Nonlinear Fourier Analysis for Two-Dimensional Ocean Surface Waves Described by the Zakharov Equation

<p>The physical hierarchy of two-dimensional ocean waves studied here consists of the 2+1 nonlinear Schrödinger equation (NLS), the Dysthe equation, the Trulsen-Dysthe equation, etc. on to the Zakharov equation. I call this the SDTDZ hierarchy. I demonstrate that the nonlinear Schrödinger equation with arbitrary potential is the natural way to treat this hierarchy, for any member of the hierarchy can be determined by an appropriate choice of the potential. Furthermore, the NLS equation with arbitrary potential can be written in terms of two bilinear forms and thereby has one and two-soliton solutions. To access the inverse scattering approach, I find a nearby equation which has N-soliton solutions: Such an equation is completely integrable by the IST on the infinite plane and by finite gap theory for periodic boundary conditions. In this way the entire SDTDZ hierarchy is closely related to a nearby integrable hierarchy which I refer to as the iSDTDZ hierarchy. Every member of this hierarchy has solutions in terms of ratios of Riemann theta functions and therefore every member has general spectral solutions in terms of quasiperiodic Fourier series. This last step occurs because ratios of theta functions are single valued, multiply periodic meromorphic functions. Once the quasiperiodic Fourier series are found, one can then invert these to determine the Riemann spectrum, namely, the Riemann matrix, wavenumbers, frequencies and phases. This means that the solutions of the nonlinear wave equations of the iSDTDZ hierarchy are generalized Fourier series indistinguishable from those of Paley and Weiner [1935] and therefore allows one to classify nonlinear wave motion in terms of a linear superposition of sine waves. How do the generalized quasiperiodic Fourier series differ from ordinary, standard periodic Fourier series? This can be seen by recognizing that the frequencies are incommensurable, and the phases can be phase locked. The nonlinear Fourier modes are Stokes waves and the coherent structure solutions are nonlinearly interacting, phase-locked Stokes waves, including breathers and superbreathers. Other types of coherent packets include fossil breathers and dromions. Techniques are developed for (1) numerical modeling of ocean waves (a fast algorithm for the Zakharov equation) and for (2) the nonlinear Fourier analysis of two-dimensional measured wave fields and space/time series (a 2D nonlinear Fourier analysis, implemented as a fast algorithm called the 2D NFFT). Examples of both applications are discussed.</p>

Quantization of Einstein-aether scalar field cosmology

We present, for the first time, the quantization process for the Einstein-aether scalar field cosmology. We consider a cosmological theory proposed as a Lorentz violating inflationary model, where the aether and scalar fields interact through the assumption that the aether action constants are ultra-local functions of the scalar field. For this specific theory there is a valid minisuperspace description which we use to quantize. For a particular relation between the two free functions entering the reduced Lagrangian the solution to the Wheeler–DeWitt equation as also the generic classical solution are presented for any given arbitrary potential function.

A New Approach to Solution of Time-Independent One-Dimensional Schrödinger Wave Equation

A new approach has been developed in this paper to solve time-independent Schrödinger wave equation for any arbitrary potential and space varying mass as well. The method is based on the state transition matrix used in the analysis of linear time-varying systems, and can determine both bound states and reflection and transmission coefficients associated with scattering problems. Numerical examples for the computation of eigenvalues and eigenmodes associated with bound states are presented for quadratic potential, quartic potential, constant potential well and arbitrary potential well with both constant and space-varying or position-dependent masses. Similarly, transmission coefficients for scattering problems without any infinite potential, and time delays for scattering problems with an infinite potential are computed for arbitrary potential wells.

A Study on The Entrepreneurial Opportunities, Global and Indian Economy in 3D Printing Sector

The main aim of the research was to explore all the opportunities, economic aspects, and challenges faced by various 3D printing industries, entrepreneurs, and consumers. 3D printing is a revolutionary digital production technique in Industry 4.0 that can transform the manufacturing sector to a new dimension and thereby creating a lot of opportunities and paved the way for economic growth both in the global and Indian industries. Moreover, the possibilities and availabilities of the 3D printing business opportunities for the entrepreneurs were studied based on their domain-specific areas such as application, manufacturing, programming, design, and development. In recent decades, most of the industries have focused on 3D printing technologies as it has arbitrary potential in all the sectors, single universal machine for producing intricate shapes and having online remote access. The present study also considered quick prototyping, fast manufacturing, along with the current novel methodologies in fabricating 3D elements. The advancements in 3D printing technology for various other industries like electrical industries in inventing receptors, circuit boards, etc. were deliberated. Thus the outcome of this specific study could be used to identify the entrepreneur company models in 3D printing and its business version by considering economic growth.

Dark Matter as a Result of Field Oscillations in the Modified Theory of Induced Gravity

The paper studies the modified theory of induced gravity (MTIG). The solutions of the MTIG equations contain two branches (stages): Einstein (ES) and “restructuring” (RS). Previously, solutions were found that the values of such parameters as the “Hubble parameter”, gravitational and cosmological “constants” at the RS stage, fluctuate near monotonously developing mean values. This article gives MTIG equations with arbitrary potential. Solutions of the equations of geodesic curves are investigated for the case of centrally symmetric space and quadratic potential at the RS stage. The oscillatory nature of the solutions leads to the appearance of a gravitational potential containing a spectrum of minima, as well as to antigravity, which is expressed by acceleration directed from the center. Such solutions lead to the distribution of the potential of the gravitational field creating an additional mass effect at large distances and are well suited for modeling the effect of dark matter in galaxies. The solutions of the equation of geodesic lines are obtained and analyzed. We found that the transition from flat asymptotics to oscillatory asymptotics at large distances from the center with a combination of the presence of antigravity zones leads to a rich variety of shapes and dynamics of geodesic curves and to the formation of complex structures.

Lifted Message Passing for Hybrid Probabilistic Inference

Lifted inference algorithms for first-order logic models, e.g., Markov logic networks (MLNs), have been of significant interest in recent years. Lifted inference methods exploit model symmetries in order to reduce the size of the model and, consequently, the computational cost of inference. In this work, we consider the problem of lifted inference in MLNs with continuous or both discrete and continuous groundings. Existing work on lifting with continuous groundings has mostly been limited to special classes of models, e.g., Gaussian models, for which variable elimination or message-passing updates can be computed exactly. Here, we develop approximate lifted inference schemes based on particle sampling. We demonstrate empirically that our approximate lifting schemes perform comparably to existing state-of-the-art for models for Gaussian MLNs, while having the flexibility to be applied to models with arbitrary potential functions.

Backstepping Control for the Schrödinger Equation with an Arbitrary Potential in a Confined Space

In this work the control design problem for the Schrödinger equation with an arbitrary potential is addressed. In particular a controller is designed which (i) for a space-dependent potential steers the state probability density function to a prescribed solution and (ii) for a space and state-dependent potential exponentially stabilizes the zero solution. The problem is addressed using a backstepping controller that steers to zero the deviation between the initial probability wave function and the target probability wave function. The exponential convergence property is rigorously established and the convergence behavior is illustrated using numerical simulations for the Morse and the Pöschl-Teller potentials as well as the semilinear Schrödinger equation with cubic potential.