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Atalay Karasu

Publications and source records attributed to Atalay Karasu.

15 recordsLinked to original sources
arXiv2016

Minimal Extension of Einstein's Theory: The Quartic Gravity

We study structure of solutions of the recently constructed minimal extensions of Einstein's gravity in four dimensions at the quartic curvature level. The extended higher derivative theory, just like Einstein's gravity, has only a massless spin-two graviton about its unique maximally symmetric vacuum. The extended theory does not admit the Schwarzschild or Kerr metrics as exact solutions, hence there is no issue of Schwarzschild type singularity but, approximately, outside a source, spherically symmetric metric with the correct Newtonian limit is recovered. We also show that for all Einstein space-times, square of the Riemann tensor (the Kretschmann scalar or the Gauss-Bonnet invariant) obeys a non-linear scalar Klein-Gordon equation.

hep-th↗
arXiv2007

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 a Backlund self-transformation are found of the new equation, and its travelling wave solutions and generalized symmetries are studied.

nlin.SI↗
arXiv2006

Gardner's deformations of the Boussinesq equations

Using the algebraic method of Gardner's deformations for completely integrable systems, we construct the recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these systems, we obtain new integrable equations adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.

nlin.SI↗
arXiv2006

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.

gr-qc↗
arXiv2002

Time Dependent Recursion Operators and Symmetries

The recursion operators and symmetries of non-autonomous, (1+1)-dimensional integrable evolution equations are considered. It has been previously observed that the symmetries of the integrable evolution equations obtained through their recursion operators do not satisfy the symmetry equations. There have been several attempts to resolve this problem. It is shown that in the case of time dependent evolution equations or time dependent recursion operators, associativity is lost. Due to this fact such recursion operators need modifications. A general formula is given for the missing term of the recursion operators. Apart from the recursion operators a method is introduced to calculate the correct symmetries. For illustrations several examples of scalar and coupled system of equations are considered.

nlin.SI↗
arXiv2002

Higher Dimensional Metrics of Colliding Gravitational Plane Waves

We give a higher even dimensional extension of vacuum colliding gravitational plane waves with the combinations of collinear and non-collinear polarized four-dimensional metric. The singularity structure of space-time depends on the parameters of the solution.

gr-qc↗
arXiv2001

Integrability of a Generalized Ito System: the Painleve Test

It is shown that a generalized Ito system of four coupled nonlinear evolution equations passes the Painleve test for integrability in five distinct cases, of which two were introduced recently by Tam, Hu and Wang. A conjecture is formulated on integrability of a vector generalization of the Ito system.

nlin.SI↗
arXiv2001

Non-autonomous Svinolupov Jordan KdV Systems

Non-autonomous Svinolupov-Jordan systems are considered. The integrability criteria of such systems are associated with the existence of recursion operators. A new non-autonomous KdV system is obtained and its recursion operator is given for all $N$. The examples for N=2 and N=3 are studied in detail. Some possible transformations are also discussed which map some systems to autonomous cases.

nlin.SI↗
arXiv2000

Some Higher Dimensional Vacuum Solutions

We study an even dimensional manifold with a pseudo-Riemannian metric with arbitrary signature and arbitrary dimensions. We consider the Ricci flat equations and present a procedure to construct solutions to some higher (even) dimensional Ricci flat field equations from the four diemnsional Ricci flat metrics. When the four dimensional Ricci flat geometry correponds to a colliding gravitational vacuum spacetime our approach provides an exact solution to the vacuum Einstein field equations for colliding graviational plane waves in an (arbitrary) even dimensional spacetime. We give explicitly higher dimensional Szekeres metrics and study their singularity behaviors.

gr-qc↗
arXiv1999

On Construction of Recursion Operators From Lax Representation

In this work we develop a general procedure for constructing the recursion operators fro non-linear integrable equations admitting Lax representation. Svereal new examples are given. In particular we find the recursion operators for some KdV-type of integrable equations.

solv-int↗
arXiv1997

Integrable Coupled KdV Systems

We give the conditions for a system of N- coupled Korteweg de Vries(KdV) type of equations to be integrable. Recursion operators of each subclasses are also given. All examples for N=2 are explicitly given.

solv-int↗