Atalay Karasu

A new integrable generalization of the Korteweg–de Vries equation

A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg–de Vries equation with a source. A Lax representation and an auto-Backlund transformation are found for the new equation, and its traveling wave solutions and generalized symmetries are studied.

On construction of recursion operators from Lax representation

In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations.

Integrable KdV systems: Recursion operators of degree four

Abstract The recursion operator and bi-Hamiltonian formulation of the Drinfeld-Sokolov system are given.

Integrable coupled KdV systems

We give the conditions for a system of N-coupled Korteweg de Vries (KdV) type of equations to be integrable. We find the recursion operators of each subclass and give all examples for N=2.

DEGENERATE SVINOLUPOV KDV SYSTEMS

Abstract We find infinitely many coupled systems of KdV type equations which are integrable. We give also their recursion operators.

Symmetric space property and an inverse scattering formulation of the SAS Einstein–Maxwell field equations

We formulate stationary axially symmetric (SAS) Einstein–Maxwell fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space. This result enables us to embed the configuration space into an eight‐dimensional flat manifold and formulate SAS Einstein–Maxwell fields as a σ‐model. We then give, in a coordinate free way, a Belinskii–Zakharov type of an inverse scattering transform technique for the field equations supplemented by a reduction scheme similar to that of Zakharov–Mikhailov and Mikhailov–Yarimchuk.

Closed timelike curves and geodesics of Godel-type metrics

It is shown explicitly that when the characteristic vector field that defines a Godel-type metric is also a Killing vector, there always exist closed timelike or null curves in spacetimes described by such a metric. For these geometries, the geodesic curves are also shown to be characterized by a lower-dimensional Lorentz force equation for a charged point particle in the relevant Riemannian background. Moreover, two explicit examples are given for which timelike and null geodesics can never be closed.

A Strange Recursion Operator for a New Integrable System of Coupled Korteweg-de Vries Equations

A recursion operator is constructed for a new integrable system of coupled Korteweg–de Vries equations by the method of gauge-invariant description of zero-curvature representations. This second-order recursion operator is characterized by unusual structure of its nonlocal part.

Gödel-type metrics in various dimensions

Godel-type metrics are introduced and used in producing charged dust solutions in various dimensions. The key ingredient is a (D − 1)-dimensional Riemannian geometry which is then employed in constructing solutions to the Einstein–Maxwell field equations with a dust distribution in D dimensions. The only essential field equation in the procedure turns out to be the source-free Maxwells equation in the relevant background. Similarly the geodesics of this type of metric are described by the Lorentz force equation for a charged particle in the lower dimensional geometry. It is explicitly shown with several examples that Godel-type metrics can be used in obtaining exact solutions to various supergravity theories and in constructing spacetimes that contain both closed timelike and closed null curves and that contain neither of these. Among the solutions that can be established using non-flat backgrounds, such as the Tangherlini metrics in (D − 1)-dimensions, there exists a class which can be interpreted as describing black-hole-type objects in a Godel-like universe.

Variable coefficient third order Korteweg-de Vries type of equations

It is shown that the integrable subclasses of the equations q,t=f(x,t)q,3 +H(x,t,q,q,1) are the same as the integrable subclasses of the equations q,t=q,3 +F(q,q,1).