Browsing by Subject "Integrability"

Now showing 1 - 6 of 6

Open Access
Boundary conditions compatible with the generalized symmetries
(1995) Gürel, T. Burak
In this work evolution type integrable equations and systems are considered. An efficient method is given to construct their boundary conditions and hence boundary value problems which are compatible with the generalized symmetries. This method is applied to some well-known nonlinear partial differential equations.
Open Access
Characteristic Lie algebra and classification of semidiscrete models
(Springer New York LLC, 2007) Habibullin, I. T.; Pekcan, A.
We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains. © Springer Science+Business Media, Inc. 2007.
Open Access
Integrability of three dimensional gravity field equations
(IOP Publishing Ltd, 2022-02-10) Gurses, Metin
We show that the tree dimensional Einstein vacuum feld equations with cosmological constant are integrable. Using the sl(2, R) valued soliton connections we obtain the metric of the spacetime in terms of the dynamical variables of the integrable nonlinear partial diferential equations. © 2021 Published under licence by IOP Publishing Ltd.
Open Access
On non-commutative integrable burgers equations
(Taylor & Francis Asia Pacific (Singapore), 2010) Gürses, M.; Karasu, A.; Turhan, R.
We construct the recursion operators for the non-commutative Burgers equations using their Lax operators. We investigate the existence of any integrable mixed version of left- and right-handed Burgers equations on higher symmetry grounds. © 2010 The Author(s).
Open Access
Sigma Models and Minimal Surfaces
(Springer Netherlands, 1998) Gürses, Metin
Correspondence is established between sigma models, minimal surfaces and the Monge-Ampére equation. The Lax pairs of the minimality condition of the minimal surfaces and the Monge-Ampére equations are given. Existence of infinitely many nonlocal conservation laws is shown and some Bäcklund transformations are also given.
Open Access
Symmetries and boundary conditions of integrable nonlinear partial differential equations
(1999) Gürel, T Burak
The solutions of initial-boundary value problems for integrable nonlinear partial differential equations have been one the most important problems in integrable systems on one hand, and on the other hand these kind problems have proved to be very hard especially when considered on half or bounded lines. The proper generalization of the Inverse Spectral Transform or any other possible method in a way that they apply on half or bounded lines, is a complicated problem itself. But one of the other obstacles is the choice of suitable boundary conditions. In this direction, there is a pioneering work of Sklyanin which has motivated us in considering the problem of establishing boundary conditions for integrable partial differential equations. To this end, we try to develop a way to find boundary conditions which would, in turn, be suitable for certain solution techniques. This could have been done in many different ways depending upon what is understood from integrability. Throughout this work, we use the phrase integrability \i\ the sense of generalized symmetries which has proved to be one of the most efficient approaches. We first give a proper definition of compatibility of boundary condition with a symmetry. Then we interpret the well known tools of symmetry approach in a different manner. These tools include the recursion operators and symmetries themselves. After some technical theorems, we pass to examples and consider many integrable equations. Furthermore, we give, a generalization of the method which makes use of the non-homogeneous symmetries. Finally we finish by some discrete equations, including the 2D Toda lattice. It is crucial to note that all the boundary conditions that have been already known to be compatible with the integrability property of the original equation, pass our criterion of compatibility.