Pedro J. Torres

Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem

This paper is devoted to study the existence of periodic solutions of the second-order equation x 00 ¼ f ðt; xÞ; where f is a Caratheodory function, by combining some new properties of Greens function together with Krasnoselskii fixed point theorem on compression and expansion of cones. As applications, we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity. In this latter case, our results cover equations with weak singularities and are compared with some recent results by I. Rachunkova´ , M. Tvrdyand I. Vrkoc˘ . r 2002 Elsevier Science (USA). All rights reserved.

Lie Symmetries and Solitons in Nonlinear Systems with Spatially Inhomogeneous Nonlinearities

Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to show that localized nonlinearities can support bound states with an arbitrary number solitons, and discuss other applications of interest to the field of nonlinear matter waves.

The Method of Moments for Nonlinear Schrödinger Equations: Theory and Applications

The method of moments in the context of nonlinear Schrodinger equations relies on defining a set of integral quantities, which characterize the solution of this partial differential equation and whose evolution can be obtained from a set of ordinary differential equations. In this paper we find all cases in which the method of moments leads to closed evolution equations, thus extending and unifying previous works in the field of applications. For some cases in which the method fails to provide rigorous information we also develop approximate methods based on it, which allow us to get some approximate information on the dynamics of the solutions of the nonlinear Schrodinger equation.

Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity

It is proved that a periodically forced second-order equation with a singular nonlinearity in the origin with linear growth in infinity possesses a T -periodic stable solution for high values of the mean value of the forcing term. The method of proof combines a rescaling argument with the analysis of the first twist coefficient of the

Periodic solutions of singular systems without the strong force condition

. We present sufficient conditions for the existence of at least a non-collision periodic solution for singular systems under weak force conditions. We deal with two different types of systems. First, we assume that the system is generated by a potential, and then we consider systems without such hypothesis. In both cases we use the same technique based on Schauder fixed point theorem. Recent results in the literature are significantly improved.

Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates

A quantized vortex dipole is the simplest vortex molecule, comprising two countercirculating vortex lines in a superfluid. Although vortex dipoles are endemic in two-dimensional superfluids, the precise details of their dynamics have remained largely unexplored. We present here several striking observations of vortex dipoles in dilute-gas Bose-Einstein condensates, and develop a vortex-particle model that generates vortex line trajectories that are in good agreement with the experimental data. Interestingly, these diverse trajectories exhibit essentially identical quasiperiodic behavior, in which the vortex lines undergo stable epicyclic orbits.

Periodic Motions of Linear Impact Oscillators via the Successor Map

We investigate the existence and multiplicity of nontrivial periodic bouncing solutions for linear and asymptotically linear impact oscillators by applying a generalized version of the Poincare--Birkhoff theorem to an adequate Poincare section called the successor map. The main theorem includes a generalization of a related result by Bonheure and Fabry and provides a sufficient condition for the existence of periodic bouncing solutions for Hills equation with obstacle at

Dynamics of a few corotating vortices in Bose-Einstein condensates.

x\not =0

Periodic Solutions of a Singular Equation With Indefinite Weight

.

Dynamics of vortex dipoles in confined Bose–Einstein condensates

We study the dynamics of small vortex clusters with a few (2-4) corotating vortices in Bose-Einstein condensates by means of experiments, numerical computations, and theoretical analysis. All of these approaches corroborate the counterintuitive presence of a dynamical instability of symmetric vortex configurations. The instability arises as a pitchfork bifurcation at sufficiently large values of the vortex system angular momentum that induces the emergence and stabilization of asymmetric rotating vortex configurations. The latter are quantified in the theoretical model and observed in the experiments. The dynamics is explored both for the integrable two-vortex particle system, where a reduction of the phase space of the system provides valuable insight, as well as for the nonintegrable three- (or more) vortex case, which additionally admits the possibility of chaotic trajectories.