1991 Summer Study Program in Geophysical Fluid Dynamics : patterns in fluid flow

The GFD program in 1991 focused on pattern forming processes in physics and geophysics. The pricipallecturer, Stephan Fauve, discussed a variety of systems, including our old favorite, Rayleigh-Bénard convection, but passing on to exotic examples such as vertically vibrated granular layers. Fauve's lectures emphasize a unified theoretical viewpoint based on symmetry arguments. Patterns produced by instabilties can be described by amplitude equations, whose form can be deduced by symmetry arguments, rather than the asymptotic expansions that have been the staple of past Summer GFD Programs. The amplitude equations are far simpler than the complete equations of motion, and symetry arguments are easier than asymptotic expansions. Symmetry arguments also explain why diverse systems are often described by the same amplitude equation. Even for granular layers, where there is not a universaly accepted continuum description, the appropnate amplitude equation can often be found using symmetry arguments and then compared with experiment. Our second speaker, Daniel Rothan, surveyed the state of the art in lattice gas computations. His lectures illustrate the great utility of these methods in simulating the flow of complex multiphase fluids, particularly at low Reynolds numbers. The lattice gas simulations reveal a complicated phenomenology much of which awaits analytic exploration. The fellowship lectures cover broad ground and reflect the interests of the staff members associated with the program. They range from the formation of sand dunes, though the theory of lattice gases, and on to two dimensional-turbulence and convection on planetary scales. Readers desiring to quote from these report should seek the permission of the authors (a partial list of electronic mail addresses is included on page v). As in previous years, these reports are extensively reworked for publication or appear as chapters in doctoral theses. The task of assembling the volume in 1991 was at first faciltated by our newly acquired computers, only to be complicated by hurricane Bob which severed electric power to Walsh Cottage in the final hectic days of the Summer.Funding was provided by the National Science Foundation through Grant No. OCE 8901012

Asymptotic stability of the fourth order ϕ4\phi^4 kink for general perturbations in the energy space

The Fourth order $\phi^4$ model generalizes the classical $\phi^4$ model of quantum field theory, sharing the same kink solution. It is also the dispersive counterpart of the well-known parabolic Cahn-Hilliard equation. Mathematically speaking, the kink is characterized by a fourth-order nonnegative linear operator with a simple kernel at the origin but no spectral gap. In this paper, we consider the kink of this theory, and prove orbital and asymptotic stability for any perturbation in the energy space

A hybrid model for large scale simulation of unsteady nonlinear waves

A hybrid model for simulating rogue waves in random seas on a large time and space scale is proposed in this thesis. It is formed by combining the derived fifth order Enhanced Nonlinear Schrödinger Equation based on Fourier transform (ENLSE-5F), the fully nonlinear Enhanced Spectral Boundary Integral (ESBI) method and its simplified version. The numerical techniques and algorithm for coupling three models on time scale are provided. Using them, and the switch between the three models during the computation is triggered automatically according to wave nonlinearities. Numerical tests are carried out and the results indicate that this hybrid model could simulate rogue waves both accurately and efficiently. In some cases showed, the hybrid model is more than 10 times faster than just using the ESBI method

Structure of New Solitary Solutions for The Schwarzian Korteweg De Vries Equation And (2+1)-Ablowitz-Kaup-Newell-Segur Equation

In this research, we introduce and represent the modified Khater method on two basic models in the optical fiber. These two models describe the dynamics of the wave movement in the optical fiber.  It is a new modification of new recent method which developed by Mostafa M. A. Khater in 2017. We implement this new modified technique on Schwarzian Korteweg de Vries equation and (2+1)-Ablowitz-Kaup-Newell-Segur equation. This modification of Khater method produces more closed solutions than many other methods. Schwarzian Korteweg de Vries (SKdV) equation has a closed relationship with (2+1)-Ablowitz- Kaup-Newell-Segur equation. Schwarzian Korteweg de Vries equation prescribes the location in a micro-segment of space and motion of the isolated waves in varied fields which localized in a tiny portion of space. It is a great and basic system in fluid mechanics, nonlinear optics, plasma physics, and quantum field theory

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time

Cauchy problem for the multi-dimensional Boussinesq type equation

AbstractThe paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation utt−Δu+Δ2u=Δσ(u). It proves that the Cauchy problem admits a global weak solution under the assumptions that σ∈C(R), σ(s) is of polynomial growth order, say p (>1), either 0⩽σ(s)s⩽β∫0sσ(τ)dτ, s∈R, where β>0 is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension N=1. In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown

Numerical and Analytical Methods in Electromagnetics

Like all branches of physics and engineering, electromagnetics relies on mathematical methods for modeling, simulation, and design procedures in all of its aspects (radiation, propagation, scattering, imaging, etc.). Originally, rigorous analytical techniques were the only machinery available to produce any useful results. In the 1960s and 1970s, emphasis was placed on asymptotic techniques, which produced approximations of the fields for very high frequencies when closed-form solutions were not feasible. Later, when computers demonstrated explosive progress, numerical techniques were utilized to develop approximate results of controllable accuracy for arbitrary geometries. In this Special Issue, the most recent advances in the aforementioned approaches are presented to illustrate the state-of-the-art mathematical techniques in electromagnetics

Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks

This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems