Sérgio H. M. Soares

Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type

We study the existence of positive solutions of Hamiltonian-type systems of second-order elliptic PDE in the whole space. The systems depend on a small parameter and involve a potential having a global well structure. We use dual variational methods, a mountain-pass type approach and Fourier analysis to prove positive solutions exist for sufficiently small values of the parameter.

A sign-changing solution for an asymptotically linear Schrödinger equation

The aim of this paper is to present a sign-changing solution for a class of radially symmetric asymptotically linear Schrodinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda’s theorem.

Existence of Solution for a Class of Quasilinear Systems

Abstract This paper proves the existence of nontrivial solution for a class of quasilinear systems on bounded domains in ℝN, N ≥ 2, whose nonlinearity has a double criticality. The proof is based on a linking theorem without the Palais-Smale condition.

Mumford-Shah algorithm applied to videokeratography image processing and consequences to refractive power values

There are many corneal diseases that can be detected using an eye-care device called videokeratograph. The videokeratograph is based on the principle of an apparatus called Placido disc and is used to precisely measure the anterior surface of the cornea. This disc contains rings alternately white and black, which are reflected on the patients cornea during the examination. The device can find anomalies by analyzing the reflected image, using image-processing algorithms. Although the efficiency of most commercial videokeratographs is acceptable, manufacturers do not disseminate to the scientific community the technique used in the image analysis algorithms. This makes it difficult for the specialized researcher in order to find better algorithms for the image-processing and, consequently, increase the instruments precision. In this work we have segmented the Placido disc in polar coordinates by implementing a diagonal section of the image, in the radial direction. The objective is to find the inflection points in the signal obtained. In this paper the signal is studied by using the Mumford-Shah segmentation method. The results are compared to those obtained with other classic methods in the literature, e.g. Marr-Hildreth filters, numerical derivative, Fourier derivative, morphological Laplacian and Canny derivative. The best result was achieved by using the Mumford-Shah functional. Using this technique it was possible to find the inflection positions with higher accuracy. The method did not detect any false inflection. Mumford-Shahs method demonstrated also a high precision in the task of eliminating noises from the original signal.

Odd homoclinic orbits for a second order Hamiltonian system

Abstract The aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.