Kadomtsev-Petviashvili equation
The Kadomtsev-Petviashvili equation (KP equation) is a nonlinear partial differential equation in two space and one time dimensions, which was established in 1970 by Boris Borissowitsch Kadomzew and Vladimir Iossifowitsch Petwiaschwili in plasma physics . It has soliton solutions similar to the Korteweg-de-Vries equation (KdV) in one spatial dimension. Like the KdV equation, the equation can be solved exactly.
description
The Kadomtsev-Petviashvili equation has the form:
The expression in brackets corresponds to the KdV equation.
- If the parameter is , one speaks of a KP equation of type 1 (KPI); treats z. B. Water waves with high surface tension .
- If the parameter is , one speaks of a KP equation of type 2 (KPII); treats z. B. Water waves with low surface tension.
The soliton solutions show different behavior in the two cases.
Originally, Kadomzew and Petwiashvili used the Kadomtsev-Petviashvili equation to describe acoustic waves in the plasma of small amplitude and large wavelength that were exposed to transverse perturbations (i.e. perturbations in the y-direction, perpendicular to the direction of propagation in the x-direction). Without transverse disturbances, the dynamics were described by the Korteweg-de-Vries equation . The authors showed that the KdV equation was stable under small transverse perturbations as long as the medium had negative dispersion (decrease in phase velocity with the wavenumber for small perturbations); this was the case of the KPII (i.e. Type 2) equation. The type 1 equation KPI, on the other hand, corresponded to the case of positive dispersion and showed instability of the KdV solitons.
Applications
Mark J. Ablowitz and Harvey Segur later described water waves with the Kadomtsev-Petviashvili equation and found applications in nonlinear optics , ferromagnetism , Bose-Einstein condensates and string theory . In the case of type 2 CP solitons, multisoliton solutions result in the form of line solitons that intersect in two dimensions (see Fig.).
The KP equation found a surprising application in algebraic geometry and complex analysis when Takahiro Shiota succeeded in solving the Schottky problem with it in 1986 . The aim is to characterize the complex tori , which are Jacobi varieties of algebraic curves . In 1976 Igor Krichever had shown that Riemann theta functions for Jacobi varieties, understood as tau functions as defined above, are solutions of the KP equation. Sergei Nowikow suspected that the reverse is also true and that the solutions of the KP equation thus solve the Schottky problem, which was proven by Shiota.
Solution method
Like the KdV equation, the KP equation can be solved by inverse scattering transformation (IST). The IST first applied Vladimir Evgenyevich Sakharov and Alexei Schabat in 1974 to more than one spatial dimension and thus demonstrated the exact solvability of the KP equation. Before that, Valery Dryuma had shown in the same year that the KP equation allows formulation as a lax pair , which was an indication of exact integrability .
The KP equation can also be solved with the direct method of Ryōgo Hirota (1971), in which a variable transformation is carried out on the tau function .
Building on the work of Mikio Satō , who in 1981 described the space of the solutions to the KP equation as an infinite-dimensional Grassmann manifold , expanded by Graeme Segal and George Wilson in 1986, the KP equation can be formulated as the simplest example of a hierarchy of nonlinear equations ( with a generalized Lax equation ) (KP hierarchy).