A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space-Fractional Gross-Pitaevskii Equation

The present work departs from an extended form of the classical multi-dimensional Gross-Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross-Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system. © 2019 Ahmed S. Hendy et al., published by Sciendo 2019

Energy conservation issues in the numerical solution of the semilinear wave equation

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the specific boundary conditions at hand. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation.Comment: 41 pages, 11 figur

Symmetries in Quantum Mechanics and Statistical Physics

This book collects contributions to the Special Issue entitled "Symmetries in Quantum Mechanics and Statistical Physics" of the journal Symmetry. These contributions focus on recent advancements in the study of PT–invariance of non-Hermitian Hamiltonians, the supersymmetric quantum mechanics of relativistic and non-relativisitc systems, duality transformations for power–law potentials and conformal transformations. New aspects on the spreading of wave packets are also discussed

Uniformly Accurate Methods for Klein-Gordon type Equations

The main contribution of this thesis is the development of a novel class of uniformly accurate methods for Klein-Gordon type equations. Klein-Gordon type equations in the non-relativistic limit regime, i.e., $c\gg 1$, are numerically very challenging to treat, since the solutions are highly oscillatory in time. Standard Gautschi-type methods suffer from severe time step restrictions as they require a CFL-condition $c^2\tau<1$ with time step size $\tau$ to resolve the oscillations. Within this thesis we overcome this difficulty by introducing limit integrators, which allows us to reduce the highly oscillatory problem to the integration of a non-oscillatory limit system. This procedure allows error bounds of order $\mathcal{O}(c^{-2}+\tau^2)$ without any step size restrictions. Thus, these integrators are very efficient in the regime $c\gg 1$. However, limit integrators fail for small values of $c$. In order to derive numerical schemes that work well for small as well as for large $c$, we use the ansatz of "twisted variables", which allows us to develop uniformly accurate methods with respect to $c$. In particular, we introduce efficient and robust uniformly accurate exponential-type integrators which resolve the solution in the relativistic regime as well as in the highly oscillatory non-relativistic regime without any step size restriction. In contrast to previous works, we do not employ any asymptotic nor multiscale expansion of the solution. Compared to classical methods our new schemes allow us to reduce the regularity assumptions as they converge under the same regularity assumptions required for the integration of the corresponding limit system. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie and Strang splitting schemes for the limit system. Moreover, we present uniformly accurate schemes for the Klein-Gordon-Schrödinger and the Klein-Gordon-Zakharov system. For all uniformly accurate integrators we establish rigorous error estimates and underline their uniform convergence property numerically

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation

Mathematical models for dispersive electromagnetic waves: an overview

In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background.Comment: 46 pages, 16 figure

Quantum brane cosmology

This thesis deals with the interaction of quantum mechanical models and cosmologies based on brane universes, an area of active theoretical speculation over the last five years.For convenience, the material has been split into two parts. Part 1 deals with a selection of background topics which are necessary and relevant to the original research. This research is presented in Part 2. In addition, some auxiliary topics, both more elementary and more advanced, are described in the appendices. The selection of background topics has been influenced by the various techniques, physical theories and mathematical technologies which play a major role in the work presented in Part 2. Although the exposition is ad hoc, an attempt has been made to systematically develop portions where the technique (or use of it) may be unfamiliar.A fairly complete treatment of the necessary mathematical scaffolding is supplied. Although important, this material is familiar or strongly mathematical, and is deferred to the appendices. This includes an elementary survey of functional analysis in Appendix A, sufficient to support a discussion of the path integral. The path integral formalism is used extensively throughout this thesis, and, where available, constitutes our preferred representation of quantum mechanics. The discussion is limited to the relevant portions of the theory: functions in Banacli spaces, and the Sturm-Liouville basis (technology which appears many times in Part 2); direct evaluation of Gaussian functional integrals, ubiquitous in field theory calculations; and ((-function regularization of the operator determinants to which such Gaussian integrals give rise, which has a direct application in Chapter 9. In Appendix B we describe the necessary framework of differential geometry which supports general relativity, and low-energy discussions of string theory. All calculations in metric gravity are based on differential geometry, together with a good proportion of the technology which buttresses quantum field theory on curved space time, string theory, and some more advanced representations of quantum mechanics (see below). All of this is used extensively throughout both parts of the thesis. We include some more advanced topological technology which supports the discussion of string compactification. General results from compactification theory, when appropriately interpreted in the brane context, contribute important stability results for zero-modes of the Kaluza—Klein fields, and provide a natural home for the spectral KK technology used (in one form or another) throughout Part 2, but most especially in Chapter 7 and Chapter 8. Einstein gravity and Yang-Mills theory are set in context as examples of connexions on fibre bundles