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A remark on the Hankel determinant formula for solutions of the Toda equation

Kenji Kajiwara1, Marta Mazzocco2 and Yasuhiro Ohta3

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We consider the Hankel determinant formula of the τ functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the τ functions in the framework of the KP theory. Similar phenomena that have been observed for the Painlevé II and IV equations are recovered. The case of finite lattice is also discussed.


PACS

05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

05.45.Yv Solitons

MSC

37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)

34E05 Asymptotic expansions

37K30 Relations with infinite-dimensional Lie algebras and other algebraic structures

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (See also 30E15)

34M55 Painlevé and other special equations; classification, hierarchies; isomonodromic deformations

Subjects

Statistical physics and nonlinear systems

Dates

Issue 42 (19 October 2007)

Received 15 January 2007, in final form 8 May 2007

Published 2 October 2007



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