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.Comment: 12 pages, final versio

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.Comment: 17pp, LateX, to be published in J.Math.Phy

Prolongation Algebra and Backlund Transformations of Drinfeld-Sokolov System of Equations

We show that the Drinfeld-Sokolov system of equations has a nontrivial prolongation structure. The closure process for prolongation algebra gives rise to the sl(4,c) algebra which is used to derive the scattering problem for the system of equations under consideration. The nontrivial new Backlund transformations and some explicit solutions are given.Comment: 13 page

Integrability of a Non-autonomous Coupled KdV System

The Painlev\'{e} property of coupled, non-autonomous Korteweg-de Vries (KdV) type of systems is studied. The conditions under which the systems pass the Painlev\'{e} test for integrability are obtained. For some of the integrable cases, exact solutions are given.Comment: 7 pages, 2 figures, to be published in IJMPC. For related figures, see http://www.metu.edu.tr/~akarasu/figures.htm

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.Comment: Proc. Workshop `Quantizaion, Dualities, and Integrable Systems,' 23-27 January 2006, Pamukkale University, Turkey; 7 pages. MSC 35Q53, 37K05, 37K10, 37K3

Singularity analysis of a spherical Kadomtsev-Petviashvili equation

The (2+1)-dimensional spherical Kadomtsev-Petviashvili (SKP) equation of J.-K. Xue [Phys. Lett. A 314:479-483 (2003)] fails the Painleve test for integrability at the highest resonance, where a nontrivial compatibility condition for recursion relations appears. This compatibility condition, however, is sufficiently weak and thus allows the SKP equation to possess an integrable (1+1)-dimensional reduction, which is detected by the Weiss method of truncated singular expansions.Comment: 7 page

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.Comment: 12 pages, 2 figures, typos corrected, version to appear in PR

Godel-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 Maxwell's 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.Comment: REVTeX4, 19 pp., no figures, improved and shortened version, note the slight change in the title [accepted for publication in Classical and Quantum Gravity

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.Comment: Latex file (amssymb), 10 page