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 . 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]
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, is used; if surface tension is strong, then . 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,[7] as well as two-dimensional matter–wave pulses in Bose–Einstein condensates.
Limiting behavior
For , typical x-dependent oscillations have a wavelength of giving a singular limiting regime as . The limit is called the dispersionless limit.[8][9][10]
If we also assume that the solutions are independent of y as , then they also satisfy the inviscid Burgers' equation:
Suppose the amplitude of oscillations of a solution is asymptotically small — — in the dispersionless limit. Then the amplitude satisfies a mean-field equation of Davey–Stewartson type.
References
- 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. doi:10.1016/j.amc.2007.01.056.
- Cheng, Y.; Li, Y. S. (1991). "The constraint of the Kadomtsev-Petviashvili equation and its special solutions". Physics Letters A. 157 (1): 22–26. doi:10.1016/0375-9601(91)90403-U.
- Ma, W. X. (2015). "Lump solutions to the Kadomtsev–Petviashvili equation". Physics Letters A. 379 (36): 1975–1978. doi:10.1016/j.physleta.2015.06.061.
- Kodama, Y. (2004). "Young diagrams and N-soliton solutions of the KP equation". Journal of Physics A: Mathematical and General. 37 (46): 11169. arXiv:nlin/0406033. doi:10.1088/0305-4470/37/46/006.
- 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. doi:10.1143/JPSJ.72.2184.
- Ablowitz, M. J.; Segur, H. (1981). Solitons and the inverse scattering transform. SIAM.
- Leblond, H. (2002). "KP lumps in ferromagnets: a three-dimensional KdV–Burgers model". Journal of Physics A: Mathematical and General. 35 (47): 10149. doi:10.1088/0305-4470/35/47/313.
- Zakharov, V. E. (1994). "Dispersionless limit of integrable systems in 2+1 dimensions". Singular limits of dispersive waves. Boston: Springer. pp. 165–174. ISBN 0-306-44628-6.
- 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. arXiv:hep-th/9410048. doi:10.1088/0305-4470/28/7/018.
- Takasaki, K.; Takebe, T. (1995). "Integrable hierarchies and dispersionless limit". Reviews in Mathematical Physics. 7 (5): 743–808. arXiv:hep-th/9405096. doi:10.1142/S0129055X9500030X.
Further reading
- 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. 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. 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. 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. doi:10.1088/0305-4470/37/28/006.
External links
- 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.