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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,
502:. 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
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420:{\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxt}u)+\lambda \partial _{yy}u=0}
195:{\displaystyle \displaystyle \partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0}
1868:
694:
in the KP equation this restriction is relaxed. Still, both in the KdV and the KP equation, waves have to travel in the positive
1118:
Wazwaz, A. M. (2007). "Multiple-soliton solutions for the KP equation by Hirota's bilinear method and by the tanh–coth method".
1836:
1619:
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Deng, S. F.; Chen, D. Y.; Zhang, D. J. (2003). "The multisoliton solutions of the KP equation with self-consistent sources".
503:
499:
460:
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24:, consisting of near-cnoidal wave trains. Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of
792:-terms enter the equation, the waves described by the KP equation behave differently in the direction of propagation (
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263:
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1747:
Xiao, T.; Zeng, Y. (2004). "Generalized
Darboux transformations for the KP equation with self-consistent sources".
286:
72:
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Cheng, Y.; Li, Y. S. (1991). "The constraint of the
Kadomtsev-Petviashvili equation and its special solutions".
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Kadomtsev, B. B.; Petviashvili, V. I. (1970). "On the stability of solitary waves in weakly dispersive media".
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293:), was introduced as an alternative model for small amplitude long waves in shallow water moving mainly in the
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Lou, S. Y.; Hu, X. B. (1997). "Infinitely many Lax pairs and symmetry constraints of the KP equation".
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459:. The BBM-KP equation provides an alternative to the usual KP equation, in a similar way that the
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Like the KdV equation, the KP equation is completely integrable. It can also be solved using the
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KP Solitons and the
Grassmannians: combinatorics and geometry of two-dimensional wave patterns
1465:
Strachan, I. A. (1995). "The Moyal bracket and the dispersionless limit of the KP hierarchy".
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247:. To be physically meaningful, the wave propagation direction has to be not-too-far from the
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Minzoni, A. A.; Smyth, N. F. (1996). "Evolution of lump solutions for the KP equation".
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Zakharov, V. E. (1994). "Dispersionless limit of integrable systems in 2+1 dimensions".
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Leblond, H. (2002). "KP lumps in ferromagnets: a three-dimensional KdV–Burgers model".
1375:
1323:"The Cauchy problem and stability of solitary-wave solutions for RLW-KP-type equations"
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1075:— in the dispersionless limit. Then the amplitude satisfies a mean-field equation of
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Takasaki, K.; Takebe, T. (1995). "Integrable hierarchies and dispersionless limit".
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In 2002, the regularized version of the KP equation, naturally referred to as the
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Kodama, Y. (2004). "Young diagrams and N-soliton solutions of the KP equation".
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Suppose the amplitude of oscillations of a solution is asymptotically small —
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498:. The BBM-KP equation can be viewed as a weak transverse perturbation of the
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in shallow water may be modeled through the
Kadomtsev–Petviashvili equation.
1826:
1347:
1322:
1180:
Ma, W. X. (2015). "Lump solutions to the
Kadomtsev–Petviashvili equation".
1697:
Nakamura, A. (1989). "A bilinear N-soliton formula for the KP equation".
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231:. The above form shows that the KP equation is a generalization to two
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1595:"Об устойчивости уединенных волн в слабо диспергирующих средах".
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33:
16:
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1364:"Convergence of solutions of the BBM and BBM-KP model equations"
1036:{\displaystyle \displaystyle \partial _{t}u+u\partial _{x}u=0.}
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direction, i.e. with only slow variations of solutions in the
605:, provided their corresponding initial data are close in
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If we also assume that the solutions are independent of
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media, as well as two-dimensional matter–wave pulses in
804:-direction tend to be smoother (be of small-deviation).
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1843:, Department of Applied Mathematics. Archived from
807:The KP equation can also be used to model waves in
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1467:Journal of Physics A: Mathematical and General
1407:Journal of Physics A: Mathematical and General
1217:Journal of Physics A: Mathematical and General
1834:
1305:Solitons and the inverse scattering transform
1302:
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853:-dependent oscillations have a wavelength of
1820:Gioni Biondini and Dmitri Pelinovsky (ed.).
