An integrable evolution equation for surface waves in deep water

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative method, an asymptotic model for small wave steepness ratio is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical results is performed.

Multiplicity of solutions for a biharmonic equation with subcritical or critical growth

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface

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

The Fokker-Planck equation for a bistable potential

The Fokker-Planck equation is studied through its relation to a Schrodinger-type equation. The advantage of this combination is that we can construct the probability distribution of the Fokker-Planck equation by using well-known solutions of the Schrodinger equation. By making use of such a combination, we present the solution of the Fokker-Planck equation for a bistable potential related to a double oscillator. Thus, we can observe the temporal evolution of the system describing its dynamic properties such as the time tau to overcome the barrier. By calculating the rates k = 1/tau as a function of the inverse scaled temperature 1/D, where D is the diffusion coefficient, we compare the aspect of the curve k x 1/D, with the ones obtained from other studies related to four different kinds of activated process. We notice that there are similarities in some ranges of the scaled temperatures, where the different processes follow the Arrhenius behavior. We propose that the type of bistable potential used in this study may be used, qualitatively, as a simple model, whose rates share common features with the rates of some single rate-limited thermally activated processes. (C) 2014 Elsevier B.V. All rights reserved.

Solution to bilharmonic equation with vanishing potential

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Complex modal analysis of a slender vertical rotor by finite elements method

In this paper, natural frequencies were analyzed (axial, torsional and flexural) and frequency response of a vertical rotor with a hard disk at the edge through the classical modal and complex analysis. The equation that rules the movement was obtained through the Lagrangian formulation. The model considered the effects of bending, torsion and axial deformation of the shaft, besides the gravitational and gyroscopic effects. The finite element method was used to discretize the structure into hollow cylindrical elements with 12 degrees of freedom. Mass, stiffness and gyroscopic matrices were explained consistently. The classical modal analysis, usually applied to stationary structures, does not consider an important characteristic of rotating machinery which are the methods of forward and backward whirl. Initially, through the traditional modal analysis, axial and torsional natural frequencies were obtained in a static shaft, since they do not suffer the influence of gyroscopic effects. Later research was performed by complex modal analysis. This type of tool, based on the use of complex coordinates to describe the dynamic behavior of rotating shaft, allows the decomposition of the system in two submodes, backward and forward. Thus, it is possible to clearly visualize that the orbit and direction of the precessional motion around the line of the rotating shaft is not deformed. A finite element program was developed using MATLAB (TM) and numerical simulations were performed to validate this model. Natural frequencies and directional frequency forced response (dFRF) were obtained using the complex modal analysis for a simple vertical rotor and also for a typical drill string used in the construction of oil wells.

Dinâmica de rotação de exoplanetas tipo terrestre com perturbação de terceiro corpo

In this work we study some topics of Celestial Mechanics, namely the problem of rigid body rotation and “spin-orbit” resonances. Emphasis is placed on the problem formulation and applications to some exoplanets with physical parameters (e.g. mass and radius) compatible with a terrestrial type constitution (e.g. rock) belonging to multiple planetary systems. The approach is both analytical and numerical. The analytical part consists of: i) the deduction of the equation of motion for the rotation problem of a spherical body with no symmetry, disturbed by a central body; ii) modeling the same problem by including a third-body in the planet-star system; iii) formulation of the concept of “spin-orbit” resonance in which the orbital period of the planet is an integer multiple of the rotation’s period. Topics of dynamical systems (e.g. equilibrium points, chaos, surface sections, etc.) will be included at this stage. In the numerical part simulations are performed with numerical models developed in the previous analytical section. As a first step we consider the orbit of the planet not perturbed by a third-body in the star-planet system. In this case the eccentricity and orbital semi-major axis of the planet are constants. Here the technique of surface sections, widely used in dynamical systems are applied. Next, we consider the action of a third body, developing a more realistic model for planetary rotation. The results in both cases are compared. Since the technique of disturbed surface sections is no longer applicable, we quantitatively evaluate the evolution of the characteristic angles of rotation (e.g. physical libration) by studying the evolution of individual orbits in the dynamically important regions of phase space, the latter obtained in the undisturbed case

Tsukamoto s Theorem in Characteristic two

In this paper it is proved that hermitian forms over quaternion division algebras over local fields of characteristic two are classified by their dimension and discriminant.

On the periodic orbits of the third-order differential equation x ′′′ − µx ′′ + x ′ − µx = εF (x, x ′ , x ′′)

In this paper we study the periodic orbits of the third-order differential equation x ′′′−µx ′′+ x ′ − µx = εF (x, x ′ , x ′′), where ε is a small parameter and the function F is of class C 2 .

Modelo de poços quânticos para a transferência de elétrons

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Equação de Fokker-Planck e enovelamento de proteínas

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

Multivacuum states in a fermionic gap equation with massive gluons and confinement

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrodinger equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

About the minimum time transference in a fractional differential equation

The use of fractional calculus when modeling phenomena allows new queries concerning the deepest parts of the physical laws involved in. Here we will be dealing with an apparent paradox in which the time of transference from zero in a system with fractional derivatives can be strictly shortened relatively to the minimal time transference done in an equivalent system in the frame of the entire derivatives.

Shallow viscous fluid heated from below and the Kadomtsev-Petviashvili equation

The non-linear evolution of nearly one-dimensional undamped waves in a viscous fluid adequately heated from below is shown to be governed by the Kadomtsev-Petviashvili equation. Its solitary-wave solution is explicitly shown. © 1990.