Nonlinear Dispersive Partial Differential Equations And Inverse Scattering

Author : Peter D. Miller,Peter A. Perry,Jean-Claude Saut,Catherine Sulem
Publisher : Springer Nature
Page : 528 pages
File Size : 46,9 Mb
Release : 2019-11-14
Category : Mathematics
ISBN : 9781493998067

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Nonlinear Dispersive Partial Differential Equations and Inverse Scattering by Peter D. Miller,Peter A. Perry,Jean-Claude Saut,Catherine Sulem Pdf

This volume contains lectures and invited papers from the Focus Program on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" held at the Fields Institute from July 31-August 18, 2017. The conference brought together researchers in completely integrable systems and PDE with the goal of advancing the understanding of qualitative and long-time behavior in dispersive nonlinear equations. The program included Percy Deift’s Coxeter lectures, which appear in this volume together with tutorial lectures given during the first week of the focus program. The research papers collected here include new results on the focusing ​nonlinear Schrödinger (NLS) equation, the massive Thirring model, and the Benjamin-Bona-Mahoney equation as dispersive PDE in one space dimension, as well as the Kadomtsev-Petviashvili II equation, the Zakharov-Kuznetsov equation, and the Gross-Pitaevskii equation as dispersive PDE in two space dimensions. The Focus Program coincided with the fiftieth anniversary of the discovery by Gardner, Greene, Kruskal and Miura that the Korteweg-de Vries (KdV) equation could be integrated by exploiting a remarkable connection between KdV and the spectral theory of Schrodinger's equation in one space dimension. This led to the discovery of a number of completely integrable models of dispersive wave propagation, including the cubic NLS equation, and the derivative NLS equation in one space dimension and the Davey-Stewartson, Kadomtsev-Petviashvili and Novikov-Veselov equations in two space dimensions. These models have been extensively studied and, in some cases, the inverse scattering theory has been put on rigorous footing. It has been used as a powerful analytical tool to study global well-posedness and elucidate asymptotic behavior of the solutions, including dispersion, soliton resolution, and semiclassical limits.

Author : Christian Klein,Jean-Claude Saut
Publisher : Springer Nature
Page : 596 pages
File Size : 53,6 Mb
Release : 2021
Category : Differential equations
ISBN : 9783030914271

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Nonlinear Dispersive Equations by Christian Klein,Jean-Claude Saut Pdf

Nonlinear Dispersive Equations are partial differential equations that naturally arise in physical settings where dispersion dominates dissipation, notably hydrodynamics, nonlinear optics, plasma physics and Bose-Einstein condensates. The topic has traditionally been approached in different ways, from the perspective of modeling of physical phenomena, to that of the theory of partial differential equations, or as part of the theory of integrable systems. This monograph offers a thorough introduction to the topic, uniting the modeling, PDE and integrable systems approaches for the first time in book form. The presentation focuses on three "universal" families of physically relevant equations endowed with a completely integrable member: the Benjamin-Ono, Davey-Stewartson, and Kadomtsev-Petviashvili equations. These asymptotic models are rigorously derived and qualitative properties such as soliton resolution are studied in detail in both integrable and non-integrable models. Numerical simulations are presented throughout to illustrate interesting phenomena. By presenting and comparing results from different fields, the book aims to stimulate scientific interactions and attract new students and researchers to the topic. To facilitate this, the chapters can be read largely independently of each other and the prerequisites have been limited to introductory courses in PDE theory.

