S. Lafortune

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.

The Gambier mapping, revisited

We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete form of this equation, the Gambier mapping, and present conditions for its integrability. Finally, we obtain the reductions of the Gambier mapping, identify their integrable forms and compute their continuous limits.

Symmetries of discrete dynamical systems involving two species

The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in other cases up to ten dimensional. The equations can describe the interaction of two long molecular chains, each involving one type of atoms.

HIGH LEWIS NUMBER COMBUSTION WAVEFRONTS: A PERTURBATIVE MELNIKOV ANALYSIS ∗

The wavefronts associated with a one‐dimensional combustion model with Arrhenius kinetics and no heat loss are analyzed within the high Lewis number perturbative limit. This situation, in which fuel diffusivity is small in comparison to that of heat, is appropriate for highly dense fluids. A formula for the wavespeed is established by a nonstandard application of Melnikov’s method and slow manifold theory from dynamical systems, and compared to numerical results. A simple characterization of the wavespeed correction is obtained: it is proportional to the ratio between the exothermicity parameter and the Lewis number. The perturbation method developed herein is also applicable to more general coupled reaction‐diffusion equations with strongly differing diffusivities. The stability of the wavefronts is also tested using a numerical Evans function method.

Integrable ultra-discrete equations and singularity analysis

Ultra-discrete equations are those in which both dependent and independent variables can be restricted to take only integer values. Recently, the authors introduced a test for integrability of ultra-discrete equations. In this paper, we use this procedure to find integrable ultra-discrete versions of the Painleve equations, some of which appear not to be known. Furthermore, we show how our procedure can be applied to equations in 1+1 dimensions. In particular, we study an integrable ultra-discrete version of the Sine–Gordon equation.

When is negativity not a problem for the ultradiscrete limit

The “ultradiscrete limit” has provided a link between integrable difference equations and cellular automata displaying soliton-like solutions. In particular, this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the “Max” operator. Although it certainly is the case that dropping these positivity conditions creates potential difficulties, it is still possible for solutions to persist under the ultradiscrete limit, even in their absence. To recognize when this will occur, one must consider whether a certain expression, involving a measure of the rates of convergence of different terms in the difference equation and their coefficients, is equal to zero. Applications discussed include the solution of elementary ordinary difference equations, a discretization of the Hirota Bilinear Difference Equation and the identification of integrals of motion for ultradiscrete equations.

Singularity confinement and algebraic integrability

Two important notions of integrability for discrete mappings, algebraic integrability and singularity confinement, have been used for discrete mappings. Algebraic integrability is related to the existence of sufficiently many conserved quantities and singularity confinement is associated with the local analysis of singularities. In this article, the relationship between these two notions is explored for birational autonomous mappings. The main result of this article is that algebraically integrable mappings are shown to have the singularity confinement property. Using this result, the proof of the nonexistence of algebraic conserved quantities for a class of discrete systems is given.

Symmetry classification of diatomic molecular chains

A symmetry classification of possible interactions in a diatomic molecular chain is provided. For nonlinear interactions the group of Lie point transformations, leaving the lattice invariant and taking solutions into solutions, is at most five-dimensional. An example is considered in which subgroups of the symmetry group are used to reduce the dynamical differential-difference equations to purely difference ones.

Superposition formulas for pseudounitary matrix Riccati equations

The purpose of this article is to derive a superposition formula for the pseudounitary matrix Riccati equation of dimension N≥2. The superposition formula will be written in closed form in terms of five particular solutions satisfying certain well‐specified conditions defining a fundamental set. Examples will be studied in order to show how the superposition formula works and how it can be used in numerical calculations.

Instability of local deformations of an elastic rod

Abstract We study the instability of pulse solutions of two coupled non-linear Klein–Gordon equations by means of Evans function techniques. The system of coupled Klein–Gordon equations considered here describes the near-threshold dynamics of a three-dimensional elastic rod with circular cross-section, subject to constant twist. We determine a condition on the speed of the traveling pulse which ensures spectral instability.