18 research outputs found
A discrete linearizability test based on multiscale analysis
In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation
Classification of discrete systems on a square lattice
We consider the classification up to a Möbius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A1, A2, and A3 linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice
Point Symmetries of Generalized Toda Field Theories II Applications of the Symmetries
The Lie symmetries of a large class of generalized Toda field theories are
studied and used to perform symmetry reduction. Reductions lead to generalized
Toda lattices on one hand, to periodic systems on the other. Boundary
conditions are introduced to reduce theories on an infinite lattice to those on
semi-infinite, or finite ones.Comment: 26 pages, no figure
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
Lie discrete symmetries of lattice equations
We extend two of the methods previously introduced to find discrete
symmetries of differential equations to the case of difference and
differential-difference equations. As an example of the application of the
methods, we construct the discrete symmetries of the discrete Painlev\'e I
equation and of the Toda lattice equation
Discrete Analog of the Burgers Equation
We propose the set of coupled ordinary differential equations
dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers
equation. We focus on traveling waves and triangular waves, and find that these
special solutions of the discrete system capture major features of their
continuous counterpart. In particular, the propagation velocity of a traveling
wave and the shape of a triangular wave match the continuous behavior. However,
there are some subtle differences. For traveling waves, the propagating front
can be extremely sharp as it exhibits double exponential decay. For triangular
waves, there is an unexpected logarithmic shift in the location of the front.
We establish these results using asymptotic analysis, heuristic arguments, and
direct numerical integration.Comment: 6 pages, 5 figure
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
Discrete Multiscale Analysis: A Biatomic Lattice System
We discuss a discrete approach to the multiscale reductive perturbative
method and apply it to a biatomic chain with a nonlinear interaction between
the atoms. This system is important to describe the time evolution of localized
solitonic excitations. We require that also the reduced equation be discrete.
To do so coherently we need to discretize the time variable to be able to get
asymptotic discrete waves and carry out a discrete multiscale expansion around
them. Our resulting nonlinear equation will be a kind of discrete Nonlinear
Schr\"odinger equation. If we make its continuum limit, we obtain the standard
Nonlinear Schr\"odinger differential equation
Partially integrable nonlinear equations with one higher symmetry
In this letter, we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function