Salvador Jiménez

A numerical scheme for the simulation of blow-up in the nonlinear Schrödinger equation

We present a self-adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear Schrodinger equations (NLSEs). As an example we study the character of the focusing singularity of the two-dimensional critical NLSE.

Spectral information retrieval from integrated broadband photodiode Martian ultraviolet measurements

We propose an algorithm to retrieve the global features of the spectral dependence of the ultraviolet (UV) irradiance from integrated, broadband UV measurements performed with a set of photodiodes with different UV filters. This fit, when applied to ground-based measurements and compared with the incident Solar spectral irradiance on the top of the atmosphere, may be used to extract the spectral dependence of the UV opacity and the most relevant parameters characterizing the scattering with atmospheric aerosol (Angstrom exponent, etc.) as well as the biological effective doses. In this way, using a set of photodiodes instead of a spectrophotometer, one may get spectral information within very low mass, package, and weight constraints, which is particularly useful for space missions. We consider its application for the rover-based exploration of the Martian ground, which is subject to daily and seasonal opacity variations.

Retrieval of ultraviolet spectral irradiance from filtered photodiode measurements

We derived a general purpose method to retrieve the ultraviolet (UV) spectral irradiance using filtered photodiodes. The algorithm retrieves through an iterative process, which is based on the application of the mean value theorem, the six unknown coefficients of an interpolating polynomial function of order 5. For illustration we retrieve the incident spectral irradiance of three representative study cases. We consider the output currents induced on six SiC filtered photodiodes by a D2 lamp in vacuum, by the same lamp with ozone absorption and by a Martian-like solar spectrum. The algorithm converges to an interpolating function that reproduces the spectrum within the range of response of the photodiodes and with an integral error <0.6% for each channel for the vacuum D2 spectrum. Since two photodiodes are centered on the ozone absorption band, this method successfully detects ozone absorption and estimates its ambient terrestrial concentration with <4% error. This algorithm will be used to retrieve the Martian surface incident spectrum with REMS UV photodiodes on board the MSL Rover mission (NASA). This method can be equivalently used with commercial photodiodes for other applications where an inexpensive, small and light detector is preferred to conventional spectroradiometers.

FRACTIONAL DUFFING'S EQUATION AND GEOMETRICAL RESONANCE

We investigate the Fractional Duffing equation in the presence of nonharmonic external perturbations. We have applied the concept of Geometrical Resonance to this equation. We have obtained the conditions that should be satisfied by the external driving forces in order to produce high-amplitude periodic oscillations avoiding chaos. We also show that, for Duffings equation with fractional damping, the perturbations that satisfy the Geometrical Resonance conditions are nonperiodic functions.

Arbitrarily large numbers of kink internal modes in inhomogeneous sine-Gordon equations

Abstract We prove analytically the existence of an infinite number of internal (shape) modes of sine-Gordon solitons in the presence of some inhomogeneous long-range forces, provided some conditions are satisfied.

Mathematics and Mars Exploration

In this study we consider modelization associated with study of solar radiation at the surface of Mars and the Martian atmosphere. In particular, we present elements concerning retrieval of the solar irradiance spectrum on the surface of Mars from data collected by arrays of photodiodes, such as those onboard the “Curiosity” MSL-rover and other missions currently under design. By using these techniques we are able to provide an approximate description of the expected measures. In this work we have also developed a new method of tomography-based signal analysis for detection of events in the Martian atmosphere boundary layer, such as dust devils. In general, this method enables detection of events that occur briefly in time and are localized in space. This tomographic method allows us to identify the presence of more dust devils than detected previously using the same data. Finally we show new scenarios of modelization through fractional differential equations associated with diffusion processes and nonlocal problems. Such approaches could be used to model complex Martian dynamics.

SOLVING TWO-POINT BOUNDARY VALUE PROBLEMS OF FRACTIONAL DIFFERENTIAL EQUATIONS VIA SPLINE COLLOCATION METHODS

In this article, we introduce spline collocation methods to solve fractional boundary value problems (BVPs). The existence and uniqueness theorem of collocation solutions is studied, and the error estimate of collocation solutions is discussed. Numerical experiments are presented to demonstrate the theoretical results.

Fate of the true-vacuum bubbles

We investigate the bounce solutions in vacuum decay problems. We show that it is possible to have a stable false vacuum in a potential that is unbounded from below.

Elements for the Retrieval of the Solar Spectrum on the Surface of Mars from an Array of Photodiodes

We present some elements concerning the retrieval of the Solar irradiance spectrum on the surface of Mars, from data collected by arrays of photodiodes, such as those aboard the “Curiosity” MSL-rover and other missions currently under design.

Elements of Newtonian Mechanics

Within the framework of Newtonian Mechanics, the dynamics of the motion of a single particle is related to the following properties: (1) the existence of either conservation or variation laws for some quantities according to certain relations, (2) the existence of symmetries and (3) the stability character of the equilibrium points of the system. Only in a few cases the analytical solution of the equations of motion is known and, for this reason, numerical simulations are usually necessary.