Theory of Nonlinear Fourier Analysis: The Construction of Quasiperiodic Fourier Series for Nonlinear Wave Motion

2020 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Kdv Equation ◽  
Wave Turbulence ◽  
Two Dimensional ◽  
Kp Equation ◽  
Wave Dynamics ◽  
Wide Range ◽  
Stokes Waves ◽  
Parallel Dynamics

Abstract I give a description of nonlinear water wave dynamics using a recently discovered tool of mathematical physics I call nonlinear Fourier analysis (NLFA). This method is based upon and is an application of a theorem due to Baker [1897, 1907] and Mumford [1984] in the field of algebraic geometry and from additional sources by the author [Osborne, 2010, 2018, 2019]. The theory begins with the Kadomtsev-Petviashvili (KP) equation, a two dimensional generalization of the Korteweg-deVries (KdV) equation: Here the NLFA method is derived from the complete integrability of the equation by finite gap theory or the inverse scattering transform for periodic/quasiperiodic boundary conditions. I first show, for a one-dimensional, plane wave solution, that the KP equation can be rotated to a solution of the KdV equation, where the coefficients of KdV are now functions of the rotation angle. I then show how the rotated KdV equation can be used to compute the spectral solutions of the KP equation itself. Finally, I write the spectral solutions of the KP equation as a finite gap solution in terms of Riemann theta functions. By virtue of the fact that I am able to write a theta function formulation of the KP equation, it is clear that the wave dynamics lie on tori and constitute parallel dynamics on the tori in the integrable cases and non-parallel dynamics on the tori for certain perturbed quasi-integrable cases. Therefore, we are dealing with a Kolmogorov-Arnold-Moser KAM theory for nonlinear partial differential wave equations. The nonlinear Fourier series have particular nonlinear Fourier modes, including: sine waves, Stokes waves and solitons. Indeed the theoretical formulation I have developed is a kind of exact two-dimensional “coherent wave turbulence” or “integrable wave turbulence” for the KP equation, for which the Stokes waves and solitons are the coherent structures. I discuss how NLFA provides a number of new tools that apply to a wide range of problems in offshore engineering and coastal dynamics: This includes nonlinear Fourier space and time series analysis, nonlinear Fourier wave field analysis, a nonlinear random phase approximation, the study of nonlinear coherent functions and nonlinear bi and tri spectral analysis.

Nonlinear Fourier Analysis with Sine Wave, Stokes Wave and Rogue Wave Basis Functions: A Paradigm Change in the Understanding of Nonlinear Waves

2020 ◽  
Author(s):  
Alfred Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Rogue Wave ◽  
Random Phase ◽  
Stokes Wave ◽  
Random Wave ◽  
Linear Dispersion ◽  
Wide Range ◽  
Stokes Waves ◽  
Wave Trains

<p>I give a new perspective for the description of nonlinear water wave trains using mathematical methods I refer to as nonlinear Fourier analysis (NLFA). I discuss how this approach holds for one-space and one time dimensions (1+1) and for two-space and one time dimensions (2+1) to all orders of approximation. I begin with the nonlinear Schroedinger (NLS) equation in 1+1 dimensions: Here the NLFA method is derived from the complete integrability of the equation by the periodic inverse scattering transform. I show how to compute the nonlinear Fourier series that exactly solve 1+1 NLS. I then show how to extend the order of 1+1 NLS to the Dysthe and the extended Dysthe equations. I also show how to include directional spreading in the formulation so that I can address the 2+1 NLS, the 2+1 Dysthe and the 2+1 Trulsen-Dysthe equations. This hierarchy of equations extends formally all the way to the Zakharov equations in the infinite order limit. Each order and extension from 1+1 to 2+1 dimensions is characterized by its own modulational dispersion relation that is required at each order of the NLFA formalism. NLFA is characterized by its own fundamental nonlinear Fourier series, which has particular nonlinear Fourier modes: sine waves, Stokes waves and breather trains. We are all familiar with sine waves (known for centuries) and Stokes waves (known since the Stokes paper in 1847). Breather trains have become known over the past three decades as a major source of rogue or freak waves in the ocean: Breather packets are known to pulse up and down during their evolution. At the moment of the maximum amplitude the largest wave in a breather packet is often referred to as a “rogue” or “freak” wave. Such extreme packets are known to be “coherent structures" so that pure linear dispersion does not occur as in a linear packet. Instead the breather packets have components that are phase locked with each other and hence remain coherent and are “long lived” just as vortices do in classical turbulence. Because the breathers live for a long time, the notion of risk based upon linear dispersion, as used in the oil and shipping industries, must be revised upwards. I discuss how to apply NLFA to (1) nonlinearly Fourier analyze time series, (2) to analyze wave fields from radar, lidar and synthetic aperture radar measurements, (3) how to treat NLFA to describe nonlinear, random wave trains using a kind of nonlinear random phase approximation and (4) how to compute the nonlinear power spectrum in terms of the parameters used to describe the rogue wave Fourier modes in a random wave train. Thus the emphasis here is to discuss a number of new tools for nonlinear Fourier analysis in a wide range of problems in the field of ocean surface waves.</p>

