A numerical method with properties of consistency in the energy domain for a class of dissipative nonlinear wave equations with applications to a Dirichlet boundary-value problem

In this work, we present a conditionally stable finite-difference scheme that consistently approximates the solution of a general class of (3+1)-dimensional nonlinear equations that generalizes in various ways the quantitative model governing discrete arrays consisting of coupled harmonic oscillators. Associated with this method, there exists a discrete scheme of energy that consistently approximates its continuous counterpart. The method has the properties that the associated rate of change of the discrete energy consistently approximates its continuous counterpart, and it approximates both a fully continuous medium and a spatially discretized system. Conditional stability of the numerical technique is established, and applications are provided to the existence of the process of nonlinear supratransmission in generalized Klein-Gordon systems and the propagation of binary signals in semi-unbounded, three-dimensional arrays of harmonic oscillators coupled through springs and perturbed harmonically at the boundaries, where the basic model is a modified sine-Gordon equation; our results show that a perfect transmission is achieved via the modulation of the driving amplitude at the boundary. Additionally, we present an example of a nonlinear system with a forbidden band-gap which does not present supratransmission, thus establishing that the existence of a forbidden band-gap in the linear dispersion relation of a nonlinear system is not a sufficient condition for the system to present supratransmission

The collision of two-kinks defects

We have investigated the head-on collision of a two-kink and a two-antikink pair that arises as a generalization of the $\phi^4$ model. We have evolved numerically the Klein-Gordon equation with a new spectral algorithm whose accuracy and convergence were attested by the numerical tests. As a general result, the two-kink pair is annihilated radiating away most of the scalar field. It is possible the production of oscillons-like configurations after the collision that bounce and coalesce to form a small amplitude oscillon at the origin. The new feature is the formation of a sequence of quasi-stationary structures that we have identified as lump-like solutions of non-topological nature. The amount of time these structures survives depends on the fine-tuning of the impact velocity.Comment: 14 pages, 9 figure

A Kolmogorov-Zakharov Spectrum in AdSAdS Gravitational Collapse

We study black hole formation during the gravitational collapse of a massless scalar field in asymptotically $AdS_D$ spacetimes for $D=4,5$. We conclude that spherically symmetric gravitational collapse in asymptotically $AdS$ spaces is turbulent and characterized by a Kolmogorov-Zakharov spectrum. Namely, we find that after an initial period of weakly nonlinear evolution, there is a regime where the power spectrum of the Ricci scalar evolves as $\omega^{-s}$ with the frequency, $\omega$, and $s\approx 1.7\pm 0.1$.Comment: 5 pages, 4 figures. v2: Typos, other initial profile considered for universality, error analysis, close to PRL versio

Breathers in oscillator chains with Hertzian interactions

We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflections are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors

The global nonlinear stability of Minkowski space. Einstein equations, f(R)-modified gravity, and Klein-Gordon fields

We study the initial value problem for two fundamental theories of gravity, that is, Einstein's field equations of general relativity and the (fourth-order) field equations of f(R) modified gravity. For both of these physical theories, we investigate the global dynamics of a self-gravitating massive matter field when an initial data set is prescribed on an asymptotically flat and spacelike hypersurface, provided these data are sufficiently close to data in Minkowski spacetime. Under such conditions, we thus establish the global nonlinear stability of Minkowski spacetime in presence of massive matter. In addition, we provide a rigorous mathematical validation of the f(R) theory based on analyzing a singular limit problem, when the function f(R) arising in the generalized Hilbert-Einstein functional approaches the scalar curvature function R of the standard Hilbert-Einstein functional. In this limit we prove that f(R) Cauchy developments converge to Einstein's Cauchy developments in the regime close to Minkowski space. Our proofs rely on a new strategy, introduced here and referred to as the Euclidian-Hyperboloidal Foliation Method (EHFM). This is a major extension of the Hyperboloidal Foliation Method (HFM) which we used earlier for the Einstein-massive field system but for a restricted class of initial data. Here, the data are solely assumed to satisfy an asymptotic flatness condition and be small in a weighted energy norm. These results for matter spacetimes provide a significant extension to the existing stability theory for vacuum spacetimes, developed by Christodoulou and Klainerman and revisited by Lindblad and Rodnianski.Comment: 127 pages. Selected chapters from a boo