nLab Kadomtsev-Petviashvili equation

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

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.

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:

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

Last revised on June 16, 2024 at 18:58:29. See the history of this page for a list of all contributions to it.