Decio Levi

Continuous symmetries of discrete equations

Abstract Lie group techniques for solving differential equations are extended to differential-difference equations. As an application, it is shown that the two-dimensional Toda lattice has an infinite dimensional symmetry group with a Kac-Moody-Virasoro Lie algebra.

Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra

The Kadomtsev–Petviashvili (KP) equation (ut+3uux/2+ 1/4 uxxx)x +3σuyy/4=0 allows an infinite‐dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group depends on three arbitrary functions of time ‘‘t’’ and is shown to be related to a subalgebra of the loop algebra A(1)4. Low‐dimensional subalgebras of the symmetry algebra are identified, specifically all those of dimension n≤3, and also a physically important six‐dimensional Lie algebra containing translations, dilations, Galilei transformations, and ‘‘quasirotations.’’ New solutions of the KP equation are obtained by symmetry reduction, using the one‐dimensional subalgebras of the symmetry algebra. These solutions contain up to three arbitrary functions of t.

Conditions for the existence of higher symmetries of evolutionary equations on the lattice

In this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential-difference equations depending only on nearest neighboring interaction. These conditions are applied to prove the existence of new integrable equations belonging to this class.

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 them to point symmetries, but to allow them to also transform the lattice.

Symmetries and conditional symmetries of differential difference equations

Two different methods of finding Lie point symmetries of differential‐difference equations are presented and applied to the two‐dimensional Toda lattice. Continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations, in particular, to differential equations of the delay type. The concept of conditional symmetries is extended from purely differential to differential‐difference equations and shown to incorporate Backlund transformations.

Nonlinear differential difference equations as Backlund transformations

Shows that the best known nonlinear differential difference equations associated with the discrete Schrodinger spectral problem and also with the discrete Zakharov-Shabat spectral problem can be interpreted as Backlund transformations for some continuous nonlinear evolution equations.

Symmetries of discrete dynamical systems

Differential–difference equations of the form un=Fn(t,un−1,un,un+1) are classified according to their continuous Lie point symmetry groups. It is shown that for nonlinear equations, the symmetry group can be at most seven‐dimensional. The integrable Toda lattice is a member of this class and has a four‐dimensional symmetry group.

LIE GROUP FORMALISM FOR DIFFERENCE EQUATIONS

The methods of Lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations. Several examples are analysed, one of them being a nonlinear difference equation. For the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit.

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.

Continuous and discrete matrix Burgers’ hierarchies

SummaryWe derive two hierarchies of matrix nonlinear evolution equations which reduce to the Burgers’ hierarchy in the scalar case and can be linearized by a matrix analogue of the Hopf-Cole transformation: for these hierarchies we display the associated class of Bäcklund transformations and show some special kinds of explicit solutions. More-over, by exploiting a discrete version of the Hopf-Cole transformation, we are also able to construct two hierarchies of linearizable nonlinear difference evolution equations and to derive for them Bäcklund trans-formations and explicit solutions.RiassuntoIn questo lavoro si derivano due gerarchie di equazioni di evoluzione nonlineari matriciali che possono essere linearizzate mediante un analogo matriciale della trasformazione di Hopf-Cole e si riducono nel caso scalare alla già nota gerarchia di Burgers. Per queste due gerarchie, come pure per le loro versioni discrete (anch’esse linearizzabili) si ottengono le trasformazioni di Bäcklund e si mostrano alcuni tipi significativi di soluzioni esplicite.