Solutions of mixed Painlevé P_III—Vmodel

Abstract

We review the construction of the mixed Painlevé P III−V system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its parameters reduces to either Painlevé P III or P V . The aim of this paper is to describe solutions of P III−V model. In particular, we determine and classify rational, power series and transcendental solutions of P III−V . A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P III−V equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order n when the parameter of P III−V equation is quantized by integer n ∈ Z.Painlevé P II equation. Painlevé P V equations appeared in the context of impenetrable Bose gas model [13]. More recently the scattering on two Aharonov-Bohm vortices was exactly solved in terms of solutions of the Painlevé III equation [7]. Ablowitz Ramani and Segur (ARS) showed a connection between the mKdV integrable model and the Painlevé P II equation when self-similarity limit was implemented. They suggested that such connection could be generalized to other integrable models [1] meaning that the self-similarity limit would lead for integrable models to equations with the Painlevè property. More recently, it was found [3] that the 2n boson integrable model obtained as particular reductions (Drinfeld-Sokolov) of KP integrable models connected to Toda lattice hierarchy gives rise to higher Painlevé equations invariant under extended affine Weyl groups. In particular, investigation of various Dirac reduction schemes applied to the 4-boson (n = 2) integrable model with Weyl symmetry structure A (1) 4 led to emergence of a new mixed P III−V model [4]. The second order P III−V equation for a canonical variable q is given by:

Symmetries and hamiltonians of Ince’s XXXVIII and XLIX equations

“…Similarly obtaining a second-order differential equation for f from (21) and defining y through f = −αy/(y − 1) will also yield equation (1). Due to the fact that addition of κwf 2 term rendered the system (20) explicitly symmetric in w, f the Hamilton equations (21) are invariant under w, f rotation here referred to as an R operation :…”

“…The present study fills this gap by studying symmetries of equations I 38 and I 49 : from Ince's list [4,5,6,7]. For the full understanding of their symmetries it is instructive to first study how their structures emerge in the context of P III−V model [3,1]. Note that Ince's equation I 38 (1) with D = 0 is equivalent to Ince's equation I 49 (2) with A = 1.…”

“…in which we recognize equation I 12 of Ince (16). The second order equation for f written in terms of y such that f = − r 0 y y − 1 leads to Ince's equation I 38 (1) with A = α 2 0 /2, B = −α 2 2 /2, C = −2ǫ 0 r 2 0 and D = 0 and thus the equation obtained in this limit is only an incomplete version of I 38 equation (1).…”

Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations

“…In Section 6, that provides an outlook, we discuss the modification of the dressing chain that maintains the Kovalevskaya-Painlevé property together with the part of its Bäcklund symmetry. This relates to the program that we have pursued in several papers [3,4,5] attempting to connect integrability to the remaining symmetry of Painlevé models that survives the partial breaking of the extended affine Weyl group A (1) N −1 by deformation terms that preserve some notion of integrability.…”

On special limits of the Mixed Painlevé PIII–VModel

“…The P III or P V Painlevé models emerge from P III−V for different values of the underlying parameters. See below the list i) − v) [4,2] for a complete summary of models that can be obtained from P III−V , their symmetries and the corresponding values of parameters. The notation W [s 1 , s 3 , π 2 ] used below denotes the symmetry group generated by s 1 , s 3 , π 2 .…”