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

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Theta Functions ◽  
Soliton Solutions ◽  
Ocean Waves ◽  
Two Dimensional ◽  
Zakharov Equation ◽  
Stokes Waves ◽  
Arbitrary Potential ◽  
Dysthe 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>

Nonlinear Fourier Analysis for Shallow Water Waves

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Wave Equations ◽  
Superposition Principle ◽  
Broad Band ◽  
Kp Equation ◽  
Linear Superposition

Abstract I consider nonlinear wave motion in shallow water as governed by the KP equation plus perturbations. I have previously shown that broad band, multiply periodic solutions of the KP equation are governed by quasiperiodic Fourier series [Osborne, OMAE 2020]. In the present paper I give a new procedure for extending this analysis to the KP equation plus shallow water Hamiltonian perturbations. We therefore have the remarkable result that a complex class of nonlinear shallow water wave equations has solutions governed by quasiperiodic Fourier series that are a linear superposition of sine waves. Such a formulation is important because it was previously thought that solving nonlinear wave equations by a linear superposition principle was impossible. The construction of these linear superpositions in shallow water in an engineering context is the goal of this paper. Furthermore, I address the nonlinear Fourier analysis of experimental data described by shallow water physics. The wave fields dealt with here are fully two-dimensional and essentially consist of the linear superposition of generalized cnoidal waves, which nonlinearly interact with one another. This includes the class of soliton solutions and their associated Mach stems, both of which are important for engineering applications. The newly discovered phenomenon of “fossil breathers” is also characterized in the formulation. I also discuss the exact construction of Morison equation forces on cylindrical piles in terms of quasiperiodic Fourier series.

Quantification of shape by use of Fourier analysis: the Mississippian blastoid genus Pentremites

Paleobiology ◽  
1977 ◽  
Vol 3 (3) ◽  
pp. 288-299 ◽  
Author(s):  
Johnny Arlton Waters
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
External Morphology ◽  
Taxonomic Character ◽  
Fossil Species ◽  
Growth Series ◽  
Wide Range ◽  
Lawrence County ◽  
Psychological Evidence ◽  
Important Character

Psychological evidence suggests that the visual outline of an object is the most important character for discriminating differences in external morphology. External morphology is an important taxonomic character in describing living and fossil species, for example, the blastoid Pentremites. Comparison of different views of the same specimen utilizing Fourier series suggests that the skeleton of Pentremites commonly is not rotationally symmetrical and that the asymmetry is not associated with any specific ray. Analysis of a growth series indicates that the amplitudes of the second and third harmonics are significantly correlated with growth, which is demonstrated to be anisometric. Definable changes in the lateral outline can be attributed to changes in harmonic amplitudes and a wide range of morphological forms comparable to known taxa can be generated by systematically varying the amplitudes of the first four harmonics. The population used in this study probably represents Pentremites robustus Lyon and is from Bangor Limestone in the abandoned Moulton Quarry, Lawrence County, Alabama.

APPLICATIONS OF MULTI-PARAMETER MARTINGALES IN FOURIER ANALYSIS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 551-568
Author(s):  
FERENC WEISZ
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Two Dimensional ◽  
Almost Everywhere Convergence ◽  
Fejér Means ◽  
Almost Everywhere ◽  
L Log L

With the help of the theory of multi-parameter martingales we prove almost everywhere convergence of the Fejér means of two-dimensional Walsh–Fourier series of f ∈ L log L.