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755:is used; if surface tension is strong, then
714:with weakly non-linear restoring forces and
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1699:Journal of the Physical Society of Japan
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1270:Journal of the Physical Society of Japan
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670:{\displaystyle y\rightarrow \pm \infty }
75:, the KP equation is usually written as
15:
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784:. Because of the asymmetry in the way
641:{\displaystyle H_{x}^{k}(\mathbb {R} )}
572:{\displaystyle H_{x}^{k}(\mathbb {R} )}
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1444:. Boston: Springer. pp. 165–174.
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979:, then they also satisfy the inviscid
972:{\displaystyle \epsilon \rightarrow 0}
935:{\displaystyle \epsilon \rightarrow 0}
909:{\displaystyle \epsilon \rightarrow 0}
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890:giving a singular limiting regime as
706:The KP equation can be used to model
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1442:Singular limits of dispersive waves
1368:Differential and Integral Equations
1303:Ablowitz, M. J.; Segur, H. (1981).
1120:Applied Mathematics and Computation
13:
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1362:Aguilar, J. B.; Tom, M.M. (2024).
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1822:"Kadomtsev–Petviashvili equation"
1806:"Kadomtsev–Petviashvili equation"
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1327:Journal of Differential Equations
800:) direction; oscillations in the
245:Korteweg–de Vries (KdV) equation
73:Vladimir Iosifovich Petviashvili
1631:Journal of Mathematical Physics
1520:Reviews in Mathematical Physics
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1433:
50:Kadomtsev–Petviashvili equation
32:. The interaction of such near-
1869:Partial differential equations
1398:
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1321:; Liu, Y.; Tom, M. M. (2002).
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1202:10.1016/j.physleta.2015.06.061
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883:{\displaystyle O(1/\epsilon )}
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842:{\displaystyle \epsilon \ll 1}
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452:{\displaystyle \lambda =\pm 1}
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264:nonlinear Schrödinger equation
224:{\displaystyle \lambda =\pm 1}
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28:(Isle of Rhé), France, in the
1:
1690:10.1016/S0165-2125(96)00023-6
1104:
504:Benjamin–Bona–Mahony equation
500:Benjamin–Bona–Mahony equation
461:Benjamin–Bona–Mahony equation
58:partial differential equation
1167:10.1016/0375-9601(91)90403-U
1068:{\displaystyle O(\epsilon )}
796:-direction) and transverse (
463:is related to the classical
260:inverse scattering transform
7:
1889:Equations of fluid dynamics
1779:10.1088/0305-4470/37/28/006
1736:Encyclopedia of Mathematics
1427:10.1088/0305-4470/35/47/313
1247:10.1088/0305-4470/37/46/006
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777:{\displaystyle \lambda =-1}
748:{\displaystyle \lambda =+1}
648:as the transverse variable
491:{\displaystyle \pm \infty }
10:
1905:
1598:Doklady Akademii Nauk SSSR
1497:10.1088/0305-4470/28/7/018
1099:Dispersionless KP equation
680:
465:Korteweg–de Vries equation
69:Boris Borisovich Kadomtsev
1550:10.1142/S0129055X9500030X
1132:10.1016/j.amc.2007.01.056
813:Bose–Einstein condensates
243:, of the one-dimensional
1841:University of Washington
1391:10.57262/die037-0304-187
1089:Novikov–Veselov equation
297:direction in 2+1 space.
289:equation (or simply the
1874:Exactly solvable models
598:{\displaystyle k\geq 1}
1348:10.1006/jdeq.2002.4171
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702:Connections to physics
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1835:Bernard Deconinck.
1771:2004JPhA...37.7143X
1719:10.1143/JPSJ.58.412
1711:1989JPSJ...58..412N
1672:1996WaMot..24..291M
1643:1997JMP....38.6401L
1610:Kodama, Y. (2017).
1589:1970SPhD...15..539K
1542:1995RvMaP...7..743T
1489:1995JPhA...28.1967S
1419:2002JPhA...3510149L
1413:(47): 10149–10161.
1339:2002JDE...185..437B
1282:2003JPSJ...72.2184D
1239:2004JPhA...3711169K
1223:(46): 11169–11190.
1194:2015PhLA..379.1975M
1159:1991PhLA..157...22C
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1879:Integrable systems
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233:spatial dimensions
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1637:(12): 6401–6427.
1621:978-981-10-4093-1
1593:. Translation of
1188:(36): 1975–1978.
1182:Physics Letters A
1147:Physics Letters A
981:Burgers' equation
819:Limiting behavior
471:variable dual to
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1847:on 2006-02-06
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1837:"The KP page"
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