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5 results on '"quad-graph equations"'

1. Integrability conditions for nonautonomous quad-graph equations

Author

Sahadevan, R., Rasin, O.G., and Hydon, P.E.

Subjects
*PHYSICS, *EQUATIONS, *ALGEBRA, *MATHEMATICS
Abstract

Abstract: This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations. [Copyright &y& Elsevier]

Published
2007
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2. A discrete Darboux–Lax scheme for integrable difference equations.

Author

Fisenko, X., Konstantinou-Rizos, S., and Xenitidis, P.

Subjects
*DIFFERENCE equations, *DARBOUX transformations, *LAX pair, *SUPERPOSITION principle (Physics), *BACKLUND transformations, *DISCRETE systems
Abstract

We propose a discrete Darboux–Lax scheme for deriving auto-Bäcklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler–Yamilov type system which is related to the nonlinear Schrödinger (NLS) equation [7]. In particular, we construct an auto-Bäcklund transformation for this discrete system, its superposition principle, and we employ them in the construction of the one- and two-soliton solutions of the Adler–Yamilov system. 02.30.Ik, 02.90.+p, 03.65.Fd 37K60, 39A36, 35Q55, 16T25. • A new method for solving integrable nonlinear partial diference equations (P Δ Es) • Systematic derivation of soliton solutions to integrable nonlinear P Δ Es • Study of an Adler-Yamilov type of system which discretises the NLS equation • Construction of Darboux and Bäcklund transformations for the Adler-Yamilov system • Derivation of one - and two-soliton solutions of the Adler-Yamilov system [ABSTRACT FROM AUTHOR]

Published
2022
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3. Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations

Author

Decio Levi and Christian Scimiterna

Subjects
quad-graph equations, linearizability, point transformations, Hopf-Cole transformations, Mathematics, QA1-939
Abstract

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.

Published
2011
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4. Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations

Author

Christian Scimiterna and Decio Levi

Subjects
Discrete mathematics, Pure mathematics, linearizability, Linearizability, Nonlinear Sciences - Exactly Solvable and Integrable Systems, lcsh:Mathematics, FOS: Physical sciences, Mathematical Physics (math-ph), Partial difference equations, lcsh:QA1-939, Set (abstract data type), Nonlinear system, point transformations, Hopf-Cole transformations, Nonlinear Sciences::Exactly Solvable and Integrable Systems, Graph (abstract data type), Point (geometry), Geometry and Topology, Exactly Solvable and Integrable Systems (nlin.SI), Analysis, Mathematical Physics, Mathematics, quad-graph equations
Abstract

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.

Published
2011

5. Integrability conditions for nonautonomous quad-graph equations

Author

Peter E. Hydon, R. Sahadevan, and Olexandr G. Rasin

Subjects
Conservation law, Pure mathematics, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Differential equation, Applied Mathematics, Existential quantification, Mathematical analysis, FOS: Physical sciences, Partial difference equations, Connection (mathematics), Completely integrable partial difference equations, Singularity, Quad-graph equations, Lax pair, Graph (abstract data type), Exactly Solvable and Integrable Systems (nlin.SI), Analysis, Conservation laws, Mathematics
Abstract

This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations., Comment: 17 pages

Published
2006
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