Rolci Cipolatti

On the existence of standing waves for a davey–stewartson system

We consider the standing waves for the Davey–Stewartson system in R2 and R3. By reducing this system to a single nonlinear equation of Schrodinger type, we study the existence, the regularity and asymptotics of ground states.

One-dimensional cubic-quintic Gross-Pitaevskii equation for Bose-Einstein condensates in a trap potential

By means of a new variational method we report a direct solution for quintic and cubic-quintic nonlinear 1D Gross-Pitaeskii equations (GPE) in a harmonic confined potential. We explore the influence of the quintic nonlinear term on the ideal 1D pure cigar-like shape model for attractive and repulsive atom-atom interaction in Bose-Einstein condensates (BEC). Also, we offer a closed analytical expression to evaluate the error produced when solely the cubic nonlinear GPE is considered in the description of 1D BEC.

Successive linear approximation for finite elasticity

A method of successive Lagrangian formulation of linear approximation for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of solving the nonlinear problem, by assuming time steps small enough and the reference configuration updated at every step, we can linearize the constitutive equation and reduce it to linear boundary value problems to be solved successively with incremental boundary data. Moreover, nearly incompressible elastic body is considered as an approximation to account for the condition of incompressibility. For the proposed method, numerical computations of pure shear of a square block for Mooney-Rivlin material are considered and the results are compared with the exact solutions. Mathematical subject classification: Primary: 65C20; Secondary: 74B20.

An inverse problem for a wave equation with arbitrary initial values and a finite time of observations

We consider a solution u(p, g, a, b) to an initial value–boundary value problem for a wave equation: and we discuss an inverse problem of determining a coefficient p(x) and a, b by observations of u(p, g, a, b)(x, t) in a neighbourhood ω of ∂Ω over a time interval (0, T) and ∂itu(p, g, a, b)(x, T0), x Ω, i = 0, 1, with T0 < T. We prove that if T − T0 and T0 are larger than the diameter of Ω, then we can choose a finite number of Dirichlet boundary inputs g1, ..., gN, so that the mapping is uniformly Lipschitz continuous with suitable Sobolev norms provided that {p, aj, bj}1 ≤ j ≤ N remains in some bounded set in a suitable Sobolev space. In our inverse problem, initial values are also unknown, and we do not assume any positivity of the initial values. Our key is a Carleman estimate and the exact controllability in a Sobolev space of higher order.

Two-photon absorption processes in semiconductor quantum dots

The two-photon absorption process is a nonlinear phenomenon that shows very low optical efficiency in bulk semiconductors. For this reason the detection of these processes becomes very difficult from the experimental point of view. Nevertheless, when dealing with semiconductor QDs with few nanometer radii, these transitions are enhanced and the detection is possible. In this contribution we present analytical calculations for the absorption coefficient in CdSe spherical QDs subjected to these second order processes, as a function of the characteristic dot parameters. The intensities of the absorption peaks, as well as the statistical treatment involving QDs ensembles, are also reported.

Bose-Einstein condensates in optical lattices: mathematical analysis and analytical approximate formulae

Abstract We show that the GPE with cubic nonlinearity, as a model for describing the one-dimensional Bose–Einstein condensates loaded into a harmonically confined optical lattice, presents a set of ground states which is orbitally stable for any value of the self-interaction (attractive and repulsive) parameter and laser intensity. We also derive a new formalism which gives explicit expressions for the minimum energy E min and the associated chemical potential μ 0 . On the basis of these formulae, we generalize the variational method to obtain approximate solutions, at any order of approximation, for E min , μ 0 and the ground state.

Identification of the collision kernel in the linear Boltzmann equation by a finite number of measurements on the boundary

In this paper we consider the inverse problem of recovering the collision kernel for the time dependent linear Boltzmann equation via a finite number of boundary measurements. We prove that this kernel can be uniquely determined by at most k measurements, provided that it belongs to a finite k-dimensional vector space.

A mathematical model for virus infection in a system of interacting computers

We introduce a simplified theoretical model to describe a virtual virus propagation process in a set of interacting computers. The propagation mechanisms considered here are those related to the reception of messages through internet as well as the ones concerning the simple exchange of files using recording devices as compact disks or the commonly used floppy disks. In spite of its inherent simplicity, this model provides a good idea of the infection process and trends. From the mathematical point of view, the nonlinear delay integral equation that we obtain here presents certain interesting features which are explored and enlightened in this paper.

On the Existence, Uniqueness and Regularity of Solutions of a Viscoelastic Stokes Problem Modelling Salt Rocks

A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of