Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations

For eta >= 0, we consider a family of damped wave equations u(u) + eta Lambda 1/2u(t) + au(t) + Lambda u = f(u), t > 0, x is an element of Omega subset of R-N, where -Lambda denotes the Laplacian with zero Dirichlet boundary condition in L-2(Omega). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space H-0(1)(Omega) x L-2(Omega) semigroups {T-eta(t) : t >= 0} which have global attractors A(eta) eta >= 0. We show that the family {A(eta)}(eta >= 0), behaves upper and lower semi-continuously as the parameter eta tends to 0(+).

On exact boundary controllability for linearly coupled wave equations

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

On asymptotic solutions of integrable wave equations

Asymptotic 'soliton train' solutions of integrable wave equations described by inverse scattering transform method with second-order scalar eigenvalue problem are considered. It is shown that if asymptotic solution can be presented as a modulated one-phase nonlinear periodic wavetrain, then the corresponding Baker-Akhiezer function transforms into quasiclassical eigenfunction of the linear spectral problem in weak dispersion limit for initially smooth pulses. In this quasiclassical limit the corresponding eigenvalues can be calculated with the use of the Bohr Sommerfeld quantization rule. The asymptotic distributions of solitons parameters obtained in this way specify the solution of the Whitham equations. (C) 2001 Elsevier B.V. B.V. All rights reserved.

Geometrical quaternionic coupling for three dimensional wave equations

The present work has the scope to show the relationship between four three-dimensional waves. This fact will be made in the form of coupling, using for it the Cauchy-Riemann conditions for quaternionic functions [#!BorgesZeMarcio!#], through certain Laplace's equation in [#!MaraoBorgesLP!#]. The coupling will relate those functions that determine the wave as well as their respective propagation speeds.

Comment on Wave functions for a Duffin-Kemmer-Petiau particle in a time-dependent potential [J. Math. Phys. 48, 073515 (2007)]

It is shown that the paper Wave functions for a Duffin-Kemmer-Petiau particle in a time-dependent potential by Merad and Bensaid [J. Math. Phys. 48, 073515 (2007)] is not correct in using inadvertently a non-Hermitian Hamiltonian in a formalism that does require Hermitian Hamiltonians.

Galilean covariance and non-relativistic Bhabha equations

We apply a five-dimensional formulation of Galilean covariance to construct non-relativistic Bhabha first-order wave equations which, depending on the representation, correspond either to the well known Dirac equation (for particles with spin 1/2) or the Duffin-Kemmer-Petiau equation (for spinless and spin 1 particles). Here the irreducible representations belong to the Lie algebra of the 'de Sitter group' in 4 + 1 dimensions, SO(5, 1). Using this approach, the non-relativistic limits of the corresponding equations are obtained directly, without taking any low-velocity approximation. As a simple illustration, we discuss the harmonic oscillator.

Classes of exact wave functions for general time-dependent Dirac Hamiltonians in 1+1 dimensions

The construction of two classes of exact solutions for the most general time-dependent Dirac Hamiltonian in 1+1 dimensions was discussed. The extension of solutions by introduction of a time-dependent mass was elaborated. The possibility of existence of a generalized Lewis-Riesenfeld invariant connected with such solutions was also analyzed.

Asymptotic approach for the nonlinear equatorial long wave interactions

In the present work we use an asymptotic approach to obtain the long wave equations. The shallow water equation is put as a function of an external parameter that is a measure of both the spatial scales anisotropy and the fast to slow time ratio. The values given to the external parameters are consistent with those computed using typical values of the perturbations in tropical dynamics. Asymptotically, the model converge toward the long wave model. Thus, it is possible to go toward the long wave approximation through intermediate realizable states. With this approach, the resonant nonlinear wave interactions are studied. To simplify, the reduced dynamics of a single resonant triad is used for some selected equatorial trios. It was verified by both theoretical and numerical results that the nonlinear energy exchange period increases smoothly as we move toward the long wave approach. The magnitude of the energy exchanges is also modified, but in this case depends on the particular triad used and also on the initial energy partition among the triad components. Some implications of the results for the tropical dynamics are disccussed. In particular, we discuss the implications of the results for El Nĩo and the Madden-Julian in connection with other scales of time and spatial variability. © Published under licence by IOP Publishing Ltd.

Galilean covariant Lagrangian models

We construct non-relativistic Lagrangian field models by enforcing Galilean covariance with a (4, 1) Minkowski manifold followed by a projection onto the (3, 1) Newtonian spacetime. We discuss scalar, Fermi and gauge fields, as well as interactions between these fields, preparing the stage for their quantization. We show that the Galilean covariant formalism provides an elegant construction of the Lagrangians which describe the electric and magnetic limits of Galilean electromagnetism. Similarly we obtain non-relativistic limits for the Proca field. Then we study Dirac Lagrangians and retrieve the Levy-Leblond wave equations when the Fermi field interacts with an Abelian gauge field.

LARGE ATOMIC and NUCLEAR THREE-BODY SYSTEMS: SCATTERING and BINDING

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Low-frequency absorption cross section of the electromagnetic waves for extreme Reissner-Nordstrom black holes in higher dimensions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

The effect of bulk light absorption on running photorefractive holograms

The influence of bulk light absorption on running photorefractive holograms is investigated. By solving the coupled wave equations we prove that the beam intensities, but not the beam phases, can be calculated by averaging the coupling constant over the crystal thickness. We show the importance of the effect by calculating the dielectric relaxation time at the crystal front, and from that the quantum efficiency from a feedback-controlled experiment with a 2.05 mm thick BTO crystal.We propose to simulate the effect of bulk light absorption by a rude estimate of the average dielectric relaxation time which is related in a simple way to the dielectric relaxation time at the crystal front, in doing so an error of less than 10% is introduced.

Galilean covariance and the Duffin-Kemmer-Petiau equation

We use a five-dimensional approach to Galilean covariance to investigate the non-relativistic Duffin-Kemmer-Petiau first-order wave equations for spinless particles. The corresponding representation is generated by five 6 × 6 matrices. We consider the harmonic oscillator as an example.

Stability of trapped Bose-Einstein condensates

In three-dimensional trapped Bose-Einstein condensate (BEC), described by the time-dependent Gross-Pitaevskii-Ginzburg equation, we study the effect of initial conditions on stability using a Gaussian variational approach and exact numerical simulations. We also discuss the validity of the criterion for stability suggested by Vakhitov and Kolokolov. The maximum initial chirp (initial focusing defocusing of cloud) that can lead a stable condensate to collapse even before the number of atoms reaches its critical limit is obtained for several specific cases. When we consider two- and three-body nonlinear terms, with negative cubic and positive quintic terms, we have the conditions for the existence of two phases in the condensate. In this case, the magnitude of the oscillations between the two phases are studied considering sufficient large initial chirps. The occurrence of collapse in a BEC with repulsive two-body interaction is also shown to be possible.

Chaotic oscillation in an attractive Bose-Einstein condensate under an impulsive force

The chaotic oscillation in an attractive Bose-Einstein condensate (BEC) under an impulsive force was discussed using mean-field Gross-Pitaevskii (GP) equation. It was found that sustained chaotic oscillation resulted in a BEC under the action of an impulsive force generated by suddenly changing the interatomic scattering length or the harmonic oscillator trapping potential. The analysis suggested that the final state interatomic attraction played an important role in the generation of the chaotic dynamics.