Ognyan Christov

On the Kolmogorov’s condition for a special case of the Kirchhoff top

In this article the classical Kirchhoff case of motion of a rigid body in an infinite ideal fluid is considered. Then for the corresponding Hamiltonian system on the zero integral level, the Kolmogorov’s condition which is important for Kolmogorov–Arnold–Moser theory is checked. In contrast to known similar results, there exists a curve in the bifurcation diagram along which the Kolmogorov’s condition vanishes for certain values of the parameters.

ON THE NONLOCAL SYMMETRIES OF THE μ-CAMASSA–HOLM EQUATION

The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm and the Hunter–Saxton equations. This equation admits quadratic pseudo-potentials which allow us to compute some first-order nonlocal symmetries. The found symmetries preserve the mean of solutions. Finally, we discuss also the associated μCH equation.

Non-Integrability of Some Higher-Order Painlev e Equations in the Sense of Liouville

In this paper we study the equation w (4) = 5w 00 (w 2 w 0 ) + 5w(w 0 ) 2 w 5 + (z + )w +; which is one of the higher-order Painlev e equations (i.e., equations in the polynomial class having the Painlev e property). Like the classical Painlev e equations, this equation admits a Hamiltonian formulation, Backlund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters = = 3k, = = 3k 1, k2 Z, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin{Morales-Ruiz{Ramis approach. Then we study the integrability of the second and third members of the PII-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.

Near Integrability in Low Dimensional Gross-Neveu Models

Abstract The low-dimensional Gross-Neveu models are studied. For the systems, related to the Lie algebras so(4), so(5), sp(4), sl(3), we prove that they have Birkhoff-Gustavson normal forms which are integrable and non-degenerate in Kolmogorov-Arnold-Moser (KAM) theory sense. Unfortunately, this is not the case for systems with three degrees of freedom, related to the Lie algebras so(6) ~ sl(4), so(7), sp(6); their Birkhoff-Gustavson normal forms are proven to be non-integrable in the Liouville sense. The last result can easily be extended to higher dimensions.

Geometric Integrability of Some Generalizations of the Camassa-Holm Equation

We study the Camassa-Holm (CH) equation and recently introduced μCH equation from the geometric point of view. We show that Kupershmidt deformations of these equations describe pseudospherical surfaces and hence are geometrically integrable.

Effective solutions of Clebsch and C. Neumann systems

We solve in Riemann theta functions the classical Clebsch system and its particular case the C Neumann system Going from a new Lax representation with rational parameter we work out the solutions Separately using relations between theta func tions we check that the corresponding expressions in theta functions satisfy Clebsch and Neumann systems Table of contents Introduction Algebro geometric integration Proof of Theorem Closed geodesics on the axial ellipsoid Introduction The equations of motion of a rigid body in an ideal uid are x x H p p x H x p H p where H is certain quadratic form in x and p A nontrivial integrable case of equations is the Clebsch case which is characterized with

Near-Integrability of Low-Dimensional Periodic Klein-Gordon Lattices

The low dimensional periodic Klein-Gordon lattices are studied for integrability. We prove that the periodic lattice with two particles and certain nonlinear potential is non integrable. However, in the cases of up to six particles, we prove that their Birkhoff-Gustavson normal forms are integrable, which allows us to apply KAM theory.

On the Number of Spikes of Solutions for a Singularly Perturbed Boundary-Value Problem

We study the stationary solutions of the famous Fisher - Kolmogorov - Petrovsky - Piscounov (FKPP) equation

On the inverse scattering approach and action-angle variables for the Dullin–Gottwald–Holm equation

u_t = D u_{x x} + \gamma u (1 - u),