68 research outputs found
Darboux integrability of trapezoidal and families of lattice equations I: First integrals
In this paper we prove that the trapezoidal and the families
of quad-equations are Darboux integrable systems. This result sheds light on
the fact that such equations are linearizable as it was proved using the
Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic
entropy, symmetries and linearization for quad equations consistent on the
cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with
some suggestions on how first integrals can be used to obtain general
solutions.Comment: 34 page
Integrability Test for Discrete Equations via Generalized Symmetries
In this article we present some integrability conditions for partial
difference equations obtained using the formal symmetries approach. We apply
them to find integrable partial difference equations contained in a class of
equations obtained by the multiple scale analysis of the general multilinear
dispersive difference equation defined on the square.Comment: Proceedings of the Symposium in Memoriam Marcos Moshinsk
Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations
In this paper we construct the autonomous quad-equations which admit as
symmetries the five-point differential-difference equations belonging to known
lists found by Garifullin, Yamilov and Levi. The obtained equations are
classified up to autonomous point transformations and some simple
non-autonomous transformations. We discuss our results in the framework of the
known literature. There are among them a few new examples of both sine-Gordon
and Liouville type equations.Comment: 27 page
Classification of five-point differential-difference equations
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 page
Lie point symmetries of differential--difference equations
We present an algorithm for determining the Lie point symmetries of
differential equations on fixed non transforming lattices, i.e. equations
involving both continuous and discrete independent variables. The symmetries of
a specific integrable discretization of the Krichever-Novikov equation, the
Toda lattice and Toda field theory are presented as examples of the general
method.Comment: 17 pages, 1 figur
The Generalized Symmetry Method for Discrete Equations
The generalized symmetry method is applied to a class of completely discrete
equations including the Adler-Bobenko-Suris list. Assuming the existence of a
generalized symmetry, we derive a few integrability conditions suitable for
testing and classifying equations of this class. Those conditions are used at
the end to test for integrability discretizations of some well-known hyperbolic
equations
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