Singularity Confinement Research Articles

Growth and integrability in discrete systems

We introduce a new discrete integrability criterion inspired from the recent findings of Ablowitz and collaborators. This criterion is based on the study of the growth of some characteristic of the solutions of a mapping, using Nevanlinna theory. Since the practical implementation of the latter does not always lead to a clear-cut answer, we complement the growth criterion by the singularity confinement property. This combination turns out to be particularly efficient. Its application allows us to recover the known forms of the discrete Painlevé equations and to show that no new ones may exist within a given parametrization.

Discrete Dynamical Systems¶Associated with Root Systems of Indefinite Type

A geometric charactrization of the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but which exhibits chaotic behavior, is presented. It is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a Cremona isometry on the Picard group of the surface. It is also shown that the group of Cremona isometries is isomorphic to an extended Weyl group of indefinite type. A method to construct the mappings associated with some root systems of indefinite type is also presented.

A geometric approach to singularity confinement and algebraic entropy

A geometric approach to the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but exhibits chaotic behavior, is presented. It is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended Weyl group of indefinite type. A method to calculate its algebraic entropy by using the theory of intersection numbers is presented.

Linearizable mappings and the low-growth criterion

We examine a family of discrete second-order systems which are integrable through reduction to a linear system. These systems were previously identified using the singularity confinement criterion. Here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates. We show that the linearisable mappings are characterised by a very special degree growth. The ones linearisable by reduction to projective systems exhibit zero growth, i.e. they behave like linear systems, while the remaining ones (derivatives of Riccati, Gambier mapping) lead to linear growth. This feature may well serve as a detector of integrability through linearisation.

Bilinear structure and determinant solution for the relativistic Lotka–Volterra equation

The relativistic Lotka–Volterra (RLV) lattice and the discrete-time relativistic Lotka–Volterra (dRLV) lattice are investigated by using the bilinear formalism. The bilinear equations for them are systematically constructed with the aid of the singularity confinement test. It is shown that the RLV lattice and dRLV lattice are decomposed into the Bäcklund transformations of the Toda lattice system. The N-soliton solutions are explicitly constructed in the form of the Casorati determinant.

Integrable systems without the Painlevé property

We examine whether the Painlev\'e property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlev\'e continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities.

Singularity structure and algebraic properties of the differential-difference Kadomtsev–Petviashvili equation

We present a study of the differential-difference Kadomtsev–Petviashvili equation, which was derived using the differential-difference version of Sato's theory. We focus on the singularity structure of the equation, in particular the Painlevé property and singularity confinement. Moreover we study its symmetries (both Lie and generalized) and derive some interesting similarity reductions.

Discrete Painlevé I and singularity confinement in projective space

We write the discrete Painlevé I equation as a polynomial map in projective space and follow the development of the singularity and the fate of the other initial value. Most of the time the initial value is at the next to leading term and we can recover it (and confine the singularity) at the (3 m+1)th iteration by imposing a condition on the de-autonomization. For m=1 one gets the usual d- P I .

Bilinearization of discrete soliton equations through the singularity confinement test

An elementary and systematic method to construct exact solutions for discrete soliton equations is presented. In this method, the singularity patterns, obtained through the singularity confinement test, give us critical information for bilinearization.

Integrability in a discrete world

We review our findings on integrable discrete systems with emphasis on the discrete integrability detector we have proposed under the name of singularity confinement. We have indeed shown, in a host of examples, that it is possible, by studying the structure of the singularities of discrete systems, to identify the integrable ones. A most important result of this approach is the discovery of discrete Painlevé equations of which a lengthy list exists today. These equations, being integrable systems, are characterised by particularly rich properties which are under active investigation. We present here an overview of these properties and stress the similarities and differences that exist between discrete and continuous Painlevé equations.

Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations

We examine the validity of the results obtained with the singularity confinement integrability criterion in the case of discrete Painlevé equations. The method used is based on the requirement of non-exponential growth of the homogeneous degree of the iterate of the mapping. We show that when we start from an integrable autonomous mapping and deautonomise it using singularity confinement the degrees of growth of the nonautonomous mapping and of the autonomous one are identical. Thus this low-growth based approach is compatible with the integrability of the results obtained through singularity confinement. The origin of the singularity confinement property and its necessary character for integrability are also analysed.

Integrability criteria for differential-difference systems: a comparison of singularity confinement and low-growth requirements

A new approach for the study of the integrability of differential-difference systems is introduced. Given a differential-difference system (of such a form that it can be iterated without necessitating the integration of a differential equation at any step) a two-stage strategy is used. First, the singularity confinement necessary integrability criterion is used in order to limit the possible choices. Once the system is sufficiently reduced, the (much stronger) requirement of nonexponential growth of the degree of the iterates of some initial condition is implemented. This method turns out to be powerful and practical. The investigation of a given class of differential-difference systems has resulted in some well known integrable systems but also to two promising integrability candidates (one of which is reduced to a known integrable case).

Discretizing families of linearizable equations

We investigate generalizations of the discrete Riccati equation as a linearizable system, to multicomponent linearizable systems. These are discretizations of nonlinear ordinary differential equations with superposition formulas. We present discrete matrix Riccati equations, projective, conformal, orthogonal and symplectic Riccati equations. Also obtained are discrete equations, based on complex orthogonal and symplectic groups, that in the continuous limit involve fourth order polynomial nonlinearities. All these equations satisfy the criterion of singularity confinement.

Singularity Confinement and Chaos in Discrete Systems

We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the analysis of the complexity (``algebraic entropy'') of the map using the growth of the degree of its iterates: integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems.

Casorati determinant solution for the discrete-time relativistic Toda lattice equation

The discrete-time relativistic Toda lattice (dRTL) equation is investigated by using the bilinear formalism. Bilinear equations are systematically constructed with the aid of the singularity confinement method. It is shown that the dRTL equation is decomposed into the Bäcklund transformations of the discrete-time Toda lattice equation. The N -soliton solution is explicitly constructed in the form of the Casorati determinant.

Constructing integrable third-order systems: the Gambier approach

We present a systematic construction of integrable third-order systems based on the coupling of an integrable second-order equation and a Riccati equation. This approach is an extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlevé method for ordinary differential equations and the singularity confinement method for mappings).

Again, linearizable mappings

We examine a family of three-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through the application of the singularity confinement criterion and are explicitly integrated.

Bilinearization of discrete soliton equations and singularity confinement

The singularity confinement method is applied to the systematic derivation of the bilinear equations for discrete soliton equations. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed.

Singularity confinement analysis of integrodifferential equations of Benjamin - Ono type

We analyse the integrability of various types of intregrodifferential equations belonging to the Benjamin - Ono family. The method used is a combination of two well known integrability detectors: the Painleve method (for continuous systems) and the singularity confinement (for discrete systems). We confirm the results of Hietarinta based on the study of multisoliton solutions. In particular, we show that the third-order extension to Benjamin - Ono that he proposed does pass the test and is thus an excellent candidate for integrability.

Bilinear structure and Schlesinger transforms of the q-P III and q-P VI equations

We show that the recently derived ( q-)discrete form of the Painlevé VI equation can be related to the discrete P III form, in particular if one uses full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both q-P III and q- P VI.