| 书目名称 | New Approaches to Nonlinear Waves | | 编辑 | Elena Tobisch | | 视频video | | | 概述 | Written by leading experts in nonlinear waves.Brings together theoretical, numerical and experimental approaches.With applications to real physical problems | | 丛书名称 | Lecture Notes in Physics | | 图书封面 |  | | 描述 | .The book details a few of the novel methods developed in the last few years for studying various aspects of nonlinear wave systems. The introductory chapter provides a general overview, thematically linking the objects described in the book. .Two chapters are devoted to wave systems possessing resonances with linear frequencies (Chapter 2) and with nonlinear frequencies (Chapter 3)..In the next two chapters modulation instability in the KdV-type of equations is studied using rigorous mathematical methods (Chapter 4) and its possible connection to freak waves is investigated (Chapter 5). .The book goes on to demonstrate how the choice of the Hamiltonian (Chapter 6) or the Lagrangian (Chapter 7) framework allows us to gain a deeper insight into the properties of a specific wave system..The final chapter discusses problems encountered when attempting to verify the theoretical predictions using numerical or laboratory experiments..All the chapters are illustrated by ample constructive examples demonstrating the applicability of these novel methods and approaches to a wide class of evolutionary dispersive PDEs, e.g. equations from Benjamin-Oro, Boussinesq, Hasegawa-Mima, KdV-type, Kl | | 出版日期 | Book 2016 | | 关键词 | Dyachenko–Zakharov Equation; Efficient Conformal Map Techniques; Euler Equations; Laboratory Scale Hydr | | 版次 | 1 | | doi | https://doi.org/10.1007/978-3-319-20690-5 | | isbn_softcover | 978-3-319-20689-9 | | isbn_ebook | 978-3-319-20690-5Series ISSN 0075-8450 Series E-ISSN 1616-6361 | | issn_series | 0075-8450 | | copyright | Springer International Publishing Switzerland 2016 |
| 1 |
Front Matter |
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Abstract
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| 2 |
,Introduction, |
Elena Tobisch |
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Abstract
In the first chapter, we throw a brief glance at the topics presented in the following chapters and their place in the context of the general theory of nonlinear wave systems with dispersion. Starting with the concept of the wave resonance, we proceed through the formalism and presently known results in the theory of discrete and kinetic wave turbulence to the list of open questions and possible theoretical generalizations. At the end of the introductory chapter, we outline a few challenging problems in the area of matching theory and experiment, generally overlooked.
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| 3 |
,The Effective Equation Method, |
Sergei Kuksin,Alberto Maiocchi |
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Abstract
In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and
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| 4 |
,On the Discovery of the Steady-State Resonant Water Waves, |
Shijun Liao,Dali Xu,Zeng Liu |
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Abstract
In 1960 Phillips gave the criterion of wave resonance and showed that the amplitude of a resonant wave component, if it is zero initially, grows linearly with time. In 1962 Benney derived evolution equations of wave-mode amplitudes and demonstrated periodic exchange of wave energy for resonant waves. However, in the past half century, the so-called steady-state resonant waves with time-independent spectrum have never been found for order higher than three, because perturbation results contain secular terms when Phillips’ criterion is satisfied so that “the perturbation theory breaks down due to singularities in the transfer functions”, as pointed out by Madsen and Fuhrman in 2012..Recently, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear problems, steady-state resonant waves have been obtained not only in deep water but also for constant water depth and even over a bottom with an infinite number of periodic ripples. In addition, steady-state resonant waves were observed experimentally in a basin at the State Key Laboratory of Ocean Engineering, Shanghai, China, showing excellent agreement with theoretical predictions..In this ch
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| 5 |
,Modulational Instability in Equations of KdV Type, |
Jared C. Bronski,Vera Mikyoung Hur,Mathew A. Johnson |
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Abstract
It is a matter of experience that nonlinear waves in a dispersive medium, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics—the amplitude, the phase, the wave number, etc.—slowly vary in large space and time scales. In the 1960s, Whitham developed an asymptotic (WKB) method to study the effects of small “modulations” on nonlinear dispersive waves. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham’s formal theory. We discuss some recent advances in the mathematical understanding of the dynamics, in particular, the instability, of slowly modulated waves for equations of KdV type.
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| 6 |
,Modulational Instability and Rogue Waves in Shallow Water Models, |
R. Grimshaw,K. W. Chow,H. N. Chan |
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Abstract
It is now well known that the focussing nonlinear Schrödinger equation allows plane waves to be modulationally unstable, and at the same time supports breather solutions which are often invoked as models for rogue waves. This suggests a direct connection between modulation instability and the existence of rogue waves. In this chapter we review this connection for a suite of long wave models, such as the Korteweg-de Vries equation, the extended Korteweg-de Vries (Gardner) equation, often used to describe surface and internal waves in shallow water, a Boussinesq equation and, also a coupled set of Korteweg-de Vries equations.
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| 7 |
,Hamiltonian Framework for Short Optical Pulses, |
Shalva Amiranashvili |
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Abstract
Physics of short optical pulses is an important and active research area in nonlinear optics. In this Chapter we theoretically consider the most extreme representatives of short pulses that contain only several oscillations of electromagnetic field. Description of such pulses is traditionally based on envelope equations and slowly varying envelope approximation, despite the fact that the envelope is not “slow” and, moreover, there is no clear definition of such a “fast” envelope. This happens due to another paradoxical feature: the standard (envelope) generalized nonlinear Schrödinger equation yields very good correspondence to numerical solutions of full Maxwell equations even for few-cycle pulses, the thing that should not be.In what follows we address ultrashort optical pulses using Hamiltonian framework for nonlinear waves. As it appears, the standard optical envelope equation is just a reformulation of general Hamiltonian equations. In a sense, no approximations are required, this is why the generalized nonlinear Schrödinger equation is so effective. Moreover, the Hamiltonian framework contributes greatly to our understanding of “fast” envelopes, ultrashort solitons, stability
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| 8 |
,Modeling Water Waves Beyond Perturbations, |
Didier Clamond,Denys Dutykh |
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Abstract
In this chapter, we illustrate the advantage of variational principles for modeling water waves from an elementary practical viewpoint. The method is based on a ‘relaxed’ variational principle, i.e., on a Lagrangian involving as many variables as possible, and imposing some suitable subordinate constraints. This approach allows the construction of approximations without necessarily relying on a small parameter. This is illustrated via simple examples, namely the Serre equations in shallow water, a generalization of the Klein–Gordon equation in deep water and how to unify these equations in arbitrary depth. The chapter ends with a discussion and caution on how this approach should be used in practice.
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| 9 |
,Quantitative Analysis of Nonlinear Water-Waves: A Perspective of an Experimentalist, |
Lev Shemer |
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Abstract
In the present review the emphasis is put on laboratory studies of propagating water waves where experiments were designed with the purpose to enable juxtaposing the measurement results with the theoretical predictions, thus providing a basis for evaluation of the domain of validity of various nonlinear theoretical model of different complexity. In particular, evolution of deterministic wave groups of different shapes and several values of characteristic nonlinearity is studied in deep and intermediate-depth water. Experiments attempting to generate extremely steep (rogue) waves are reviewed in greater detail. Relation between the kinematics of steep nonlinear waves and incipient breaking is considered. Discussion of deterministic wave systems is followed by review of laboratory experiments on propagation of numerous realizations of random wave groups with different initial spectra. The experimental results are compared with the corresponding Monte-Carlo numerical simulations based on different models.
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| 10 |
Back Matter |
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Abstract
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发表于 2025-3-21 16:10:21
