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Hierarchies of nonlinear integrable q-difference equations from series of Lax pairs

Mike Hay

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We present two hierarchies of nonlinear, integrable q-difference equations, one of which includes a q-difference form of each of the second and fifth Painlevé equations, \qtw and \qfi , the other includes \qth . All the equations have multiple free parameters. A method to calculate a 2 × 2 Lax pair for each equation in the hierarchy is also given.


PACS

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

02.30.Ik Integrable systems

02.30.Gp Special functions

02.30.Hq Ordinary differential equations

MSC

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

37K60 Lattice dynamics (See also 37L60)

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

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

33E17 Painlevé-type functions

Subjects

Mathematical physics

Statistical physics and nonlinear systems

Dates

Issue 34 (24 August 2007)

Received 27 May 2007, in final form 17 July 2007

Published 7 August 2007



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