The Legacy Of The Inverse Scattering Transform In Applied Mathematics

Author : J. L. Bona,Roy Choudhury,David Kaup,Ams-IMS-Siam Joint Summer Research Conference on the Legacy of Inverse Scattering Transfor
Publisher : American Mathematical Soc.
Page : 338 pages
File Size : 54,6 Mb
Release : 2002
Category : Science
ISBN : 9780821831618

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The Legacy of the Inverse Scattering Transform in Applied Mathematics by J. L. Bona,Roy Choudhury,David Kaup,Ams-IMS-Siam Joint Summer Research Conference on the Legacy of Inverse Scattering Transfor Pdf

Swift progress and new applications characterize the area of solitons and the inverse scattering transform. There are rapid developments in current nonlinear optical technology: Larger intensities are more available; pulse widths are smaller; relaxation times and damping rates are less significant. In keeping with these advancements, exactly integrable soliton equations, such as $3$-wave resonant interactions and second harmonic generation, are becoming more and more relevant in experimental applications. Techniques are now being developed for using these interactions to frequency convert high intensity sources into frequency regimes where there are no lasers. Other experiments involve using these interactions to develop intense variable frequency sources, opening up even more possibilities. This volume contains new developments and state-of-the-art research arising from the conference on the ""Legacy of the Inverse Scattering Transform"" held at Mount Holyoke College (South Hadley, MA). Unique to this volume is the opening section, ""Reviews"". This part of the book provides reviews of major research results in the inverse scattering transform (IST), on the application of IST to classical problems in differential geometry, on algebraic and analytic aspects of soliton-type equations, on a new method for studying boundary value problems for integrable partial differential equations (PDEs) in two dimensions, on chaos in PDEs, on advances in multi-soliton complexes, and on a unified approach to integrable systems via Painleve analysis. This conference provided a forum for general exposition and discussion of recent developments in nonlinear waves and related areas with potential applications to other fields. The book will be of interest to graduate students and researchers interested in mathematics, physics, and engineering.

Author : Mark J. Ablowitz,Harvey Segur
Publisher : SIAM
Page : 433 pages
File Size : 44,5 Mb
Release : 2006-05-15
Category : Mathematics
ISBN : 9780898714777

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Solitons and the Inverse Scattering Transform by Mark J. Ablowitz,Harvey Segur Pdf

A study, by two of the major contributors to the theory, of the inverse scattering transform and its application to problems of nonlinear dispersive waves that arise in fluid dynamics, plasma physics, nonlinear optics, particle physics, crystal lattice theory, nonlinear circuit theory and other areas. A soliton is a localised pulse-like nonlinear wave that possesses remarkable stability properties. Typically, problems that admit soliton solutions are in the form of evolution equations that describe how some variable or set of variables evolve in time from a given state. The equations may take a variety of forms, for example, PDEs, differential difference equations, partial difference equations, and integrodifferential equations, as well as coupled ODEs of finite order. What is surprising is that, although these problems are nonlinear, the general solution that evolves from almost arbitrary initial data may be obtained without approximation.

Author : J. Bee Bednar
Publisher : SIAM
Page : 310 pages
File Size : 50,9 Mb
Release : 1983-01-01
Category : Mathematics
ISBN : 0898711908

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Conference on Inverse Scattering--Theory and Application by J. Bee Bednar Pdf

Author : W. Eckhaus,A.M. van Harten
Publisher : Elsevier
Page : 221 pages
File Size : 48,5 Mb
Release : 2011-08-18
Category : Mathematics
ISBN : 0080871615

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The Inverse Scattering Transformation and The Theory of Solitons by W. Eckhaus,A.M. van Harten Pdf

The Inverse Scattering Transformation and The Theory of Solitons

Author : Khosrow Chadan,David Colton,William Rundell,Lassi P?iv?rinta
Publisher : SIAM
Page : 208 pages
File Size : 54,8 Mb
Release : 1997-01-01
Category : Mathematics
ISBN : 0898719712

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An Introduction to Inverse Scattering and Inverse Spectral Problems by Khosrow Chadan,David Colton,William Rundell,Lassi P?iv?rinta Pdf

Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.

