Nalini Joshi

Painlevé property of general variable-coefficient versions of the Korteweg-de Vries and non-linear Schrödinger equations

Abstract General variable-coefficient versions of the Korteweg-de Vries (KdV) and non-linear Schrodinger (NLS) equations are shown to posses the Painleve property when their time-dependent coefficient functions are related by respective constraints. Under these constraints, found previously by Grimshaw in another context, the equations can be mapped to their well-known constant-coefficient versions. Transformations mapping the variable-coefficient versions to other modifications of the KdV and the NLS equations are discussed.

Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

The second Painlev? hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well known second Painlev? equation, . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlev? analysis to ordinary differential equations. We extend these techniques in order to derive auto-B?cklund transformations for the second Painlev? hierarchy. We also derive a number of other B?cklund transformations, including a B?cklund transformation onto a hierarchy of equations, and a little known B?cklund transformation for itself. We then use our results on B?cklund transformations to obtain, for each member of the hierarchy, a sequence of special integrals.

On Boutroux's Tritronquée solutions of the first Painlevé equation

The triply truncated solutions of the first Painleve equation were specified by Boutroux in his famous paper of 1913 as those having no poles (of large modulus) except in one sector of angle 2π/5. There are five such solutions and each of them can be obtained from any other one by applying a certain symmetry transformation. One of these solutions is real on the real axis. We found a characteristic property of this solution, different from the asymptotic description given by Boutroux. This allows us to estimate numerically the position of its real pole and zero closest to the origin. We also study properties of asymptotic series for truncated solutions.

On a Schwarzian PDE associated with the KdV hierarchy

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Mobius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equa- tion for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the full Painleve VI equation, i.e. with four arbitary parameters.

Nonlinear Nonautonomous Discrete Dynamical Systems from a General Discrete Isomonodromy Problem

Nonlinear nonautonomous discrete dynamical systems (DDS) whose continuum limits are the well-known Painlevé equations, have recently arisen in models of quantum gravity. The Painlevé equations are believed integrable because each is the isomonodromy condition for an associated linear differential equation. However, not every DDS with an integrable continuum limit is necessarily integrable. Which of the many discrete versions of the Painlevé equations inherit their integrability is not known. How to derive all their integrable discrete versions is also not known. We provide a systematic method of attacking these questions by giving a general discrete isomonodromy problem. Discrete versions of the first and second Painlevé equations are deduced from this general problem.

On the linearization of the Painlevé III-VI equations and reductions of the three-wave resonant system

We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3×3 matrix Fuchs-Garnier pairs for the third, fourth, and fifth Painleve equations, together with the previously known Fuchs-Garnier pair for the sixth Painleve equation. These Fuchs-Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2×2 matrix Fuchs-Garnier pairs obtained by Jimbo and Miwa [Physica D 2, 407–448 (1981)]. As an application of the 3×3 matrix pairs, we found an integral autotransformation for the standard Fuchs-Garnier pair for the fifth Painleve equation. It generates an Okamoto-like Backlund transformation for the fifth Painleve equation. Another application is an integral transformation relating two different 2×2 matrix Fuchs-Garnier pairs for the third Painleve equation.

A Lax pair for a lattice modified KdV equation, reductions to q-Painlevé equations and associated Lax pairs

We present a new, nonautonomous Lax pair for a lattice nonautomous modified Korteweg–deVries equation and show that it can be consistently extended multidimensionally, a property commonly referred to as being consistent around a cube. This nonautonomous equation is reduced to a series ofq-discrete Painlev´ e equations, and Lax pairs for the reduced equations are found. A 2 × 2 Lax pair is given for a qPIII with multiple parameters and, also, for versions of qPII and qPV, all for the first time. PACS number: 02.30.Ik

How to detect integrability in cellular automata

Ultra-discrete equations are generalized cellular automata in the sense that the dependent (and independent) variables take only integer values. We present a new method for identifying integrable ultra-discrete equations which is the equivalent of the singularity confinement property for difference equations and the Painleve property for differential equations. Using this criterion, we find integrable ultra-discrete equations which include the ultra-discrete Painleve equations.

q-difference equations of KdV type and Chazy-type second-degree difference equations

By imposing special compatible similarity constraints on a class of integrable partial

The discrete first, second and thirty-fourth Painlevé hierarchies

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