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Sep 23, 2013
09/13

by
Decio Levi; Matteo Petrera

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We expand a partial difference equation (P$\Delta$E) on multiple lattices and obtain the P$\Delta$E which governs its far field behaviour. The perturbative--reductive approach is here performed on well known nonlinear P$\Delta$Es, both integrable and non integrable. We study the cases of the lattice modified Korteweg--de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra--Kac--Van Moerbeke (VKVM) equation and a non integrable lattice KdV equation. Such reductions allow us to...

Source: http://arxiv.org/abs/math-ph/0510084v1

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Jul 19, 2013
07/13

by
Christian Scimiterna; Decio Levi

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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...

Source: http://arxiv.org/abs/1005.5288v2

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Sep 19, 2013
09/13

by
Decio Levi; Matteo Petrera

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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.

Source: http://arxiv.org/abs/math-ph/0701079v1

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Sep 21, 2013
09/13

by
Decio Levi; Christian Scimiterna

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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.

Source: http://arxiv.org/abs/1108.3648v1

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Jul 20, 2013
07/13

by
Decio Levi; Pavel Winternitz

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We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the "first differential approximation" is invariant under the same Lie group as the original ordinary differential equation.

Source: http://arxiv.org/abs/1304.7016v1

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Sep 22, 2013
09/13

by
Decio Levi; Christian Scimiterna

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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.

Source: http://arxiv.org/abs/1302.0154v1

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Sep 21, 2013
09/13

by
Christian Scimiterna; Decio Levi

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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 transformation

Source: http://arxiv.org/abs/1301.2426v1

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Sep 23, 2013
09/13

by
Decio Levi; Christian Scimiterna

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In this article we show that the complex Burgers and the Kundu--Eckhaus equations are related by a Miura transformation. We use this relation to discretize the Kundu--Eckhaus equation.

Source: http://arxiv.org/abs/0904.4844v1

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Jun 30, 2018
06/18

by
Decio Levi; Miguel A. Rodriguez

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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...

Topics: Mathematics, Numerical Analysis, Mathematical Physics

Source: http://arxiv.org/abs/1407.0838

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Sep 22, 2013
09/13

by
Decio Levi; Miguel A. Rodríguez

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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.

Source: http://arxiv.org/abs/math-ph/0307011v1

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Jun 30, 2018
06/18

by
Decio Levi; Luigi Martina; Pavel Winternitz

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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 subalgebra $ SL_x \lf 2 , \mathbb{R} \rg \otimes SL_y \lf 2 ,...

Topics: Mathematics, Numerical Analysis, Nonlinear Sciences, Exactly Solvable and Integrable Systems,...

Source: http://arxiv.org/abs/1407.4043

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Sep 21, 2013
09/13

by
Decio Levi; Sébastien Tremblay; Pavel Winternitz

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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...

Source: http://arxiv.org/abs/0709.3238v1

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Sep 17, 2013
09/13

by
Decio Levi; Piergiulio Tempesta; Pavel Winternitz

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We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schr\"{o}dinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space-time. In this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case. Moreover, systems that are integrable,...

Source: http://arxiv.org/abs/nlin/0305047v1

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Sep 18, 2013
09/13

by
Decio Levi; Zora Thomova; Pavel Winternitz

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We define infinitesimal contact transformations for ordinary difference schemes as transformations that depend on $K+1$ lattice points $(K \geq 1)$ and can be integrated to form a local or global Lie group. We then prove that such contact transformations do not exist.

Source: http://arxiv.org/abs/1102.2949v1

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Jun 27, 2018
06/18

by
Decio Levi; Luigi Martina; Pavel Winternitz

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The main purpose of this article is to show how symmetry structures in partial 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 compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of...

Topics: Nonlinear Sciences, Exactly Solvable and Integrable Systems, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1504.01953

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Jun 29, 2018
06/18

by
Giorgio Gubbiotti; Christian Scimiterna; Decio Levi

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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.

Topics: Exactly Solvable and Integrable Systems, Nonlinear Sciences, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1603.07930

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Sep 21, 2013
09/13

by
Decio Levi; Sébastien Tremblay; Pavel Winternitz

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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.

Source: http://arxiv.org/abs/0709.3112v1

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Sep 22, 2013
09/13

by
Decio Levi; Piergiulio Tempesta; Pavel Winternitz

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We show that the algebraic aspects of Lie symmetries and generalized symmetries in nonrelativistic and relativistic quantum mechanics can be preserved in linear lattice theories. The mathematical tool for symmetry preserving discretizations on regular lattices is the umbral calculus.

Source: http://arxiv.org/abs/hep-th/0310013v1

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Jun 28, 2018
06/18

by
Giorgio Gubbiotti; Christian Scimiterna; Decio Levi

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In this paper we study the integrability of a class of nonlinear non autonomous quad graph equations compatible around the cube introduced by Boll. 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 pass the algebraic entropy test.

Topics: Mathematics, Exactly Solvable and Integrable Systems, Mathematical Physics, Nonlinear Sciences

Source: http://arxiv.org/abs/1510.07175

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6.0

Jun 28, 2018
06/18

by
Giorgio Gubbiotti; Decio Levi; Christian Scimiterna

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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.

Topics: Mathematics, Exactly Solvable and Integrable Systems, Mathematical Physics, Nonlinear Sciences

Source: http://arxiv.org/abs/1512.01967

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Sep 23, 2013
09/13

by
Xiaoda Ji; Decio Levi; Matteo Petrera

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We expand a discrete--time lattice sine--Gordon equation on multiple lattices and obtain the partial difference equation which governs its far field behaviour. Such reduction allow us to obtain a new completely discrete nonlinear Schr\"oedinger (NLS) type equation.