Aromatic Volatile Composition of Celery and Celeriac Cultivars

HortScience ◽  
1990 ◽  
Vol 25 (5) ◽  
pp. 556-559 ◽  
Author(s):  
Fredy Van Wassenhove ◽  
Patrick Dirinck ◽  
Georges Vulsteke ◽  
Niceas Schamp
Keyword(s):  
Volatile Compounds ◽  
Chromatographic Method ◽  
Apium Graveolens ◽  
Volatile Components ◽  
Two Dimensional ◽  
Qualitative Composition ◽  
Volatile Composition ◽  
Wide Range ◽  
Gas Chromatographic Method ◽  
Identification And Quantification

A two-dimensional capillary gas chromatographic method was developed to separate and quantify aromatic volatiles of celery in one analysis. The isolation, identification, and quantification of the volatile compounds of four cultivars of blanching celery (Apium graveolens L. var. dulce) and six cultivars of celeriac (Apium graveolens L. var. rapaceum) are described. The qualitative composition of Likens-Nickerson extracts of both cultivars is similar. The concentration of terpenes and phthalides, the key volatile components, found in various cultivars of both celery and celeriac varied over a wide range.

scholarly journals Prediction of Wide Range Two-Dimensional Refractivity Using an IDW Interpolation Method from High-Altitude Refractivity Data of Multiple Meteorological Observatories

2021 ◽  
Vol 11 (4) ◽  
pp. 1431
Author(s):  
Sungsik Wang ◽  
Tae Heung Lim ◽  
Kyoungsoo Oh ◽  
Chulhun Seo ◽  
Hosung Choo
Keyword(s):  
High Altitude ◽  
Korean Peninsula ◽  
Interpolation Method ◽  
Atmospheric Condition ◽  
Propagation Path ◽  
Two Dimensional ◽  
Data Set ◽  
Wide Range ◽  
Atmospheric Data ◽  
Terrain Surface

This article proposes a method for the prediction of wide range two-dimensional refractivity for synthetic aperture radar (SAR) applications, using an inverse distance weighted (IDW) interpolation of high-altitude radio refractivity data from multiple meteorological observatories. The radio refractivity is extracted from an atmospheric data set of twenty meteorological observatories around the Korean Peninsula along a given altitude. Then, from the sparse refractive data, the two-dimensional regional radio refractivity of the entire Korean Peninsula is derived using the IDW interpolation, in consideration of the curvature of the Earth. The refractivities of the four seasons in 2019 are derived at the locations of seven meteorological observatories within the Korean Peninsula, using the refractivity data from the other nineteen observatories. The atmospheric refractivities on 15 February 2019 are then evaluated across the entire Korean Peninsula, using the atmospheric data collected from the twenty meteorological observatories. We found that the proposed IDW interpolation has the lowest average, the lowest average root-mean-square error (RMSE) of ∇M (gradient of M), and more continuous results than other methods. To compare the resulting IDW refractivity interpolation for airborne SAR applications, all the propagation path losses across Pohang and Heuksando are obtained using the standard atmospheric condition of ∇M = 118 and the observation-based interpolated atmospheric conditions on 15 February 2019. On the terrain surface ranging from 90 km to 190 km, the average path losses in the standard and derived conditions are 179.7 dB and 182.1 dB, respectively. Finally, based on the air-to-ground scenario in the SAR application, two-dimensional illuminated field intensities on the terrain surface are illustrated.

scholarly journals Is contact angle a cause or an effect? – A cautionary tale

2020 ◽  
Vol 146 ◽  
pp. 03004
Author(s):  
Douglas Ruth
Keyword(s):  
Contact Angle ◽  
Solid Surface ◽  
Contact Angles ◽  
Two Dimensions ◽  
Experimental Results ◽  
Flow In Porous Media ◽  
Two Dimensional ◽  
Wide Range ◽  
Fluid Drop ◽  
Surrounding Fluid

