Backlund transformations for discrete Painleve equations: Discrete P-II-P-V

Cataloged from PDF version of article.Transformation properties of discrete Painleve´ equations are investigated by using an algorithmic method. This method yields explicit transformations which relates the solutions of discrete Painleve´ equations, discrete PII–PV, with different values of parameters. The particular solutions which are expressible in terms of the discrete analogue of the classical special functions of discrete Painleve´ equations can also be obtained from these transformations. 2005 Elsevier Ltd. All rights reserved

Schlessinger Transformations for Painleve VI equation

Cataloged from PDF version of article.A method to obtain the Schlesinger transformations for Painlevi VI equation is given. The procedure involves formulating a Riemann-Hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same. 0 1995 American Institute of Physics

Second-order second-degree Painleve equations related with Painleve I-IV equations and Fuchsian-type transformations

Cataloged from PDF version of article.One-to-one correspondence between the Painlevé I-VI equations and certain second-order second-degree equations of Painlevé type is investigated. The transformation between the Painlevé equations and second-order second-degree equations is the one involving the Fuchsian-type equation. © 1999 American Institute of Physics

Third order differential equations with fixed critical points

Cataloged from PDF version of article.The singular point analysis of third order ordinary differential equations which are algebraic in y and y′ is presented. Some new third order ordinary differential equations that pass the Painlevé test as well as the known ones are found. © 2008 Elsevier Inc. All rights reserved

A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also devise a method to derive explicit general solutions of the nonlinear ODEs. Interestingly, many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models.Comment: 12 pages. J. Phys. A: Math. Gen. 39 (2006) in pres

Square lattice Ising model susceptibility: Series expansion method and differential equation for χ(3)\chi^{(3)}

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by $\chi^{(3)}$, the ``three-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi$. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation $\chi^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^{2})$, where $s =sh (2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.Comment: 28 pages, no figur

A Lagrangian Description of the Higher-Order Painlev\'e Equations

We derive the Lagrangians of the higher-order Painlev\'e equations using Jacobi's last multiplier technique. Some of these higher-order differential equations display certain remarkable properties like passing the Painlev\'e test and satisfy the conditions stated by Jur\'a$\check{s}$, (Acta Appl. Math. 66 (2001) 25--39), thus allowing for a Lagrangian description.Comment: 16 pages, to be published in Applied Mathematics and Computatio

On the solvability of the discrete second Painleve equation

Cataloged from PDF version of article.The inverse monodromy method for studying the Riemann-Hilbert problem associated with classical Painlevé equations is applied to the discrete second Painlevé equation. © 2011 IOP Publishing Ltd

Non-polynomial fourth order equations which pass the Painlevé test

The singular point analysis of fourth order ordinary differential equations in the non-polynomial class are presented. Some new fourth order ordinary differential equations which pass the Painlevé test as well as the known ones are found. © 2005 Verlag der Zeitschrift für Naturforschung, Tübingen