Crossing
swells, consisting of near-cnoidal wave trains. Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of
Île de Ré (Isle of Rhé), France, in the
Atlantic Ocean. The interaction of such near-
solitons in shallow water may be modeled through the Kadomtsev–Petviashvili equation.
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.
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]
where . 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 . 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 -based
Sobolev space for all , provided their corresponding initial data are close in as the transverse variable .[8]
History
Boris Kadomtsev.
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).
For , typical x-dependent oscillations have a wavelength of giving a singular limiting regime as . The limit is called the
dispersionless limit.[10][11][12]
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.
^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.
^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.
Bibcode:
2003JPSJ...72.2184D.
doi:
10.1143/JPSJ.72.2184.
^Ablowitz, M. J.; Segur, H. (1981). Solitons and the inverse scattering transform. SIAM.
^Zakharov, V. E. (1994). "Dispersionless limit of integrable systems in 2+1 dimensions". Singular limits of dispersive waves. Boston: Springer. pp. 165–174.
ISBN0-306-44628-6.
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.
ISBN978-981-10-4093-1.
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.