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

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

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

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

Variable Coefficient Third Order KdV Type of Equations

We show that the integrable subclassess of a class of third order non-autonomous equations are identical with the integrable subclassess of the autonomous ones.Comment: Latex file , 15 page

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.Comment: 4 pages RevTex

Degenerate Svinolupov KdV systems

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

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.Comment: Latex File (AMS format), 23 pages, to be published in Journal of Mathematical Physic

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.Comment: 13 pages, 2 figure