersionless limit of the KP hierarchy . Journal of Physics A: Mathematical and General . 28 . 7 . 1967 . 10.1088/0305-4470/28/7/018 . hep-th/9410048 . 1995JPhA...28.1967S . 15334780 .
12. Takasaki . K. . Takebe . T. . 1995 . Integrable hierarchies and dispersionless limit . Reviews in Mathematical Physics . 7 . 5 . 743–808 . 10.1142/S0129055X9500030X . hep-th/9405096 . 1995RvMaP...7..743T . 17351327 .