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 $r_{min} r_{max}$ acquires a new constant value $G^{prime}sim Gr_{max}/r_{min}$. 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 $r_{max}$ 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 $rgg 1/mu$ the Coulomb and Newton forces acquire the behaviour $sim 1/r$ (instead of $1/r^{2}$)

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 $r\gg 1/\mu$ the Coulomb and Newton forces acquire the behaviour $\sim 1/r$ (instead of $1/r^{2}$).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 $r_{\min} r_{\max}$ acquires a new constant value $G^{\prime}\sim Gr_{\max}/r_{\min}$. 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 $r_{\max}$ 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 $\Sigma\times {\bf R}$ (where $\Sigma$ 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 $\Sigma=S^2$ these link invariants are pure numbers and are equal to Reshetikhin-Turaev link invariants.Comment: 39, latex, 7 figure