Author : Fioralba Cakoni,David Colton,Houssem Haddar
Publisher : SIAM
Page : 259 pages
File Size : 52,5 Mb
Release : 2022-12-07
Category : Mathematics
ISBN : 9781611977424

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Inverse Scattering Theory and Transmission Eigenvalues by Fioralba Cakoni,David Colton,Houssem Haddar Pdf

Inverse scattering theory is a major theme in applied mathematics, with applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting challenges in the development of efficient inversion algorithms. A further complication is that anisotropic materials cannot be uniquely determined from given scattering data. In the first edition of Inverse Scattering Theory and Transmission Eigenvalues, the authors discussed methods for determining the support of inhomogeneous media from measured far field data and the role of transmission eigenvalue problems in the mathematical development of these methods. In this second edition, three new chapters describe recent developments in inverse scattering theory. In particular, the authors explore the use of modified background media in the nondestructive testing of materials and methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data. They also examine nonscattering wave numbers—a subset of transmission eigenvalues—using techniques taken from the theory of free boundary value problems for elliptic partial differential equations and discuss the dualism of scattering poles and transmission eigenvalues that has led to new methods for the numerical computation of scattering poles. This book will be of interest to research mathematicians and engineers and physicists working on problems in target identification. It will also be useful to advanced graduate students in many areas of applied mathematics.

Author : Fioralba Cakoni,David Colton,Houssem Haddar
Publisher : SIAM
Page : 203 pages
File Size : 55,5 Mb
Release : 2016-10-28
Category : Mathematics
ISBN : 9781611974461

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Inverse Scattering Theory and Transmission Eigenvalues by Fioralba Cakoni,David Colton,Houssem Haddar Pdf

Inverse scattering theory is a major theme of applied mathematics, and it has applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting particular problems in the development of efficient inversion algorithms. Although linearized models continue to play an important role in many applications, an increased need to focus on problems in which multiple scattering effects cannot be ignored has led to a central role for nonlinearity, and the possibility of collecting large amounts of data over limited regions of space means that the ill-posed nature of the inverse scattering problem has become a problem of central importance.? Initial efforts to address the nonlinear and the ill-posed nature of the inverse scattering problem focused on nonlinear optimization methods. While efficient in many situations, strong a priori information is necessary for their implementation. This problem led to a qualitative approach to inverse scattering theory in which the amount of a priori information is drastically reduced, although at the expense of only obtaining limited information about the values of the constitutive parameters. This qualitative approach (the linear sampling method, the factorization method, the theory of transmission eigenvalues, etc.) is the theme of Inverse Scattering Theory and Transmission Eigenvalues.? The authors begin with a basic introduction to the theory, then proceed to more recent developments, including a detailed discussion of the transmission eigenvalue problem; present the new generalized linear sampling method in addition to the well-known linear sampling and factorization methods; and in order to achieve clarification of presentation, focus on the inverse scattering problem for scalar homogeneous media.?

Author : David H. Sattinger,C. A. Tracy,Stephanos Venakides
Publisher : American Mathematical Soc.
Page : 133 pages
File Size : 49,5 Mb
Release : 1991
Category : Mathematics
ISBN : 9780821851296

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Inverse Scattering and Applications by David H. Sattinger,C. A. Tracy,Stephanos Venakides Pdf

This book contains the proceedings of an AMS-IMS-SIAM Summer Institute on Inverse Scattering and Applications, held in Amherst, Massachusetts, in June 1990. The papers cover the current state of the art in inverse conductivity problems, applications of inverse scattering theory to integrable systems, three-dimensional inverse scattering, inverse monodromy problems, and nonlinear waves, among other topics. Intended for researchers working in inverse scattering theory, inverse conductance problems, and completely integrable systems, this book presents results by some of the major experts in the field.

Author : Fioralba Cakoni,David Colton
Publisher : Springer Science & Business Media
Page : 232 pages
File Size : 50,9 Mb
Release : 2005-12-29
Category : Mathematics
ISBN : 9783540312307

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Qualitative Methods in Inverse Scattering Theory by Fioralba Cakoni,David Colton Pdf

Inverse scattering theory has been a particularly active and successful field in applied mathematics and engineering for the past twenty years. The increasing demands of imaging and target identification require new powerful and flexible techniques besides the existing weak scattering approximation or nonlinear optimization methods. One class of such methods comes under the general description of qualitative methods in inverse scattering theory. This textbook is an easily-accessible "class-tested" introduction to the field. It is accessible also to readers who are not professional mathematicians, thus making these new mathematical ideas in inverse scattering theory available to the wider scientific and engineering community.

