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A PDE modeling a reaction–diffusion physical system is discretized using its conditional symmetries. Discretization is carried out using two specific conditional symmetries. Explicit solutions of the difference equation are constructed when the symmetry is projective.

In this article we show that we can carry out the symmetry preserving discretization of the Boussinesq equation with respect to three of its more significant conditional symmetries. We perform the symmetry reduction of the obtained nonlinear discrete schemes with respect to the conditional symmetries and obtain the reduced discrete equations which unlike in the continuous case, are not always reducible to second order difference equations. A numerical comparison with the exact continuous solution given by Weierstrass elliptic functions is carried out.

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., 14 pages, 4 figures

Nonlinear PDEs having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples of new equations, constructed starting from the conditional symmetries of Boussinesq, are presented and discussed thoroughly to show and clarify the methodology introduced.

Nonlinear PDE's having {\bf given} conditional symmetries are constructed. They are obtained starting from the invariants of the "conditional symmetry" generator and imposing the extra condition given by the characteristic of the symmetry. Several of examples starting from the Boussinesq and including non-autonomous Korteweg-De Vries like equations are given to showcase the methodology introduced., 12 pages

An analysis of the spinless (1+1) D free-particle Salpeter equation is presented. Future investigations are suggested. İ Springer Nature Singapore Pte Ltd. 2018.

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra $o(3,1)$ as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane $E_2$. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a discretisation procedure developed earlier, we present a difference scheme that is invariant under the group $O(3,1)$ and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under $O(3,1)$ and is itself invariant under a subgroup of $O(3,1)$, namely the $O(2)$ rotations of the Euclidean plane.

The relativistic wave equations currently used in physics are the Dirac equation and the Klein-Gordon equation. Only recently, the Salpeter equation has stimulated the interest of the scientific community. In this paper a new relativistic wave equation for particles with non-zero rest mass, the Pearcey equation, is presented. There is great formal similarity between the Pearcey equation, the Schrödinger equation and the Salpeter equation. In particular, the Pearcey equation allows us to understand the onset of the relativistic behaviour in a similar way as the Salpeter equation. It is possibile also to keep the parallelism between optics and quantum mechanics at which we are used in non relativistic quantum mechanics.

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 Physics

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.

We consider a difference-difference Davey-Stewartson system together with its bilinear structure. We write some new Gram-type determinantal solutions taking into account a set of Jacobi identities for determinants. A bilinear Backlund transformation is constructed and consequently a Lax pair for the discrete system is derived.

We propose a differential difference equation in ${\mathcal R}^1\times {\mathcal Z}^2$ and study it by Hirota's bilinear method. This equation has a singular continuum limit into a system which admits the reduction to the Davey-Stewartson equation. The solutions of this discrete DS system are characterized by Casorati and Grammian determinants. Based on the bilinear form of this discrete DS system, we construct the bilinear B\"{a}cklund transformation which enables us to obtain its Lax pair., Comment: 12 pages, 2 figures

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., 17 pages. 1 figure

FELs based on SASE are believed to be powerful tools to explore the frontiers of basic sciences, from physics to chemistry to biology. Intense R&D programs have started in the USA and Europe in order to understand the SASE physics and to prove the feasibility of these sources. The allocation of considerable resources in the Italian National Research Plan (PNR) brought about the formation of a CNR–ENEA–INFN–University of Roma “Tor Vergata” study group. A conceptual design study has been developed and possible schemes for linac sources have been investigated, leading to the SPARX proposal. We report in this paper the results of a preliminary start to end simulation concerning one option we are considering based on an S-band normal conducting linac with high-brightness photoinjector integrated in an RF compressor.

We discuss the transverse evolution of the electromagnetic field in high gain free electron lasers and analyse the existence of soliton like solutions.

The main purpose of this article is to show how symmetry structures in par- tial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are com- pared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite- dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point in- variant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.

In this paper we construct the class of equations associated with a discrete spectral problem of third order. Among the equations of this class we find an integrable differential equation on the lattice with exponential nonlinearity. By considering its associated Darboux transformation we construct an explicit solution.

In this paper, we study the integrability of a class of nonlinear non-autonomous quad graph equations compatible around the cube introduced by Boll in the framework of the generalized Adler, Bobenko, and Suris (ABS) classification. We show that all these equations possess three-point generalized symmetries which are subcases of either the Yamilov discretization of the Krichever–Novikov equation or of its non-autonomous extension. We also prove that all those symmetries are integrable as they pass the algebraic entropy test.

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the discrete Sawada-Kotera equations. The resulting list contains 17 equations some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries., Comment: 29 pages

The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries. We do however construct a difference system approximating the Liouville equation that is invariant under the maximal finite subgroup ##IMG## [http://ej.iop.org/images/1751-8121/48/2/025204/jpa504755ieqn1.gif] {$S{{L}_{x}}(2,mathbb{R})otimes S{{L}_{y}}(2,mathbb{R})$} . The invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard (noninvariant) scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries.

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.

In this paper we study one aspect of the continuous symmetries of the Toda equation. Namely, we establish a correspondence between Backlund transformations and continuous symmetries of the Toda equation. A symmetry transformation acting on a solution of the Toda equation can be seen as a superposition of Backlund transformations. Conversely, a Backlund transformation can be written, at least formally, as a composition of infinitely many higher symmetry transformations. This result reinforces the opinion that the presence of sufficiently many continuous symmetries for discrete equations is an indication of their integrability.

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., submitted to J.Phys. A 10 Latex pages

A method is presented for calculating the Lie point symmetries of difference equations with one, or several, independent variables. The equations are given on a priori specified lattices. The Lie transformations act on the lattice, as well as on the equation. The transformations take solutions into solutions and can be used to perform symmetry reduction.

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.

The Lie algebra L(h) of point symmetries of a discrete analogue of the nonlinear Schrodinger equation (NLS) is described. In the continuous limit, the discrete equation is transformed into the NLS, while the structure of the Lie algebra changes: a contraction occurs with the lattice spacing h as the contraction parameter. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symmetry algebra of the NLS.

We discuss the role of dilation symmetries for differential difference equations depending on nearest-neighbour interactions. In particular, we show that for a simple class of differential difference equations of this kind, symmetries which depend linearly on lime are only compatible with linearizable equations.

Non linearities, due to the intensity growth, govern the saturation mechanisms in free electron laser devices. A toy model to describe the dynamical behavior of the device from small signal regime to saturation is obtained from the Ginzburg–Landau equation. It is shown that a proper extension of the model suggests the existence of solitary optical pulses.