arXiv subjects
Publications and source records attributed to D. Levi.
We use a discrete multiscale analysis to study the asymptotic integrability of differential-difference equations. In particular, we show that multiscale perturbation techniques provide an analytic tool to derive necessary integrability conditions for two well-known discretizations of the nonlinear Schroedinger equation.
The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying equations of this class. Those conditions are used at the end to test for integrability discretizations of some well-known hyperbolic equations.
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation $[\hat P,\hat M]=1$. In ordinary quantum mechanics $\hat P$ is the derivative and $\hat M$ the coordinate operator. Here we shall realize $\hat P$ as a second order differential operator and $\hat M$ as a first order integral one. We show that this makes it possible to solve large classes of differential and integro-differential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing so called flatenned beams in laser theory
We show that the umbral correspondence between differential equations can be achieved by means of a suitable transformation preserving the algebraic structure of the problems. We present the general properties of these transformations, derive explicit examples and discuss them in the case of the Appèl and Sheffer polynomial families. We apply these transformations to non-linear equations, and discuss how the relevant solutions should be interpreted.
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.
It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations (e.g. scaling and/or translations) which are not symmetries of the considered equation. In this note we extend the approach to asymptotic symmetries for the analysis of PDEs, to the case 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.
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation and to solve a linear ordinary second order q-difference equation
Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice.
The discrete heat equation is worked out in order to illustrate the search of symmetries of difference equations. It is paid an special attention to the Lie structure of these symmetries, as well as to their dependence on the derivative discretization. The case of q-symmetries for discrete equations in a q-lattice is also considered.
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.
A modification of the symmetry approach for the classification of integrable differential-difference equations of the form $$ u_{n,t} = f_n(u_{n-1}, u_n, u_{n+1}), $$ where $n$ is a discrete integer variable, is presented (the well-known Volterra and Toda equations can be written in this form). If before, in the framework of the symmetry approach, only equations similar to $$ u_{n,t} = f(u_{n-1}, u_n, u_{n+1}), $$ i.e. defined by a function $f$, were considered, now we have an infinite set $f_n$ of a priori quite different functions.