Linearization through symmetries for discrete equations

We show that one can define through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete Cole-Hopf transformation. If the equation is linearizable the symmetry provides the linearizing transformation. At the end we present few examples of applications for equations defined on four lattice points

Continuous Symmetries of the Lattice Potential KdV Equation

In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg-de Vries (lpKdV) equation. Using its associated spectral problem we construct the soliton solutions and the Lax technique enables us to provide infinite sequences of generalized symmetries. Finally, using a discrete symmetry of the lpKdV equation, we construct a large class of non-autonomous generalized symmetries.Comment: 20 pages, submitted to Jour. Phys.

Classification of discrete equations linearizable by point transformation on a square lattice

We provide a complete set of linearizability conditions for nonlinear partial difference equations de- fined on four points and, using them, we classify all linearizable multilinear partial difference equations defined on four points up to a Mobious transformatio

Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations

In this paper we propose some linearizability tests of partial difference equations on a quad-graph given by one point, two points and generalized Hopf-Cole transformations. We apply the so obtained tests to a set of nontrivial examples

C-Integrability Test for Discrete Equations via Multiple Scale Expansions

In this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential difference equation. In this case the equation satisfies the $A_1$, $A_2$ and $A_3$ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schr\"odinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable

On the construction of partial difference schemes II: discrete variables and Schwarzian lattices

In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing an arbitrary partial differential equation on an arbitrary lattice. An open problem is the meaning of a lattice which does not satisfy the Clairaut--Schwarz--Young theorem. To analyze it we apply the procedure on a simple example, the potential Burgers equation with two different lattices, an orthogonal lattice which is invariant under the symmetries of the equation and satisfies the commutativity of the partial difference operators and an exponential lattice which is not invariant and does not satisfy the Clairaut--Schwarz--Young theorem. A discussion on the numerical results is also presented showing the different behavior of both schemes for two different exact solutions and their numerical approximations.Comment: 14 pages, 4 figure

Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

We discuss the non autonomous nonlinear partial difference equations belonging to Boll classification of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its B\"acklund transformations and Lax pairs. By carrying out the algebraic entropy calculations we show that the $H^4$ trapezoidal and the $H^6$ families are linearizable and in a few examples we show how we can effectively linearize them

Lie symmetries of multidimensional difference equations

A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be used to perform symmetry reduction. The method generalizes one presented in a recent publication for the case of ordinary difference equations. In turn, it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

Lie point symmetries and ODEs passing the Painlev\'e test

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlev\'e property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlev\'e transcendents only $P_{III}$, $P_V$ and $P_{VI}$ have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlev\'e transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlev\'e test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents

On Partial Differential and Difference Equations with Symmetries Depending on Arbitrary Functions

In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show on a few examples, both in partial differential and partial difference equations when this happens. Moreover we show that the infinitesimal generators of generalized symmetries depending on arbitrary functions, both for continuous and discrete equations, effectively play the role of master symmetries