Source: http://arxiv.org/abs/math-ph/0511006v1

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Sep 19, 2013
09/13

by
Decio Levi; Matteo Petrera; Christian Scimiterna

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In this paper we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized symmetries and master symmetries. We finally show that we can use master symmetries of the lSKdV equation to construct non-autonomous non-integrable generalized symmetries.

Source: http://arxiv.org/abs/math-ph/0701044v1

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Sep 21, 2013
09/13

by
Decio Levi; Matteo Petrera; Christian Scimiterna

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In this letter we present an analytic evidence of the non-integrability of the discrete nonlinear Schroedinger equation, a well-known discrete evolution equation which has been obtained in various contexts of physics and biology. We use a reductive perturbation technique to show an obstruction to its integrability.

Source: http://arxiv.org/abs/0808.0837v1

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5.0

Jun 28, 2018
06/18

by
Giorgio Gubbiotti; Christian Scimiterna; Decio Levi

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In this paper we introduce a non autonomous generalization of the $\QV$ equation introduced by Viallet. All the equations of Boll's classification appear in it for special choices of the parameters. Using the algebraic entropy test we infer that the equation should be integrable and with the aid of a formula introduced by Xenitidis we find its three point generalized symmetries.

Topics: Mathematics, Exactly Solvable and Integrable Systems, Mathematical Physics, Nonlinear Sciences

Source: http://arxiv.org/abs/1512.00395

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132

Jul 20, 2013
07/13

by
Gegenhasi; Xing-Biao Hu; Decio Levi

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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...

Source: http://arxiv.org/abs/nlin/0604066v1

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Sep 23, 2013
09/13

by
Decio Levi; Zora Thomova; Pavel Winternitz

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We define a class of transformations of the dependent and independent variables in an ordinary difference scheme. The transformations leave the solution set of the system invariant and reduces to a group of contact transformations in the continuous limit. We use a simple example to show that the class is not empty and that such "contact transformations for discrete systems" genuinely exist.

Source: http://arxiv.org/abs/1110.3409v1

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Jun 29, 2018
06/18

by
Alfred Michel Grundland; Decio Levi; Luigi Martina

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This paper is devoted to a study of the connections between three different analytic descriptions for the immersion functions of 2D-surfaces corresponding to the following three types of symmetries: gauge symmetries of the linear spectral problem, conformal transformations in the spectral parameter and generalized symmetries of the associated integrable system. After a brief exposition of the theory of soliton surfaces and of the main tool used to study classical and generalized Lie symmetries,...

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1603.07634

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Sep 23, 2013
09/13

by
Decio Levi; Pavel Winternitz; Ravil I. Yamilov

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A symmetry classification is performed for a class of differential-difference equations depending on 9 parameters. A 6-parameter subclass of these equations is an integrable discretization of the Krichever-Novikov equation. The dimension $n$ of the Lie point symmetry algebra satisfies $1 \le n \le 5$. The highest dimensions, namely $n=5$ and $n=4$ occur only in the integrable cases.

Source: http://arxiv.org/abs/1110.5021v1

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Sep 24, 2013
09/13

by
Natig Atakishiyev; Pedro Franco; Decio Levi; Orlando Ragnisco

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We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform.

Source: http://arxiv.org/abs/1112.2073v2

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Sep 20, 2013
09/13

by
Decio Levi; Matteo Petrera; Christian Scimiterna; Ravil Yamilov

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We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schr\"odinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to B\"acklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS...

Source: http://arxiv.org/abs/0802.1850v3

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Jun 30, 2018
06/18

by
Decio Levi; Eugenio Ricca; Zora Thomova; Pavel Winternitz

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The symmetry algebra of the differential--difference equation $$\dot u_n = [P(u_n)u_{n+1}u_{n-1} + Q(u_n)(u_{n+1}+u_{n-1})+ R(u_n)]/(u_{n+1}-u_{n-1}),$$ where $P$, $Q$ and $R$ are arbitrary analytic functions is shown to have the dimension $1 \le \mbox{dim}L \le 5$. When $P$, $Q$ and $R$ are specific second order polynomials in $u_n$ (depending on 6 constants) this is the integrable discretization of the Krichever--Novikov equation. We find 3 cases when the arbitrary functions are not...

Topics: Nonlinear Sciences, Mathematics, Exactly Solvable and Integrable Systems, Mathematical Physics,...

Source: http://arxiv.org/abs/1401.6991

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Sep 23, 2013
09/13

by
Rafael Hernandez Heredero; Decio Levi; Matteo Petrera; Christian Scimiterna

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We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation of the form of the nonlinear Schrodinger (NLS) equation. If the starting lattice equation is integrable then the resulting NLS equation turns out to be integrable, while if the starting equation is linearizable we get a linear Schrodinger equation. On the...

Source: http://arxiv.org/abs/0710.5299v2

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Sep 21, 2013
09/13

by
Rafael Hernandez Heredero; Decio Levi; Matteo Petrera; Christian Scimiterna

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We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.

Source: http://arxiv.org/abs/0709.3704v1

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Sep 21, 2013
09/13

by
Rafael Hernandez Heredero; Decio Levi; Matteo Petrera; Christian Scimiterna

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We present a discrete multiscale expansion of the lattice potential Korteweg-de Vries (lpKdV) equation on functions of infinite order of slow-varyness. To do so we introduce a formal expansion of the shift operator on many lattices holding at all orders. The lowest secularity condition from the expansion of the lpKdV equation gives a nonlinear lattice equation, depending on shifts of all orders, of the form of the nonlinear Schr\"odinger (NLS) equation

Source: http://arxiv.org/abs/0706.1046v1