Propagation of Multidimensional Nonlinear Waves and Kinematical Conservation Laws PDF Download

Author: Phoolan Prasad
Publisher: Springer
ISBN: 9811075816
Category : Mathematics
Languages : en
Pages : 165

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Book Description
This book formulates the kinematical conservation laws (KCL), analyses them and presents their applications to various problems in physics. Finally, it addresses one of the most challenging problems in fluid dynamics: finding successive positions of a curved shock front. The topics discussed are the outcome of collaborative work that was carried out mainly at the Indian Institute of Science, Bengaluru, India. The theory presented in the book is supported by referring to extensive numerical results. The book is organised into ten chapters. Chapters 1–4 offer a summary of and briefly discuss the theory of hyperbolic partial differential equations and conservation laws. Formulation of equations of a weakly nonlinear wavefront and those of a shock front are briefly explained in Chapter 5, while Chapter 6 addresses KCL theory in space of arbitrary dimensions. The remaining chapters examine various analyses and applications of KCL equations ending in the ultimate goal-propagation of a three-dimensional curved shock front and formation, propagation and interaction of kink lines on it.

Author: Prasad
Publisher: CRC Press
ISBN: 9780582072534
Category : Mathematics
Languages : en
Pages : 140

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Book Description
Phoolan Prasad's book contains theoretical developments in the study of the propagation of a curved nonlinear wave front and shock front, particularly in the caustic region. It should be an invaluable reference source for researchers in nonlinear waves; fluid dynamics (especially gas dynamics); mathematical physics; aeronautical, chemical and mechanical engineering.

Author: Julian L. Davis
Publisher: Springer Science & Business Media
ISBN: 1461238862
Category : Science
Languages : en
Pages : 396

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Book Description
The purpose of this volume is to present a clear and systematic account of the mathematical methods of wave phenomena in solids, gases, and water that will be readily accessible to physicists and engineers. The emphasis is on developing the necessary mathematical techniques, and on showing how these mathematical concepts can be effective in unifying the physics of wave propagation in a variety of physical settings: sound and shock waves in gases, water waves, and stress waves in solids. Nonlinear effects and asymptotic phenomena will be discussed. Wave propagation in continuous media (solid, liquid, or gas) has as its foundation the three basic conservation laws of physics: conservation of mass, momentum, and energy, which will be described in various sections of the book in their proper physical setting. These conservation laws are expressed either in the Lagrangian or the Eulerian representation depending on whether the boundaries are relatively fixed or moving. In any case, these laws of physics allow us to derive the "field equations" which are expressed as systems of partial differential equations. For wave propagation phenomena these equations are said to be "hyperbolic" and, in general, nonlinear in the sense of being "quasi linear" . We therefore attempt to determine the properties of a system of "quasi linear hyperbolic" partial differential equations which will allow us to calculate the displacement, velocity fields, etc.

Author: Sathyanarayanan Chandramouli
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 0

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Book Description
In this thesis, we develop and study two distinct problems in the field of nonlinear waves. The first part of the thesis is connected to the development of a computational algorithm that preserves underlying structure of the simulated initial boundary value problem in the form of multiple global conservation laws or dissipation rate equations. \\\\begin{itemize}\\\\item The time-dependent spectral renormalization (TDSR) method was introduced by Cole and Musslimani as a viable method to numerically solve initial boundary value problems. An important and novel aspect of the TDSR scheme is its ability to incorporate physics in the form of conservation laws or dissipation rate equations. However, the method was restricted to enforce the conservation or dissipation rate of just one quantity. The present work significantly extends the computational features of the algorithm with the (i) incorporation of multiple conservation laws and/or dissipation rate equations, (ii) ability to enforce versatile boundary conditions, and (iii) higher-order time integration strategies. The TDSR method is applied on several prototypical evolution equations of physical significance. Examples include the Korteweg-de Vries (KdV), multi-dimensional nonlinear Schr\\\\"odinger (NLS) and the Allen-Cahn equations. The work was published in Nonlinearity \\\\cite{chandramouli2022time}. \\\\end{itemize} The second half of the thesis identifies a broad class of novel, \\\\textit{non-centered} Riemann problems in optical media with externally imposed gain and loss distributions. Thereafter, we shed light on some unique features that arise from step-like distributions in such spatially inhomogeneous media. Our work thus is an important contribution to the field of non-Hermitian dispersive hydrodynamics. \\\\begin{itemize} \\\\item Dispersive hydrodynamics, the study of nonlinear dispersive wave dynamics in fluid-like media, is an active research area that combines mathematical analysis with computational and laboratory experiments. To date, most of the research in this area has been focused on wave phenomena in (i) bulk media, in which case the underlying governing equations are of constant coefficients type, or (ii) inhomogeneous environments, where now the evolution equations contain, for example, a real-valued external potential. In the latter case, the presence of inhomogeneity (in general) hinders the formulation of a Riemann problem due to the lack of plane wave-type solutions of constant intensity (or density). However such waves can exist in non-Hermitian media, as was demonstrated for the nonlinear Schrödinger (NLS) equation with a Wadati-type complex external potential. Inspired by the above-mentioned discussions, in this paper, the notion of non-Hermitian dispersive hydrodynamics and its associated non-Hermitian Riemann problems are introduced. Starting from the defocusing (repulsive) NLS equation in the presence of generic smooth complex external potentials, a new set of hydrodynamic-like equations are obtained. They differ from their classical counterparts (without an external potential), by the presence of additional source terms that alter the density and momentum equations. When restricted to a class of Wadati-type complex potentials, this new non-Hermitian hydrodynamic system admits constant intensity/density solutions. This in turn, allows one to formulate an exact centered (or non-centered) Riemann problem involving a step-like initial condition that connects two exact constant density states. A broad class of non-Hermitian potentials that lead to modulationally stable constant intensity states are identified. These results are subsequently used to numerically solve the associated non-Hermitian Riemann problem for various initial conditions. Due to the lack of translation symmetry, the resulting long-time dynamics show a strong dependence on the location of the step relative to the gain-loss distribution. This is in sharp contrast to the classical NLS Riemann problem (in the absence of potential), where the dynamics are generally independent of the step location. This fact leads to {a diverse array of} wave pattern dynamics that are otherwise absent. In particular, various novel gain-loss generated near-field features are observed, which in turn drive the optical flows in the far-field. {These far-field non-Hermitian counter-flows could be comprised of various rich nonlinear wave phenomena, including DSW-DSW, DSW-rarefaction, and soliton-DSW interactions. A manuscript containing the results has been submitted to Nonlinearity \\\\cite{chandramouli2023nonHermitian}.} \\\\end{itemize}

