364 research outputs found
Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations
We prove that the attractor of the 1D quintic complex Ginzburg-Landau
equation with a broken phase symmetry has strictly positive space-time entropy
for an open set of parameter values. The result is obtained by studying chaotic
oscillations in grids of weakly interacting solitons in a class of
Ginzburg-Landau type equations. We provide an analytic proof for the existence
of two-soliton configurations with chaotic temporal behavior, and construct
solutions which are closed to a grid of such chaotic soliton pairs, with every
pair in the grid well spatially separated from the neighboring ones for all
time. The temporal evolution of the well-separated multi-soliton structures is
described by a weakly coupled lattice dynamical system (LDS) for the
coordinates and phases of the solitons. We develop a version of normal
hyperbolicity theory for the weakly coupled LDSs with continuous time and
establish for them the existence of space-time chaotic patterns similar to the
Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory
may be of independent interest, the main difficulty addressed in the paper
concerns with lifting the space-time chaotic solutions of the LDS back to the
initial PDE. The equations we consider here are space-time autonomous, i.e. we
impose no spatial or temporal modulation which could prevent the individual
solitons in the grid from drifting towards each other and destroying the
well-separated grid structure in a finite time. We however manage to show that
the set of space-time chaotic solutions for which the random soliton drift is
arrested is large enough, so the corresponding space-time entropy is strictly
positive
Modification of Newton's law of gravity at very large distances
We discuss a Modified Field Theory (MOFT) in which the number of fields can vary. It is shown that when the number of fields is conserved MOFT reduces to the standard field theory but interaction constants undergo an additional renormalization and acquire a dependence on spatial scales. In particular, the renormalization of the gravitational constant leads to the deviation of the law of gravity from the Newtons law in some range of scales acquires a new constant value . From the dynamical standpoint this looks as if every point source is surrounded with a halo of dark matter. It is also shown that if the maximal scale is absent, the homogeneity of the dark matter in the Universe is consistent with a fractal distribution of baryons in space, in which the luminous matter is located on thin two-dimensional surfaces separated by empty regions of ever growing size
Corrections to the Newton and Coulomb potentials caused by effects of spacetime foam
We use an extended quantum field theory (EQFT) hep-th/9911168 to explore possible observational effects of the spacetime. It is shown that as it was expected the spacetime foam can provide quantum bose fields with a cutoff at very small scales, if the energy of zero - point fluctuations of fields is taken into account. It is also shown that EQFT changes the behaviour of massless fields at very large scales (in the classical region). We show that as the Coulomb and Newton forces acquire the behaviour (instead of )
Corrections to the Newton and Coulomb potentials caused by effects of spacetime foam
We use an extended quantum field theory (EQFT) hep-th/9911168 to explore
possible observational effects of the spacetime. It is shown that as it was
expected the spacetime foam can provide quantum bose fields with a cutoff at
very small scales, if the energy of zero - point fluctuations of fields is
taken into account. It is also shown that EQFT changes the behaviour of
massless fields at very large scales (in the classical region). We show that as
the Coulomb and Newton forces acquire the behaviour
(instead of ).Comment: Latex, 4 page
Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model
We give an analytic (free of computer assistance) proof of the existence of a
classical Lorenz attractor for an open set of parameter values of the Lorenz
model in the form of Yudovich-Morioka-Shimizu. The proof is based on detection
of a homoclinic butterfly with a zero saddle value and rigorous verification of
one of the Shilnikov criteria for the birth of the Lorenz attractor; we also
supply a proof for this criterion. The results are applied in order to give an
analytic proof of the existence of a robust, pseudohyperbolic strange attractor
(the so-called discrete Lorenz attractor) for an open set of parameter values
in a 4-parameter family of three-dimensional Henon-like diffeomorphisms
Oscillating mushrooms: adiabatic theory for a non-ergodic system
Can elliptic islands contribute to sustained energy growth as parameters of a
Hamiltonian system slowly vary with time? In this paper we show that a mushroom
billiard with a periodically oscillating boundary accelerates the particle
inside it exponentially fast. We provide an estimate for the rate of
acceleration. Our numerical experiments confirms the theory. We suggest that a
similar mechanism applies to general systems with mixed phase space.Comment: final revisio
Remarks on 2-dimensional HQFT's
We introduce and study algebraic structures underlying 2-dimensional Homotopy
Quantum Field Theories (HQFTs) with arbitrary target spaces. These algebraic
structures are formalized in the notion of a twisted Frobenius algebra. Our
work generalizes results of Brightwell, Turner, and the second author on
2-dimensional HQFTs with simply-connected or aspherical targets.Comment: 22 pages, 14 figures. In this version we added a detailed proof for
Theorem 3.3 and made some minor corrections
Generators of groups of Hamitonian maps
We prove that analytic Hamiltonian dynamics on tori, annuli, or Euclidean space can be approximated by a composition of nonlinear shear maps where each of the shears depends only on the position or only on the momentum
On modification of the Newton's law of gravity at very large distances
We discuss a Modified Field Theory (MOFT) in which the number of fields can vary. It is shown that when the number of fields is conserved MOFT reduces to the standard field theory but interaction constants undergo an additional renormalization and acquire a dependence on spatial scales. In particular, the renormalization of the gravitational constant leads to the deviation of the law of gravity from the Newton's law in some range of scales acquires a new constant value . From the dynamical standpoint this looks as if every point source is surrounded with a halo of dark matter. It is also shown that if the maximal scale is absent, the homogeneity of the dark matter in the Universe is consistent with a fractal distribution of baryons in space, in which the luminous matter is located on thin two-dimensional surfaces separated by empty regions of ever growing size
Link Invariants and Combinatorial Quantization of Hamiltonian Chern-Simons Theory
We define and study the properties of observables associated to any link in
(where is a compact surface) using the
combinatorial quantization of hamiltonian Chern-Simons theory. These
observables are traces of holonomies in a non commutative Yang-Mills theory
where the gauge symmetry is ensured by a quantum group. We show that these
observables are link invariants taking values in a non commutative algebra, the
so called Moduli Algebra. When these link invariants are pure
numbers and are equal to Reshetikhin-Turaev link invariants.Comment: 39, latex, 7 figure
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