A Miura of the Painlevé I Equation and Its Discrete Analogs

We present Miura transformations for the continuous and several discrete Painlev\'e I equations. In the case of the continuous P_I, we use the Hamiltonian formulation of the Painlev\'e equations and show that there exists a Miura transformation between P_I and the binomial, second degree, equation of Cosgrove SD_V. In the case of the discrete P_I's we obtain two different kinds of Miuras. One kind relates a d-P_I to some other d-P_I while the other leads to discrete four-point equations which are the discrete analogs of the derivative of Cosgrove's equation SD_V.     

A study of the discrete Pv equation: Miura transformations and particular solutions

We derive the form of the Miura transformation of the discrete P_v equation and show that it is indeed an auto-Bäcklund transformation, i.e. it relates the discrete P_v to itself. Using this auto-Bäcklund, we obtain the Schlesinger transformations of discrete P_v which relate the solution for one set of the parameters of the equation to that of another set of neighbouring parameters. Finally, we obtain particular solutions of the discrete P_v (i.e. solutions that exist only for some specific values of the parameters). These solutions are of two types: solutions involving the confluent hypergeometric function (on codimension-one submanifold of parameters) and rational solutions (on codimension-two submanifold of parameters).     

Special functions as solutions to discrete Painlevé equations

We present results on special solutions of discrete Painlevé equations. These solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second-order (discrete) equations. These linear equations define the discrete analogues of special functions.     

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.     

The existence of infinitely many Lie-Bäcklund symmetries for a new derivative nonlinear schrödinger equation

The new derivative nonlinear Schrödinger equation considered by Chen et al., is shown to possess strong and hereditary symmetries, and hence infinitely many commuting Lie-Bäcklund (L-B) symmetries. Further, we derive the corresponding constants of motion, which are in involution.     

Schlesinger transforms for the discrete painlevé IV equation

We derive an auto-Bäcklund transformation for the discrete Painlevé IV equation and use it in order to derive Schlesinger transformations for the same equation as well as particular solutions in perfect analogy to the continuous case.     

Constructing discrete Painlevé equations: from E8(1) to A1(1) and back

The ‘restoration method’ is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that are equivalent to it by homographic transformations. Discrete Painlevé equations are then obtained by deautonomising the various mappings thus obtained. We apply the restoration method to two challenging examples, one of which does not lead to a QRT mapping at the autonomous limit but we verify that even in that case our method is indeed still applicable. For one of the equations we derive we also show how, starting from a form where the independent variable advances one step at a time, we can obtain versions that correspond to multiple-step evolutions.     

Motion of strings, embedding problem and soliton equations

The motion of a flexible string of constant length in E ³ in interaction, corresponding to a variety of physical situations, is considered. It is pointed out that such a system could be studied in terms of the embedding problem in differential geometry, either as a moving helical space curve in E ³ or by the embedding equations of two dimensional surfaces in E ³. The resulting integrability equations are identifiable with standard soliton equations such as the non-linear Schrödinger, modified K-dV, sine-Gordon, Lund-Regge equations, etc. On appropriate reductions the embedding equations in conjunction with suitable local space-time and/or gauge symmetries reproduce the AKNS-type eigenvalue equations and Riccati equations associated with soliton equations. The group theoretical properties follow naturally from these studies. Thus the above procedure gives a simple geometric interpretation to a large class of the soliton possessing nonlinear evolution equations and at the same time solves the underlying string equations.     

From Integrable Lattices to Non-QRT Mappings

Second-order mappings obtained as reductions of integrable lattice equations are generally expected to have integrals that are ratios of biquadratic polynomials, i.e., to be of QRT-type. In this paper we find reductions of integrable lattice equations that are not of this type. The mappings we consider are exact reductions of integrable lattice equations proposed by Adler et al. [Comm Math Phys 233: 513, 2003]. Surprisingly, we found that these mappings possess invariants that are of the type originally studied by Hirota et al. [J Phys A 34: 10377, 2001]. Moreover, we show that several mappings obtained are linearisable and we present their linearisation.