nLab Kadomtsev-Petviashvili equation

Overview

The Kadomtsev-Petviashvili equation (KP) is a particular nonlinear evolution equation for fluid dynamics which is completely integrable and has soliton solutions.

the KP equation belongs to an infinite hierarchy of completely integrable quadratic partial differential equations. This is the Kadomtsev-Petviashvili hierarchy generalizing the KdV hierarchy. A linear combination of Schur functions satisfies the KP hierarchy iff its coefficients satisfy the Pluecker relations.

Related entries

• Sato Grassmannian

• Hirota equation

• Witten conjecture

References

The original article:

• B. B. Kadomtsev, V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (1970) 539-541 [mathnet:eng/dan35447]

See also:

• Wikipedia, Kadomtsev-Petviashvili equation

• Etsuro Date, Michio Jimbo, Masaki Kashiwara, Tetsuji Miwa, Operator approach to the Kadomtsev-Petviashvili equation – Transformation groups for soliton equations. III, J. Phys. Soc. Jpn. 50 (1980) 3806-3812 [doi:10.1143/JPSJ.50.3806]

• Masaki Kashiwara, Tetsuji Miwa, The τ-function of the Kadomtsev-Petviashvili equation transformation groups for soliton equations, I, Proc. Japan Acad. Ser. A Math. Sci. 57(7): 342–347 (1981) doi

• I.P. Goulden, D.M. Jackson, The KP hierarchy, branched covers, and triangulations, Adv. Math. 219:3 (2008) 932–951 doi

• V. G. Kac, J. W. van de Leur, The $n$-component KP hierarchy and representation theory, J. Math. Phys. 44, 3245–3293 (2003) doi

Relation to positive Grassmannian and cluster algebras

• Yuji Kodama, Lauren Williams, KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. 108 (22) 8984–8989 arxiv/1105.4170 doi

• Yuji Kodama, Lauren Williams, KP solitons and total positivity for the Grassmannian, Invent. math. 198, 637–699 (2014) doi arXiv:1106.0023