The most influential parameter on the behavior of two-component flow in porous media is “wettability”. When wettability is being characterized, the most frequently used parameter is the “contact angle”. When a fluid-drop is placed on a solid surface, in the presence of a second, surrounding fluid, the fluid-fluid surface contacts the solid-surface at an angle that is typically measured through the fluid-drop. If this angle is less than 90°, the fluid in the drop is said to “wet” the surface. If this angle is greater than 90°, the surrounding fluid is said to “wet” the surface. This definition is universally accepted and appears to be scientifically justifiable, at least for a static situation where the solid surface is horizontal. Recently, this concept has been extended to characterize wettability in non-static situations using high-resolution, two-dimensional digital images of multi-component systems. Using simple thought experiments and published experimental results, many of them decades old, it will be demonstrated that contact angles are not primary parameters – their values depend on many other parameters. Using these arguments, it will be demonstrated that contact angles are not the cause of wettability behavior but the effect of wettability behavior and other parameters. The result of this is that the contact angle cannot be used as a primary indicator of wettability except in very restricted situations. Furthermore, it will be demonstrated that even for the simple case of a capillary interface in a vertical tube, attempting to use simply a two-dimensional image to determine the contact angle can result in a wide range of measured values. This observation is consistent with some published experimental results. It follows that contact angles measured in two-dimensions cannot be trusted to provide accurate values and these values should not be used to characterize the wettability of the system.

scholarly journals Robust anomalous metallic states and vestiges of self-duality in two-dimensional granular In-InOx composites

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Xinyang Zhang ◽  
Bar Hen ◽  
Alexander Palevski ◽  
Aharon Kapitulnik
Keyword(s):  
Magnetic Field ◽  
Metallic Phase ◽  
Broadband Noise ◽  
Two Dimensional ◽  
Conventional Theory ◽  
Logarithmic Divergence ◽  
Self Duality ◽  
Low Field ◽  
Wide Range ◽  
Low Temperature Resistance

AbstractMany experiments investigating magnetic-field tuned superconductor-insulator transition (H-SIT) often exhibit low-temperature resistance saturation, which is interpreted as an anomalous metallic phase emerging from a ‘failed superconductor’, thus challenging conventional theory. Here we study a random granular array of indium islands grown on a gateable layer of indium-oxide. By tuning the intergrain couplings, we reveal a wide range of magnetic fields where resistance saturation is observed, under conditions of careful electromagnetic filtering and within a wide range of linear response. Exposure to external broadband noise or microwave radiation is shown to strengthen the tendency of superconductivity, where at low field a global superconducting phase is restored. Increasing magnetic field unveils an ‘avoided H-SIT’ that exhibits granularity-induced logarithmic divergence of the resistance/conductance above/below that transition, pointing to possible vestiges of the original emergent duality observed in a true H-SIT. We conclude that anomalous metallic phase is intimately associated with inherent inhomogeneities, exhibiting robust behavior at attainable temperatures for strongly granular two-dimensional systems.

scholarly journals Energetics of mesoscale cell turbulence in two-dimensional monolayers

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Shao-Zhen Lin ◽  
Wu-Yang Zhang ◽  
Dapeng Bi ◽  
Bo Li ◽  
Xi-Qiao Feng
Keyword(s):  
Kinetic Energy ◽  
Tumor Metastasis ◽  
Cell Types ◽  
Boltzmann Distribution ◽  
Scaling Exponent ◽  
Biological Processes ◽  
Two Dimensional ◽  
Cell Monolayers ◽  
Wide Range ◽  
Epithelial Cell Monolayers

AbstractInvestigation of energy mechanisms at the collective cell scale is a challenge for understanding various biological processes, such as embryonic development and tumor metastasis. Here we investigate the energetics of self-sustained mesoscale turbulence in confluent two-dimensional (2D) cell monolayers. We find that the kinetic energy and enstrophy of collective cell flows in both epithelial and non-epithelial cell monolayers collapse to a family of probability density functions, which follow the q-Gaussian distribution rather than the Maxwell–Boltzmann distribution. The enstrophy scales linearly with the kinetic energy as the monolayer matures. The energy spectra exhibit a power-decaying law at large wavenumbers, with a scaling exponent markedly different from that in the classical 2D Kolmogorov–Kraichnan turbulence. These energetic features are demonstrated to be common for all cell types on various substrates with a wide range of stiffness. This study provides unique clues to understand active natures of cell population and tissues.

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