Author : Felipe Linares
Publisher : Unknown
Page : 301 pages
File Size : 49,7 Mb
Release : 2014
Category : Electronic
ISBN : 0387849165

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Introduction to Nonlinear Dispersive Equations by Felipe Linares Pdf

Author : Mark J. Ablowitz,Harvey Segur
Publisher : SIAM
Page : 433 pages
File Size : 51,5 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 : Spencer P Kuo
Publisher : World Scientific
Page : 198 pages
File Size : 42,6 Mb
Release : 2023-06-26
Category : Science
ISBN : 9781800614055

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Nonlinear Waves And Inverse Scattering Transform by Spencer P Kuo Pdf

Nonlinear waves are essential phenomena in scientific and engineering disciplines. The features of nonlinear waves are usually described by solutions to nonlinear partial differential equations (NLPDEs). This book was prepared to familiarize students with nonlinear waves and methods of solving NLPDEs, which will enable them to expand their studies into related areas. The selection of topics and the focus given to each provide essential materials for a lecturer teaching a nonlinear wave course.Chapter 1 introduces 'mode' types in nonlinear systems as well as Bäcklund transform, an indispensable technique to solve generic NLPDEs for stationary solutions. Chapters 2 and 3 are devoted to the derivation and solution characterization of three generic nonlinear equations: nonlinear Schrödinger equation, Korteweg-de Vries (KdV) equation, and Burgers equation. Chapter 4 is devoted to the inverse scattering transform (IST), addressing the initial value problems of a group of NLPDEs. In Chapter 5, derivations and proofs of the IST formulas are presented. Steps for applying IST to solve NLPDEs for solitary solutions are illustrated in Chapter 6.

Author : Lokenath Debnath
Publisher : Springer Science & Business Media
Page : 602 pages
File Size : 52,9 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9781489928467

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Nonlinear Partial Differential Equations for Scientists and Engineers by Lokenath Debnath Pdf

This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from diverse fields. Methods and properties of solutions, along with their physical significance, help make the book more useful for a diverse readership. The book is an exceptionally complete text/reference for graduates, researchers, and professionals in mathematics, physics, and engineering.

Author : Lokenath Debnath
Publisher : World Scientific
Page : 683 pages
File Size : 51,7 Mb
Release : 1992-09-09
Category : Electronic
ISBN : 9789814554961

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Nonlinear Dispersive Wave Systems by Lokenath Debnath Pdf

This book brings together a comprehensive account of major developments in the theory and applications of nonlinear dispersive waves, nonlinear water waves, KdV and nonlinear Schrodinger equations, Davey-Stewartson equation, Benjamin-Ono equation and nonlinear instability phenomena. In order to give the book a wider readership, chapters have been written by internationally known researchers who have made significant contributions to nonlinear waves and nonlinear instability. This volume will be invaluable to applied mathematicians, physicists, geophysicists, oceanographers, engineering scientists, and to anyone interested in nonlinear dynamics.

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 : 43,6 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 : Pham Loi Vu
Publisher : CRC Press
Page : 198 pages
File Size : 44,8 Mb
Release : 2019-11-11
Category : Mathematics
ISBN : 9781000709063

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Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations by Pham Loi Vu Pdf

Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations is devoted to inverse scattering problems (ISPs) for differential equations and their application to nonlinear evolution equations (NLEEs). The book is suitable for anyone who has a mathematical background and interest in functional analysis, partial differential equations, equations of mathematical physics, and functions of a complex variable. This book is intended for a wide community working with inverse scattering problems and their applications; in particular, there is a traditional community in mathematical physics. In this monograph, the problems are solved step-by-step, and detailed proofs are given for the problems to make the topics more accessible for students who are approaching them for the first time. Features • The unique solvability of ISPs are proved. The scattering data of the considered inverse scattering problems (ISPs) are described completely. • Solving the associated initial value problem or initial-boundary value problem for the nonlinear evolution equations (NLEEs) is carried out step-by-step. Namely, the NLEE can be written as the compatibility condition of two linear equations. The unknown boundary values are calculated with the help of the Lax (generalized) equation, and then the time-dependent scattering data (SD) are constructed from the initial and boundary conditions. • The potentials are recovered uniquely in terms of time-dependent SD, and the solution of the NLEEs is expressed uniquely in terms of the found solutions of the ISP. • Since the considered ISPs are solved well, then the SPs generated by two linear equations constitute the inverse scattering method (ISM). The application of the ISM to solving the NLEEs is consistent and is effectively embedded in the schema of the ISM.

