Francisco Rus

Padé numerical method for the Rosenau-Hyman compacton equation

Three implicit finite difference methods based on Pade approximations in space are developed for the Rosenau-Hyman K(n,n) equation. The analytical solutions and their invariants are used to assess the accuracy of these methods. Shocks which develop after the interaction of compactons are shown to be independent of the numerical method and its parameters indicating that their origin may not be numerical. The accuracy in long-time integrations of high-order Pade methods is shown.

Self-similar radiation from numerical Rosenau-Hyman compactons

The numerical simulation of compactons, solitary waves with compact support, is characterized by the presence of spurious phenomena, as numerically induced radiation, which is illustrated here using four numerical methods applied to the Rosenau-Hyman K(p,p) equation. Both forward and backward radiations are emitted from the compacton presenting a self-similar shape which has been illustrated graphically by the proper scaling. A grid refinement study shows that the amplitude of the radiations decreases as the grid size does, confirming its numerical origin. The front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters. The amplitude of the radiations decreases exponentially in time, being characterized by a nearly constant scaling exponent. An ansatz for both the backward and forward radiations corresponding to a self-similar function characterized by the scaling exponent is suggested by the present numerical results.

Integrating real-time analysis in a component model for embedded systems

Component-based development is a key technology in the development of software for modern embedded systems. However, standard component models and tools are not suitable for this type of systems, since they do not explicitly address real-time, memory or cost constraints. This work presents a new predictable component model for embedded systems together with a set of tools to support it. The environment allows for the development of new components that can be assembled to build complete embedded applications, including hardware interaction. The main contribution of the environment is its support for real-time analysis at the component and application level. The analysis is achieved by combining component meta-information in the form of an abstract behaviour model and a method to measure worst-case execution times in the final platform.

A repository of equations with cosine/sine compactons

Nonlinear evolution equations with cosine/sine compacton solutions are reviewed, including the Rosenau-Hyman equation and generalizations of Korteweg-de Vries, Camassa-Holm, Boussinesq, Benjamin-Bona-Mahony, Klein-Gordon and other equations. Each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined. Several equations claimed in the literature to be different among them are found to be equivalent.

Numerical methods based on modified equations for nonlinear evolution equations with compactons

Abstract Compactons are traveling wave solutions with compact support resulting from the balance of both nonlinearity and nonlinear dispersion. Numerical methods with second-, fourth-, sixth-, and eighth-order approximations to the spatial derivatives obtained by means of the method of modified equations applied to the Ismail–Taha finite difference scheme for the Rosenau–Hyman equation are developed. The whole set of methods is compared among them in accuracy, invariant conservation, and in compacton collisions. The best method, among those studied, in terms of the tradeoff between accuracy and computational cost is determined.

Adiabatic perturbations for compactons under dissipation and numerically-induced dissipation

Compacton propagation under dissipation shows amplitude damping and the generation of tails. The numerical simulation of compactons by means of dissipative schemes also show the same behaviors. The truncation error terms of a numerical method can be considered as a perturbation of the original partial differential equation and perturbation methods can be applied to its analysis. For dissipative schemes, or when artificial dissipation is added, the adiabatic perturbation method yields evolution equations for the amplitude loss in the numerical solution and the amplitude of the numerically-induced tails. In this paper, such methods are applied to the K(2,2) Rosenau-Hyman equation, showing a very good agreement between perturbative and numerical results.

UM-RTCOM: An analyzable component model for real-time distributed systems

Component-based development is a key technology in the development of software for modern real-time systems. However, standard component models and tools are not suitable for this type of system, since they do not explicitly address real time, memory or cost constraints. This paper presents a new predictable component model for real-time systems (UM-RTCOM) together with a set of tools to support it. The environment allows new components to be developed which can then be assembled to build complete applications, including hardware interaction. The model includes support for real-time analysis at the component and application level. The analysis is achieved by combining component meta-information in the form of an abstract behaviour model and a method to measure worst-case execution times in the final platform. Additionally, we propose an implementation model based on RT-CORBA where the developer uses the UM-RTCOM components and a set of tools to map these elements to elements of the desired platform. In order to apply our proposals, we have used the model and tools in real applications specifically in the context of nuclear power plant simulators.

Removing trailing tails and delays induced by artificial dissipation in Padé numerical schemes for stable compacton collisions

The numerical simulation of colliding solitary waves with compact support arising from the Rosenau-Hyman K(n,n) equation requires the addition of artificial dissipation for stability in the majority of methods. The price to pay is the appearance of trailing tails, amplitude damping, and delays as the solution evolves. These undesirable effects can be corrected by properly counterbalancing two sources of artificial dissipation; this procedure is designed by using the slow time evolution of the parameters of the solitary waves under the presence of the dissipation determined by means of adiabatic perturbation methods. The validity of the tail removal methodology is demonstrated on a Pade numerical scheme. The tails are completely removed leaving only a small compact ripple at the original position of their front, and the numerical stability of the scheme under compacton collisions is preserved, as shown by extensive numerical experiments for several values of n.

Numerical interactions between compactons and kovatons of the Rosenau–Pikovsky K(cos) equation

A numerical study of the nonlinear wave solutions of the Rosenau–Pikovsky K(cos) equation is presented. This equation supports at least two kinds of solitary waves with compact support: compactons of varying amplitude and speed both bounded and kovatons which have the maximum compacton amplitude but arbitrary width. A new Pade numerical method is used to simulate the propagation and, with small artificial viscosity added, the interaction between these kind of solitary waves. Several numerically induced phenomena that appear while propagating these compact travelling waves are discussed quantitatively, including self-similar forward and backward wavepackets. The collisions of compactons and kovatons show new phenomena such as the inversion of compactons and the generation of pairwise ripples decomposing into small compacton–anticompacton pairs.

Solitary Waves on Graphene Superlattices

This chapter reviews the basic theoretical aspects of the propagation of solitary electromagnetic waves in graphene superlattices, a one atom thick sheet of graphene deposited on a superlattice, made by several periodically alternating layers of SiO\(_2\) and h-BN. The electronic band structure of graphene and the techniques of band gap engineering are briefly presented. The analysis of the electronic properties of graphene superlattices by using both the transfer matrix method and the Kronig–Penny model are summarized. The nonlinear wave equation for the vector potential of the electromagnetic wave field is derived. This graphene superlattice equation (GSLeq) generalizes the sine-Gordon equation (sGeq). Hence, it also has kink and antikink solutions propagating at a constant speed. There is no closed-form expression for their shape. A straightforward asymptotic method is applied in order to analytically approximate its shape. The interactions of kinks and antikinks is studied by using a numerical method, the Strauss–Vazquez, which is a conservative, finite difference scheme. This numerical method is second-order accurate in both space and time, and nonlinearly stable, exactly conserving a discrete energy. Extensive numerical results for the kink–antikink interactions are presented as a function of a asymptotic parameter. For small values of this parameter, the interaction is apparently elastic, without noticeable radiation, being very similar to that expected for the sGeq. For large values of the asymptotic parameter, the inelasticity of the interaction results in the emission of wavepackets of radiation. In summary, the whole set of results suggest that the GSLeq behaves as a nearly integrable perturbation of the sGeq. Consequently, graphene superlattices can be used to study nonlinear wave phenomena with electromagnetic waves in the THz scale.