Author : Alexander G. Ramm
Publisher : Belhaven
Page : 400 pages
File Size : 50,6 Mb
Release : 1992
Category : Mathematics
ISBN : UOM:39015022259785

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Multidimensional Inverse Scattering Problems by Alexander G. Ramm Pdf

Author : Fioralba Cakoni,David Colton
Publisher : Springer Science & Business Media
Page : 300 pages
File Size : 42,9 Mb
Release : 2013-10-28
Category : Mathematics
ISBN : 9781461488279

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A Qualitative Approach to Inverse Scattering Theory by Fioralba Cakoni,David Colton Pdf

Inverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. SIAM Review, 2006

Author : Dilip N. Ghosh Roy,L. S. Couchman
Publisher : Academic Press
Page : 336 pages
File Size : 54,8 Mb
Release : 2001-10-04
Category : Computers
ISBN : 0080546137

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Inverse Problems and Inverse Scattering of Plane Waves by Dilip N. Ghosh Roy,L. S. Couchman Pdf

The purpose of this text is to present the theory and mathematics of inverse scattering, in a simple way, to the many researchers and professionals who use it in their everyday research. While applications range across a broad spectrum of disciplines, examples in this text will focus primarly, but not exclusively, on acoustics. The text will be especially valuable for those applied workers who would like to delve more deeply into the fundamentally mathematical character of the subject matter. Practitioners in this field comprise applied physicists, engineers, and technologists, whereas the theory is almost entirely in the domain of abstract mathematics. This gulf between the two, if bridged, can only lead to improvement in the level of scholarship in this highly important discipline. This is the book's primary focus.

Author : Richard Beals,Percy Deift,Carlos Tomei
Publisher : American Mathematical Soc.
Page : 209 pages
File Size : 51,8 Mb
Release : 2015-03-02
Category : Mathematics
ISBN : 9781470420543

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Direct and Inverse Scattering on the Line by Richard Beals,Percy Deift,Carlos Tomei Pdf

This book deals with the theory of linear ordinary differential operators of arbitrary order. Unlike treatments that focus on spectral theory, this work centers on the construction of special eigenfunctions (generalized Jost solutions) and on the inverse problem: the problem of reconstructing the operator from minimal data associated to the special eigenfunctions. In the second order case this program includes spectral theory and is equivalent to quantum mechanical scattering theory; the essential analysis involves only the bounded eigenfunctions. For higher order operators, bounded eigenfunctions are again sufficient for spectral theory and quantum scattering theory, but they are far from sufficient for a successful inverse theory. The authors give a complete and self-contained theory of the inverse problem for an ordinary differential operator of any order. The theory provides a linearization for the associated nonlinear evolution equations, including KdV and Boussinesq. The authors also discuss Darboux-Bäcklund transformations, related first-order systems and their evolutions, and applications to spectral theory and quantum mechanical scattering theory. Among the book's most significant contributions are a new construction of normalized eigenfunctions and the first complete treatment of the self-adjoint inverse problem in order greater than two. In addition, the authors present the first analytic treatment of the corresponding flows, including a detailed description of the phase space for Boussinesq and other equations. The book is intended for mathematicians, physicists, and engineers in the area of soliton equations, as well as those interested in the analytical aspects of inverse scattering or in the general theory of linear ordinary differential operators. This book is likely to be a valuable resource to many. Required background consists of a basic knowledge of complex variable theory, the theory of ordinary differential equations, linear algebra, and functional analysis. The authors have attempted to make the book sufficiently complete and self-contained to make it accessible to a graduate student having no prior knowledge of scattering or inverse scattering theory. The book may therefore be suitable for a graduate textbook or as background reading in a seminar.

Author : S. Novikov,S.V. Manakov,L.P. Pitaevskii,V.E. Zakharov
Publisher : Springer Science & Business Media
Page : 298 pages
File Size : 51,5 Mb
Release : 1984-05-31
Category : Mathematics
ISBN : 0306109778

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Theory of Solitons by S. Novikov,S.V. Manakov,L.P. Pitaevskii,V.E. Zakharov Pdf

Author : Wiktor Eckhaus
Publisher : Unknown
Page : 222 pages
File Size : 43,8 Mb
Release : 1981
Category : Electronic
ISBN : 0444861661

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The inverse scattering transformation and the theory of solitons by Wiktor Eckhaus Pdf