Author : M. Burak Erdoğan,Nikolaos Tzirakis
Publisher : Cambridge University Press
Page : 203 pages
File Size : 53,7 Mb
Release : 2016-05-12
Category : Mathematics
ISBN : 9781107149045

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Dispersive Partial Differential Equations by M. Burak Erdoğan,Nikolaos Tzirakis Pdf

Introduces nonlinear dispersive partial differential equations in a detailed yet elementary way without compromising the depth and richness of the subject.

Author : Herbert Koch,Daniel Tataru,Monica Vişan
Publisher : Springer
Page : 310 pages
File Size : 52,6 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9783034807364

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Dispersive Equations and Nonlinear Waves by Herbert Koch,Daniel Tataru,Monica Vişan Pdf

The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.​

Author : Mark J. Ablowitz,P. A. Clarkson
Publisher : Cambridge University Press
Page : 532 pages
File Size : 53,7 Mb
Release : 1991-12-12
Category : Mathematics
ISBN : 9780521387309

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Solitons, Nonlinear Evolution Equations and Inverse Scattering by Mark J. Ablowitz,P. A. Clarkson Pdf

This book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.

Author : Basil Nicolaenko,Darryl D. Holm,James M. Hyman
Publisher : American Mathematical Soc.
Page : 494 pages
File Size : 42,7 Mb
Release : 1986
Category : Mathematics
ISBN : 0821811258

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Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1 by Basil Nicolaenko,Darryl D. Holm,James M. Hyman Pdf

Focusing on the increased interplay of theoretical advances in nonlinear hyperbolic systems, completely integrable systems, and evolutionary systems of nonlinear partial differential equations, this title contains papers grouped in sections: integrable systems, hyperbolic systems, variational problems, evolutionary systems, and dispersive systems.

Author : Luis A. Caffarelli,François Golse,Yan Guo,Carlos E. Kenig,Alexis Vasseur
Publisher : Springer Science & Business Media
Page : 156 pages
File Size : 55,8 Mb
Release : 2012-02-02
Category : Mathematics
ISBN : 9783034801911

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Nonlinear Partial Differential Equations by Luis A. Caffarelli,François Golse,Yan Guo,Carlos E. Kenig,Alexis Vasseur Pdf

The book covers several topics of current interest in the field of nonlinear partial differential equations and their applications to the physics of continuous media and particle interactions. It treats the quasigeostrophic equation, integral diffusions, periodic Lorentz gas, Boltzmann equation, and critical dispersive nonlinear Schrödinger and wave equations. The book describes in a careful and expository manner several powerful methods from recent top research articles.

Author : Basil Nicolaenko,Darryl D. Holm,James M. Hyman,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 490 pages
File Size : 40,6 Mb
Release : 1986-12-31
Category : Mathematics
ISBN : 082189689X

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Nonlinear Systems of Partial Differential Equations in Applied Mathematics by Basil Nicolaenko,Darryl D. Holm,James M. Hyman,American Mathematical Society Pdf

These two volumes of 47 papers focus on the increased interplay of theoretical advances in nonlinear hyperbolic systems, completely integrable systems, and evolutionary systems of nonlinear partial differential equations. The papers both survey recent results and indicate future research trends in these vital and rapidly developing branches of PDEs. The editor has grouped the papers loosely into the following five sections: integrable systems, hyperbolic systems, variational problems, evolutionary systems, and dispersive systems. However, the variety of the subjects discussed as well as their many interwoven trends demonstrate that it is through interactive advances that such rapid progress has occurred. These papers require a good background in partial differential equations. Many of the contributors are mathematical physicists, and the papers are addressed to mathematical physicists (particularly in perturbed integrable systems), as well as to PDE specialists and applied mathematicians in general.