Author: A.G. Kulikovskii
Publisher: CRC Press
ISBN: 1000443485
Category : Mathematics
Languages : en
Pages : 256

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Book Description
Nonlinear Waves in Elastic Media explores the theoretical results of one-dimensional nonlinear waves, including shock waves, in elastic media. It is the first book to provide an in-depth and comprehensive presentation of the nonlinear wave theory while taking anisotropy effects into account. The theory is completely worked out and draws on 15 years of research by the authors, one of whom also wrote the 1965 classic Magnetohydrodynamics. Nonlinear Waves in Elastic Media emphasizes the behavior of quasitransverse waves and analyzes arbitrary discontinuity disintegration problems, illustrating that the solution can be non-unique - a surprising result. The solution is shown to be especially interesting when anisotropy and nonlinearity effects interact, even in small-amplitude waves. In addition, the text contains an independent mathematical chapter describing general methods to study hyperbolic systems expressing the conservation laws. The theoretical results described in Nonlinear Waves in Elastic Media allow, for the first time, discovery and interpretation of many new peculiarities inherent to the general problem of discontinuous solutions and so provide a valuable resource for advanced students and researchers involved with continuum mechanics and partial differential equations.

Author: C.I. Christov
Publisher: Springer Science & Business Media
ISBN: 1461200954
Category : Mathematics
Languages : en
Pages : 274

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Book Description
This book gives an overview ofthe current state of nonlinear wave mechanics with emphasis on strong discontinuities (shock waves) and localized self preserving shapes (solitons) in both elastic and fluid media. The exposition is intentionallyat a detailed mathematical and physical level, our expectation being that the reader will enjoy coming to grips in a concrete manner with advances in this fascinating subject. Historically, modern research in nonlinear wave mechanics began with the famous 1858 piston problem paper of Riemann on shock waves and con tinued into the early part of the last century with the work of Hadamard, Rankine, and Hugoniot. After WWII, research into nonlinear propagation of dispersive waves rapidly accelerated with the advent of computers. Works of particular importance in the immediate post-war years include those of von Neumann, Fermi, and Lax. Later, additional contributions were made by Lighthill, Glimm, Strauss, Wendroff, and Bishop. Dispersion alone leads to shock fronts of the propagating waves. That the nonlinearity can com pensate for the dispersion, leading to propagation with a stable wave having constant velocity and shape (solitons) came as a surprise. A solitary wave was first discussed by J. Scott Russell in 1845 in "Report of British Asso ciations for the Advancement of Science. " He had, while horseback riding, observed a solitary wave travelling along a water channel and followed its unbroken progress for over a mile.

Author:
Publisher: Elsevier
ISBN: 0080957803
Category : Mathematics
Languages : en
Pages : 381

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Book Description
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression. - Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering

Author: Spencer P Kuo
Publisher: World Scientific
ISBN: 9811231656
Category : Science
Languages : en
Pages : 206

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Book Description
Waves are essential phenomena in most scientific and engineering disciplines, such as electromagnetism and optics, and different mechanics including fluid, solid, structural, quantum, etc. They appear in linear and nonlinear systems. Some can be observed directly and others are not. The features of the waves are usually described by solutions to either linear or nonlinear partial differential equations, which are fundamental to the students and researchers.Generic equations, describing wave and pulse propagation in linear and nonlinear systems, are introduced and analyzed as initial/boundary value problems. These systems cover the general properties of non-dispersive and dispersive, uniform and non-uniform, with/without dissipations. Methods of analyses are introduced and illustrated with analytical solutions. Wave-wave and wave-particle interactions ascribed to the nonlinearity of media (such as plasma) are discussed in the final chapter.This interdisciplinary textbook is essential reading for anyone in above mentioned disciplines. It was prepared to provide students with an understanding of waves and methods of solving wave propagation problems. The presentation is self-contained and should be read without difficulty by those who have adequate preparation in classic mechanics. The selection of topics and the focus given to each provide essential materials for a lecturer to cover the bases in a linear/nonlinear wave course.

Author: Alan Jeffrey
Publisher:
ISBN:
Category : Magnetohydrodynamics
Languages : en
Pages : 388

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Book Description

Author: Guy Boillat
Publisher: Springer
ISBN: 3540495657
Category : Mathematics
Languages : en
Pages : 149

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Book Description
These lecture notes of the courses presented at the first CIME session 1994 by leading scientists present the state of the art in recent mathematical methods in Nonlinear Wave Propagation.