arXiv subjects
Publications and source records attributed to V. C. C. Alves.
The Painlevé equations are here connected to other classes of equations with the Painlevé Property (Ince's equations) by the same degeneracy procedure that connects the Painlevé equations (coalescence). These Ince's equations here are also connected among themselves like in the traditional Painlevé's coalescence cascade. Such degeneracy is considered also for the special equations, symmetric equations and Bäcklund transformations.
We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant Painlevé equations. The $A^{(1)}_{N-1}$ Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal gradation.
We extend Painlevé IV model by adding quadratic terms to its Hamiltonian obtaining two classes of models (coalescence and deformation) that interpolate between Painlevé IV and II equations for special limits of the underlying parameters. We derive the underlying Bäcklund transformations, symmetry structure and requirements to satisfy Painlevé property.
We discuss symmetries of Hamiltonians of I$_{38}$ and I$_{49}$ equations that appear on Ince's list of fifty second-order differential equations with Painlevé property. This study is informed by structure of Weyl symmetries of Painlevé P$_{III}$ and mixed Painlevé P$_{III-V}$ equations and provides insights into differences between the symmetries of Painlevé equations and symmetries of solvable equations on Ince's list.
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 $n \in \mathbb{Z}$.