38
38

Sep 21, 2013
09/13

by
Christian Scimiterna; Decio Levi

texts

#
eye 38

#
favorite 0

#
comment 0

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

78
78

Sep 23, 2013
09/13

by
Decio Levi; Christian Scimiterna

texts

#
eye 78

#
favorite 0

#
comment 0

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

54
54

Sep 21, 2013
09/13

by
Decio Levi; Christian Scimiterna

texts

#
eye 54

#
favorite 0

#
comment 0

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

114
114

Jul 19, 2013
07/13

by
Christian Scimiterna; Decio Levi

texts

#
eye 114

#
favorite 0

#
comment 0

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

47
47

Sep 22, 2013
09/13

by
Decio Levi; Christian Scimiterna

texts

#
eye 47

#
favorite 0

#
comment 0

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

5
5.0

Jun 28, 2018
06/18

by
Giorgio Gubbiotti; Christian Scimiterna; Decio Levi

texts

#
eye 5

#
favorite 0

#
comment 0

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

57
57

Sep 21, 2013
09/13

by
Decio Levi; Matteo Petrera; Christian Scimiterna

texts

#
eye 57

#
favorite 0

#
comment 0

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

6
6.0

Jun 29, 2018
06/18

by
Giorgio Gubbiotti; Christian Scimiterna; Decio Levi

texts

#
eye 6

#
favorite 0

#
comment 0

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

50
50

Sep 19, 2013
09/13

by
Decio Levi; Matteo Petrera; Christian Scimiterna

texts

#
eye 50

#
favorite 0

#
comment 0

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

3
3.0

Jun 28, 2018
06/18

by
Giorgio Gubbiotti; Christian Scimiterna; Decio Levi

texts

#
eye 3

#
favorite 0

#
comment 0

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

6
6.0

Jun 28, 2018
06/18

by
Giorgio Gubbiotti; Decio Levi; Christian Scimiterna

texts

#
eye 6

#
favorite 0

#
comment 0

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

58
58

Sep 20, 2013
09/13

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

texts

#
eye 58

#
favorite 0

#
comment 0

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

53
53

Sep 23, 2013
09/13

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

texts

#
eye 53

#
favorite 0

#
comment 0

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

57
57

Sep 21, 2013
09/13

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

texts

#
eye 57

#
favorite 0

#
comment 0

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

68
68

Sep 21, 2013
09/13

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

texts

#
eye 68

#
favorite 0

#
comment 0

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