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S. Lafortune

Publications and source records attributed to S. Lafortune.

17 recordsLinked to original sources
arXiv2009

The higher-dimensional Ablowitz-Ladik model: from (non-)integrability and solitary waves to surprising collapse properties and more exotic solutions

We propose a consideration of the properties of the two-dimensional Ablowitz-Ladik discretization of the ubiquitous nonlinear Schrodinger (NLS) model. We use singularity confinement techniques to suggest that the relevant discretization should not be integrable. More importantly, we identify the prototypical solitary waves of the model and examine their stability, illustrating the remarkable feature that near the continuum limit, this discretization leads to the absence of collapse and complete spectral wave stability, in stark contrast to the standard discretization of the NLS. We also briefly touch upon the three-dimensional case and generalizations of our considerations therein, and also present some more exotic solutions of the model, such as exact line solitons and discrete vortices.

nlin.PS↗
arXiv2006

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 non-standard 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.

nlin.PS↗
arXiv2003

Singularity confinement and algebraic integrability

Two important notions of integrability for discrete mappings are algebraic integrability and singularity confinement, have been used for discrete mappings. Algebraic integrability is related to the existence of sufficiently many conserved quantities whereas singularity confinement is associated with the local analysis of singularities. In this paper, the relationship between these two notions is explored for birational autonomous mappings. Two types of results are obtained: first, algebraically integrable mappings are shown to have the singularity confinement property. Second, a proof of the non-existence of algebraic conserved quantities of discrete systems based on the lack of confinement property is given.

nlin.SI↗
arXiv2002

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.

nlin.SI↗
arXiv2001

A Study of the Continuous and Discrete Gambier Systems

We present a systematic study of the Gambier system, which in the continuous case is given by two Riccati equations in cascade. We derive the condition for its integrability and show that the generic Gambier system contains one free function. We also derive the Schlesinger transformations for this system which allows in principle the systematic construction of the integrable cases. The above procedure is carried over to a discrete setting. We show thus how the discrete Gambier system can be expressed as a system of two homographic mappings in cascade. The integrable cases are obtained through the singularity confinement discrete integrability criterion. Finally the discrete Schlesinger transformations are also derived giving a handle to the construction of the integrable Gambier mapping.

nlin.SI↗
arXiv2001

Linearisable systems and the Gambier approach

A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the coupling of a Riccati equation with second order Painlevé equations. This is done in both continuous and discrete cases.

nlin.SI↗
arXiv2001

Blending two discrete integrability criteria: singularity confinement and algebraic entropy

We confront two integrability criteria for rational mappings. The first is the singularity confinement based on the requirement that every singularity, spontaneously appearing during the iteration of a mapping, disappear after some steps. The second recently proposed is the algebraic entropy criterion associated to the growth of the degree of the iterates. The algebraic entropy results confirm the previous findings of singularity confinement on discrete Painlevé equations. The degree-growth methods are also applied to linearisable systems. The result is that systems integrable through linearisation have a slower growth than systems integrable through isospectral methods. This may provide a valuable detector of not just integrability but also of the precise integration method. We propose an extension of the Gambier mapping in $N$ dimensions. Finally a dual strategy for the investigation of the integrability of discrete systems is proposed based on both singularity confinement and the low growth requirement.

nlin.SI↗
arXiv2001

Discrete systems related to some equations of the Painlevé-Gambier classification

We derive integrable discrete systems which are contiguity relations of two equations in the Painlevé-Gambier classification depending on some parameter. These studies extend earlier work where the contiguity relations for the six transcendental Painlevé equations were obtained. In the case of the Gambier equation we give the contiguity relations for both the continuous and the discrete system.

nlin.SI↗
arXiv2001

Linearisable 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.

nlin.SI↗
arXiv2000

Point Symmetries of Generalized Toda Field Theories

A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point symmetries of these theories are found for an infinite, semi-infinite and finite number of fields. Special attention is accorded to conformal invariance and its breaking.

nlin.SI↗
arXiv1998

Discrete and Continuous Linearizable Equations

We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one and reduce the system to a single differential equation. This equation is of the form of those singled-out by Painlevé in his quest for integrable forms. In the discrete case, we extend previous results of ours showing that, again by elimination of variables, the general projective system can be written as a mapping for a single variable. We show that this mapping is a member of the family of multilinear systems (which is not integrable in general). The continuous limit of multilinear mappings is also discussed.

solv-int↗
arXiv1998

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 10 dimensional. The equations can describe the interaction of two long molecular chains, each involving one type of atoms.

solv-int↗
arXiv1998

Solutions to the Optical Cascading Equations

Group theoretical methods are used to study the equations describing χ^{(2)}:χ^{(2)} cascading. The equations are shown not to be integrable by inverse scattering techniques. On the other hand, these equations do share some of the nice properties of soliton equations. Large families of explicit analytical solutions are obtained in terms of elliptic functions. In special cases, these periodic solutions reduce to localized ones, i.e., solitary waves. All previously known explicit solutions are recovered, and many additional ones are obtained

solv-int↗
arXiv1998

Schlesinger Transformations for Linearisable Equations

We introduce the Schlesinger transformations of the Gambier equation. The latter can be written, in both the continuous and discrete cases, as a system of two coupled Riccati equations in cascade involving an integer parameter n. In the continuous case the parameter appears explicitly in the equation while in the discrete case it corresponds to the number of steps for singularity confinement. Two Schlesinger transformations are obtained relating the solutions for some value $n$ to that corresponding to either n+1 or n+2.

solv-int↗
arXiv1998

Again, Linearizable Mappings

We examine a family of 3-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.

solv-int↗
arXiv1998

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.

solv-int↗