18 research outputs found
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
Non-Integrability of Some Higher-Order Painlev\'e Equations in the Sense of Liouville
In this paper we study the equation 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, B\"acklund transformations and families of
rational and special functions. We prove that this equation considered as a
Hamiltonian system with parameters , , , 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 -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
On the integrability of a system describing the stationary solutions in Bose--Fermi mixtures
We study the integrability of a Hamiltonian system describing the stationary
solutions in Bose--Fermi mixtures in one dimensional optical lattices. We prove
that the system is integrable only when it is separable. The proof is based on
the Differential Galois approach and Ziglin-Morales-Ramis method.Comment: 21 page
On the KAM - Theory Conditions for the Kirchhoff Top
* Partially supported by Grant MM523/95 with Ministry of Science and Technologies.In this paper 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 KAM theory conditions
are checked. In contrast to the known similar results, there exists a curve
in the bifurcation diagram along which the Kolmogorov’s condition vanishes
for certain values of the parameters
Canonically Conjugate Variables for the μCH Equation
2010 Mathematics Subject Classification: 35Q35, 37K10.We consider the μCH equation which arises as an asymptotic rotator equation in a liquid crystal with a preferred direction if one takes into account the reciprocal action of dipoles on themselves. This equation is closely related to the periodic Camassa–Holm and the Hunter-Saxton equations. The μCH equation is also integrable and bi-Hamiltonian, that is, it is Hamiltonian with respect to two compatible Poisson brackets. We give a set of conjugated variables for both brackets.This work is partially supported by grant 193/2011 of Sofia University
Near-integrability of periodic Klein-Gordon lattices
In this paper we study the Klein-Gordon (KG) lattice with periodic boundary
conditions. It is an degrees of freedom Hamiltonian system with linear
inter-site forces and nonlinear on-site potential, which here is taken to be of
the form. First, we prove that the system in consideration is
non-integrable in Liuville sense. The proof is based on the Morales-Ramis
theory. Next, we deal with the resonant Birkhoff normal form of the KG
Hamiltonian, truncated to order four. Due to the choice of potential, the
periodic KG lattice shares the same set of discrete symmetries as the periodic
Fermi-Pasta-Ulam (FPU) chain. Then we show that the above normal form is
integrable. To do this we utilize the results of B. Rink on FPU chains. If
is odd this integrable normal form turns out to be KAM nondegenerate
Hamiltonian. This implies the existence of many invariant tori at low-energy
level in the dynamics of the periodic KG lattice, on which the motion is
quasi-periodic. We also prove that the KG lattice with Dirichlet boundary
conditions (that is, with fixed endpoints) admits an integrable, KAM
nondegenerated normal forth order form, which in turn shows that almost all
low-energetic solutions of KG lattice with fixed endpoints are quasi-periodic.Comment: arXiv admin note: text overlap with arXiv:1710.0413
Perturbations of Systems Describing the Motion of a Particle in Central Fields
The present paper deals with the KAM-theory conditions for systems describing the motion of a particle in central field