Integrable Hierarchies and Dispersionless Limit

Takasaki, Kanehisa; Takebe, Takashi

doi:10.1142/s0129055x9500030x

Cited by 390 publications

(718 citation statements)

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“…We shall only write down a few examples of these dispersionless equations; we refer the reader to e.g. [17,18,20] for a complete and more rigorous description of the dispersionless hierarchy. Let us simply mention that, as mentioned above, we are dealing with a classical limit, that is the commutators become of order ≡ 1/N , and their leading term defines a Poisson bracket.…”

Section: Classical/dispersionless Limitmentioning

confidence: 99%

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HCIZ integral and 2D Toda lattice hierarchy

Zinn-Justin¹

2002

Nuclear Physics B

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The expression of the large $N$ Harish Chandra--Itzykson--Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large $N$ HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.Comment: 20 pages. various minor improvements/correction

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Section: Classical/dispersionless Limitmentioning

confidence: 99%

“…The second relation we use the dispersionless limit of the Fay identity [18], which we shall not prove here and write under the form (see [25]):…”

Section: Derivatives Of F With Respect To the T Qmentioning

confidence: 99%

HCIZ integral and 2D Toda lattice hierarchy

Zinn-Justin¹

2002

Nuclear Physics B

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“….) is the tau-function of the dispersionless KP hierarchy [23,24]. When the reality condition Re T k = 0 is imposed, this function admits a nice geometric/electrostatic description given in [18] for the particular case σ = 1/π.…”

Section: Free Fermions and Tau-functionsmentioning

confidence: 99%

Canonical and Grand Canonical Partition Functions of Dyson Gases as Tau-Functions of Integrable Hierarchies and Their Fermionic Realization

Zabrodin

2010

Complex Anal. Oper. Theory

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The partition function for a canonical ensemble of 2D Coulomb charges in a background potential (the Dyson gas) is realized as a vacuum expectation value of a group-like element constructed in terms of free fermionic operators. This representation provides an explicit identification of the partition function with a taufunction of the 2D Toda lattice hierarchy. Its dispersionless (quasiclassical) limit yields the tau-function for analytic curves encoding the integrable structure of the inverse potential problem and parametric conformal maps. A similar fermionic realization of partition functions for grand canonical ensembles of 2D Coulomb charges in the presence of an ideal conductor is also suggested. Their representation as Fredholm determinants is given and their relation to integrable hierarchies, growth problems and conformal maps is discussed.

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“…Following the above idea in the opposite direction one is able to construct, with the aid of shift operators, a dispersionful analogue of the Whitham hierarchy. Very recently Takasaki considered dispersionless limit of multi-component KP hierarchy with charges (lattice variables), that can be treated as a generalization of the Toda hierarchy [13]. In this sense the original Toda hierarchy is equivalent to the two-component KP hierarchy.…”

Section: Introductionmentioning

confidence: 99%