About: Tau function (integrable systems)

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Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form. The term Tau function, or -function, was first used systematically by Mikio Sato and his students in the specific context of the Kadomtsev–Petviashvili (or KP) equation, and related integrable hierarchies. It is a central ingredient in the theory of solitons. Tau functions also appear as matrix model partition functions in the spectral theory of Random Matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to mo

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rdfs:label	Tau function (integrable systems) (en)
rdfs:comment	Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form. The term Tau function, or -function, was first used systematically by Mikio Sato and his students in the specific context of the Kadomtsev–Petviashvili (or KP) equation, and related integrable hierarchies. It is a central ingredient in the theory of solitons. Tau functions also appear as matrix model partition functions in the spectral theory of Random Matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to mo (en)
foaf:depiction
dcterms:subject	Generating functions Partition functions Integrable systems Dynamical systems Special functions Mathematical physics
Wikipage page ID	65820239 (xsd:integer)
Wikipage revision ID	1123856986 (xsd:integer)
Link from a Wikipage to another Wikipage	Quantum field theory Quantum mechanics Schur polynomial Enumerative geometry Normal ordering Generating functions Peter Lax ELSV formula Integrable system Kadomtsev-Petviashvili equation Period mapping Generating function Graeme Segal Branched covering Moduli spaces Monodromy Theta function Plucker embedding Combinatorics Hamilton–Jacobi equation Overdetermined system Stokes parameters Divisor Exterior algebra Ferdinand Georg Frobenius Partition (number theory) Partition function (statistical mechanics) Differential of the first kind Dirac sea Partition functions Fock space Isospectral Kadomtsev–Petviashvili equation Isomonodromic deformation Statistical mechanics Integrable systems Abel–Jacobi map Dynamical systems Special functions Mathematical physics Schottky problem Plucker coordinates Solitons Grassmann manifold Schlesinger equations Mikio Sato Random matrix Hamilton's principal function Fredholm integral operator Young diagram Plucker relations
sameAs	Tau function (integrable systems) Tau function (integrable systems)
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has abstract	Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form. The term Tau function, or -function, was first used systematically by Mikio Sato and his students in the specific context of the Kadomtsev–Petviashvili (or KP) equation, and related integrable hierarchies. It is a central ingredient in the theory of solitons. Tau functions also appear as matrix model partition functions in the spectral theory of Random Matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to moduli spaces of Riemann surfaces, and enumeration of branched coverings, or so-called Hurwitz numbers. In the Hamilton-Jacobi approach to Liouville integrable Hamiltonian systems, Hamilton's principal function, evaluated on the level surfaces of a complete set of Poisson commuting invariants, plays a rôle similar to the -function, serving both as a complete solution of the Hamilton-Jacobi equation and a canonical generating functionto linearizing coordinates. (en)
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foaf:isPrimaryTopicOf	wikipedia-en:Tau_function_(integrable_systems)
is Link from a Wikipage to another Wikipage of	Integrable system Grassmannian Tau function
is Wikipage disambiguates of	Tau function
is foaf:primaryTopic of	wikipedia-en:Tau_function_(integrable_systems)

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