Lie Symmetries and Exact Solutions of First Order Difference Schemes

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.Comment: Minor changes and journal-re

Lie point symmetries of difference equations and lattices

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations

Discrete derivatives and symmetries of difference equations

We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation.Comment: submitted to J.Phys. A 10 Latex page

Multiscale expansion and integrability properties of the lattice potential KdV equation

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007 Conferenc

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

Adaptive Design of Excitonic Absorption in Broken-Symmetry Quantum Wells

Adaptive quantum design is used to identify broken-symmetry quantum well potential profiles with optical response properties superior to previous ad-hoc solutions. This technique performs an unbiased stochastic search of configuration space. It allows us to engineer many-body excitonic wave functions and thus provides a new methodology to efficiently develop optimized quantum confined Stark effect device structures.Comment: 4 pages, 3 encapsulated postscript figure

Integrability of Differential-Difference Equations with Discrete Kinks

In this article we discuss a series of models introduced by Barashenkov, Oxtoby and Pelinovsky to describe some discrete approximations to the \phi^4 theory which preserve travelling kink solutions. We show, by applying the multiple scale test that they have some integrability properties as they pass the A_1 and A_2 conditions. However they are not integrable as they fail the A_3 conditions.Comment: submitted to the Proceedings of the workshop "Nonlinear Physics: Theory and Experiment.VI" in a special issue di Theoretical and Mathematical Physic

Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV

We consider multiple lattices and functions defined on them. We introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices. We use these results to perform the multiple--scale reduction of the lattice potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur

Energy flow of moving dissipative topological solitons

We study the energy flow due to the motion of topological solitons in nonlinear extended systems in the presence of damping and driving. The total field momentum contribution to the energy flux, which reduces the soliton motion to that of a point particle, is insufficient. We identify an additional exchange energy flux channel mediated by the spatial and temporal inhomogeneity of the system state. In the well-known case of a DC external force the corresponding exchange current is shown to be small but non-zero. For the case of AC driving forces, which lead to a soliton ratchet, the exchange energy flux mediates the complete energy flow of the system. We also consider the case of combination of AC and DC external forces, as well as spatial discretization effects.Comment: 24 pages, 5 figures, submitted to Chao