Universal invariance properties of Painlevé analysis and Bäcklund transformation in nonlinear partial differential equations

We define two transformations, independent of the polynomial PDE, making all the compatibility conditions at resonances and all the equations defining the Bäcklund transformation invariant by homographic transformation of the expansion function. The resulting dramatic shortening of invariant equations makes their solution feasible by hand, leading to new analytic solutions.     

New Bäcklund transformations for the third and fourth Painlevé equations to equations of second order and higher degree

We give new Bäcklund transformations for the third and fourth Painlevé equations, to equations of second order and higher degree.     

On a geometrical property of the Bäcklund transformation of the sine-Gordon equation

We show how a solution of the Bäcklund transformation equations for the sine-Gordon equation determines an explicit isometry from the underlying surface to the upper half plane. We then use the action on the upper half to find explicitly the general solution of the equations.     

Parameterless discrete Painlevé equations and their Miura relations

We present a study of discrete Painlevé equations which do not have any parameter, apart from those that can be removed by the appropriate scaling. We find four basic equations of this type as well as several more related to the basic ones by Miura transformations, which we derive explicitly. We obtain also the continuous limits of the basic parameterless equations and show that two of them are the discrete analogues of both the continuous Painlevé I and the zero-parameter Painlevé III.     

Interrelations of discrete Painlevé equations through limiting procedures

We study the discrete Painlevé equations associated to the affine Weyl group which can be obtained by the implementation of a special limits of -associated equations. This study is motivated by the existence of two -associated discrete both having a double ternary dependence in their coefficients and which have not been related before. We show here that two equations correspond to two different limits of a -associated discrete Painlevé equation. Applying the same limiting procedures to other -associated equations we obtained several -related equations most of which have not been previously derived.     

A new family of discrete Painlevé equations and associated linearisable systems

We derive discrete systems which result from a second, not studied up to now, form of the q-P_VI equation. The derivation is based on two different procedures: “limits” and “degeneracies”. We obtain several new discrete Painlevé equations along with some linearisable systems. The parallel between the results for the standard form of q-P_VI and those of the new one is also established.     

Bäcklund transformations for certain rational solutions of Painlevé VI

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of A⁵. We also show that this A⁵ root lattice can be related to the F₄⁽¹⁾ root lattice. We thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized by the elements of this F₄⁽¹⁾ root lattice.     

The Miura transformation and Lie-Bäcklund algebras of exactly solvable equations

All the one-dimensional one-component local evolution equations connected via the Miura transformation are found. Exactly solvable equations and their Lie-Bäcklund algebras are shown to generate interesting transformations of infinite classes of evolution equations.     

Auto-Bäcklund transformations for a matrix partial differential equation

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.     

A new method to test discrete Painlevé equations

Necessary discretization rules to preserve the Painlevé property are stated. A new method is added to the discrete Painlevé test, which perturbs the continuum limit and generates infinitely many no-log conditions.     

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.     

Bilinear Bcklund transformation and explicit solutions for a nonlinear evolution equation

The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The B\"{a}cklund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.     

The first and second Painlevé equations of higher order and some relations between them

The first and second Painlevé equations of higher order are introduced. The relations between the Korteweg-de Vries hierarchies and their singular manifold equations are presented. These identities are used to search for the relations between the first and the second Painlevé equations of higher order.     

Painlevé analysis and hamiltonian structure of a new space-dependent KdV equation

We deduce the Lax pair for a new space-dependent KdV equation, , via the technique of Painlevé analysis. From it, infinitely many conservation laws are deduced and the symplectic structure is obtained.     

The Painlevé property,a modified Boussinesq equation and a modified Kadomtsev-Petviashvili equation

It is shown that the modified Boussinesq equation and the modified Kadomtsev-Petviashvili equation, both of which are known to be completely integrable, possess the Painlev'e property for differential equations as defined by Weiss, Tabor and Carnevale.     

Discrete dressing transformations and Painlevé equations

We propose a discrete analog of the dressing transformation. Our starting point is a variant of the quotient-difference algorithm which, in this case, corresponds to a linear problem with shifts in the eigenvalues. The proper periodicity conditions lead to one-dimensional systems which are discrete Painlevé equations. We obtain thus the alternate d-P_II equation and a novel form for the discrete P_IV equation.     

Quantum curves and q-deformed Painlevé equations

Letters in Mathematical Physics - We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus...