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This book presents a detailed mathematical analysis of scattering theory, obtains soliton solutions, and analyzes soliton interactions, both scalar and vector.
Book Synopsis Discrete and Continuous Nonlinear Schrödinger Systems by : M. J. Ablowitz
Download or read book Discrete and Continuous Nonlinear Schrödinger Systems written by M. J. Ablowitz and published by Cambridge University Press. This book was released on 2004 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a detailed mathematical analysis of scattering theory, obtains soliton solutions, and analyzes soliton interactions, both scalar and vector.
This book constitutes the first effort to summarize a large volume of results obtained over the past 20 years in the context of the Discrete Nonlinear Schrödinger equation and the physical settings that it describes.
Book Synopsis The Discrete Nonlinear Schrödinger Equation by : Panayotis G. Kevrekidis
Download or read book The Discrete Nonlinear Schrödinger Equation written by Panayotis G. Kevrekidis and published by Springer Science & Business Media. This book was released on 2009-07-07 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the first effort to summarize a large volume of results obtained over the past 20 years in the context of the Discrete Nonlinear Schrödinger equation and the physical settings that it describes.
It is well known that the Fourier transform can be used to solve initial value problems (IVPs) for linear partial dierential equations (PDEs). It is also well known that a special class of nonlinear PDEs exists for which a nonlinear analogue of this technique exists, called the inverse scattering transform (IST). Equations of this type, usually called integrable systems, exhibit a surprisingly rich and beautiful mathematical structure. A large body of knowledge has been accumulated on these systems over the last forty years. In particular, the solution of IVPs for integrable nonlinear PDEs in one spatial and one temporal dimension was developed in the 1970's under the assumption of rapidly decaying initial conditions at infinity, hereafter referred to as zero boundary conditions (ZBCs). On the other hand, IVPs in which the initial condition satisfies non-zero boundary conditions (NZBCs) at infinity are much less well characterized. So are boundary value problems (BVPs).^In particular, BVPs for integrable nonlinear PDEs can only be linearized for a special kind ofboundary conditions (BCs); such BCs are then called linearizable. Both of these kinds of problems (namely, IVPs with NZBCs and BVPs) are still the object of active research. This thesis is devoted to both kinds of problems. Specifically, the thesis contains the following original contributions:I. In Chapter 3 we revisit various problems for the focusing nonlinear Schrödinger (NLS) equation with ZBCs at infinity. Explicitly, we present a detailed discussion of: (i) double poles in the scattering problem within the framework of the Riemann-Hilbert formalism; (ii) BVPs on the half line with linearizable BCs at the origin, including self-symmetric eigenvalues and the reflection-induced position shift. II. In Chapter 4 we develop a method to solve BVPs for the Ablowitz-Ladik system on the natural numbers with linearizable BCs at the origin.^We do so by constructing a suitable Bäcklund transformation. Importantly, this method allows us to deal eciently with self-symmetric eigenvalues. As a result, we completely classify the solutions to the BVP of the AL system with a linearizable BC at the origin. III. In Chapter 5 we develop a method to solve BVPs for the defocusing NLS equation on the half line with NZBCs at infinity and linearizable BCs at the origin. As with the Ablowitz-Ladik system, we do so by constructing a suitable Bäcklund transformation, and we use it to completely characterize the BVP, including the self-symmetric eigenvalues. IV. In Chapter 6 we provide a detailed comparison of two dierent approaches to the IST for the defocusing vector NLS (VNLS) equation with NZBCs at infinity.^After briefly reviewing the standard IST approach developed in [65] for the two-component VNLS equation and the new approach to IST used in [66] for the multi-component VNLS equation, we show how the new approach relates to the standard one for both the scalar NLS equation and the two-component VNLS equation. These results serve both to obtain a better understanding on the new approach, and as a preparatory step to obtain explicit soliton solutions in the multi-component case.
Book Synopsis Boundary Value Problems for Discrete and Continuous Nonlinear Schrödinger Equations℗ by : Anh Minh Bui
Download or read book Boundary Value Problems for Discrete and Continuous Nonlinear Schrödinger Equations℗ written by Anh Minh Bui and published by . This book was released on 2012 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is well known that the Fourier transform can be used to solve initial value problems (IVPs) for linear partial dierential equations (PDEs). It is also well known that a special class of nonlinear PDEs exists for which a nonlinear analogue of this technique exists, called the inverse scattering transform (IST). Equations of this type, usually called integrable systems, exhibit a surprisingly rich and beautiful mathematical structure. A large body of knowledge has been accumulated on these systems over the last forty years. In particular, the solution of IVPs for integrable nonlinear PDEs in one spatial and one temporal dimension was developed in the 1970's under the assumption of rapidly decaying initial conditions at infinity, hereafter referred to as zero boundary conditions (ZBCs). On the other hand, IVPs in which the initial condition satisfies non-zero boundary conditions (NZBCs) at infinity are much less well characterized. So are boundary value problems (BVPs).^In particular, BVPs for integrable nonlinear PDEs can only be linearized for a special kind ofboundary conditions (BCs); such BCs are then called linearizable. Both of these kinds of problems (namely, IVPs with NZBCs and BVPs) are still the object of active research. This thesis is devoted to both kinds of problems. Specifically, the thesis contains the following original contributions:I. In Chapter 3 we revisit various problems for the focusing nonlinear Schrödinger (NLS) equation with ZBCs at infinity. Explicitly, we present a detailed discussion of: (i) double poles in the scattering problem within the framework of the Riemann-Hilbert formalism; (ii) BVPs on the half line with linearizable BCs at the origin, including self-symmetric eigenvalues and the reflection-induced position shift. II. In Chapter 4 we develop a method to solve BVPs for the Ablowitz-Ladik system on the natural numbers with linearizable BCs at the origin.^We do so by constructing a suitable Bäcklund transformation. Importantly, this method allows us to deal eciently with self-symmetric eigenvalues. As a result, we completely classify the solutions to the BVP of the AL system with a linearizable BC at the origin. III. In Chapter 5 we develop a method to solve BVPs for the defocusing NLS equation on the half line with NZBCs at infinity and linearizable BCs at the origin. As with the Ablowitz-Ladik system, we do so by constructing a suitable Bäcklund transformation, and we use it to completely characterize the BVP, including the self-symmetric eigenvalues. IV. In Chapter 6 we provide a detailed comparison of two dierent approaches to the IST for the defocusing vector NLS (VNLS) equation with NZBCs at infinity.^After briefly reviewing the standard IST approach developed in [65] for the two-component VNLS equation and the new approach to IST used in [66] for the multi-component VNLS equation, we show how the new approach relates to the standard one for both the scalar NLS equation and the two-component VNLS equation. These results serve both to obtain a better understanding on the new approach, and as a preparatory step to obtain explicit soliton solutions in the multi-component case.
This is the first of two Euroconferences aimed at addressing the issues of Nonlinearity and Disorder. The 1995 Euroconference was devoted to the mathematical, numerical and experimental studies related to the Klein-Gordon and Schrödinger systems. The Euroconference was organized around main lectures in each area to introduce the main concepts and stimulate discussions. The mathematical studies covered the functional anlaysis and stochastic techniques applied to the general Klein-Gordon and Schrödinger wave equations. Also a panoramic view of the numerical schemes was presented to simulate the above equations, as well as an overview of the applications of such systems in the areas of condensed matter, optical physics, new materials and biophysics. Special attention was devoted to the discrete Schrödinger and Klein-Gordon systems and their applications.
Book Synopsis Nonlinear Klein-gordon And Schrodinger Systems: Theory And Applications by : Luis Vazquez
Download or read book Nonlinear Klein-gordon And Schrodinger Systems: Theory And Applications written by Luis Vazquez and published by World Scientific. This book was released on 1996-06-20 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first of two Euroconferences aimed at addressing the issues of Nonlinearity and Disorder. The 1995 Euroconference was devoted to the mathematical, numerical and experimental studies related to the Klein-Gordon and Schrödinger systems. The Euroconference was organized around main lectures in each area to introduce the main concepts and stimulate discussions. The mathematical studies covered the functional anlaysis and stochastic techniques applied to the general Klein-Gordon and Schrödinger wave equations. Also a panoramic view of the numerical schemes was presented to simulate the above equations, as well as an overview of the applications of such systems in the areas of condensed matter, optical physics, new materials and biophysics. Special attention was devoted to the discrete Schrödinger and Klein-Gordon systems and their applications.
Book Synopsis Discrete and Continuous Dynamical Systems by :
Download or read book Discrete and Continuous Dynamical Systems written by and published by . This book was released on 2009 with total page 728 pages. Available in PDF, EPUB and Kindle. Book excerpt:
This book commemorates the 60th birthday of Dr. Wim van Horssen, a specialist in nonlinear dynamic and wave processes in solids, fluids and structures. In honor of Dr. Horssen’s contributions to the field, it presents papers discussing topics such as the current problems of the theory of nonlinear dynamic processes in continua and structures; applications, including discrete and continuous dynamic models of structures and media; and problems of asymptotic approaches.
Book Synopsis Nonlinear Dynamics of Discrete and Continuous Systems by : Andrei K. Abramian
Download or read book Nonlinear Dynamics of Discrete and Continuous Systems written by Andrei K. Abramian and published by Springer Nature. This book was released on 2020-11-02 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book commemorates the 60th birthday of Dr. Wim van Horssen, a specialist in nonlinear dynamic and wave processes in solids, fluids and structures. In honor of Dr. Horssen’s contributions to the field, it presents papers discussing topics such as the current problems of the theory of nonlinear dynamic processes in continua and structures; applications, including discrete and continuous dynamic models of structures and media; and problems of asymptotic approaches.
Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications presents a collection of breakthrough research portraying the odyssey of light from optical solitons to optical rogue waves in nonlinear optical fibers. The book provides a simple yet holistic view on the theoretical and application-oriented aspects of light, with a special focus on the underlying nonlinear phenomena. Exploring the very frontiers of light-wave technology, the text covers the basics of nonlinear fiberoptics and the dynamics of electromagnetic pulse propagation in nonlinear waveguides. It also highlights some of the latest advances in nonlinear optical fiber technology, discussing hidden symmetry reductions and Ablowitz–Kaup–Newell–Segur (AKNS) hierarchies for nonautonomous solitons, state-of-the-art Brillouin scattering applications, backpropagation, and the concept of eigenvalue communication—a powerful nonlinear digital signal processing technique that paves the way to overcome the current limitations of traditional communications methods in nonlinear fiber channels. Key chapters study the feasibility of the eigenvalue demodulation scheme based on digital coherent technology by throwing light on the experimental study of the noise tolerance of the demodulated eigenvalues, investigate matter wave solitons and other localized excitations pertaining to Bose–Einstein condensates in atom optics, and examine quantum field theory analogue effects occurring in binary waveguide arrays, plasmonic arrays, etc., as well as their ensuing nonlinear wave propagation. Featuring a foreword by Dr. Akira Hasegawa, the father of soliton communication systems, Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications serves as a curtain raiser to usher in the photonics era. The technological innovations at the core of the book form the basis for the next generation of ultra-high speed computers and telecommunication devices.
Book Synopsis Odyssey of Light in Nonlinear Optical Fibers by : Kuppuswamy Porsezian
Download or read book Odyssey of Light in Nonlinear Optical Fibers written by Kuppuswamy Porsezian and published by CRC Press. This book was released on 2017-12-19 with total page 531 pages. Available in PDF, EPUB and Kindle. Book excerpt: Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications presents a collection of breakthrough research portraying the odyssey of light from optical solitons to optical rogue waves in nonlinear optical fibers. The book provides a simple yet holistic view on the theoretical and application-oriented aspects of light, with a special focus on the underlying nonlinear phenomena. Exploring the very frontiers of light-wave technology, the text covers the basics of nonlinear fiberoptics and the dynamics of electromagnetic pulse propagation in nonlinear waveguides. It also highlights some of the latest advances in nonlinear optical fiber technology, discussing hidden symmetry reductions and Ablowitz–Kaup–Newell–Segur (AKNS) hierarchies for nonautonomous solitons, state-of-the-art Brillouin scattering applications, backpropagation, and the concept of eigenvalue communication—a powerful nonlinear digital signal processing technique that paves the way to overcome the current limitations of traditional communications methods in nonlinear fiber channels. Key chapters study the feasibility of the eigenvalue demodulation scheme based on digital coherent technology by throwing light on the experimental study of the noise tolerance of the demodulated eigenvalues, investigate matter wave solitons and other localized excitations pertaining to Bose–Einstein condensates in atom optics, and examine quantum field theory analogue effects occurring in binary waveguide arrays, plasmonic arrays, etc., as well as their ensuing nonlinear wave propagation. Featuring a foreword by Dr. Akira Hasegawa, the father of soliton communication systems, Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications serves as a curtain raiser to usher in the photonics era. The technological innovations at the core of the book form the basis for the next generation of ultra-high speed computers and telecommunication devices.
The lectures that comprise this volume constitute a comprehensive survey of the many and various aspects of integrable dynamical systems. The present edition is a streamlined, revised and updated version of a 1997 set of notes that was published as Lecture Notes in Physics, Volume 495. This volume will be complemented by a companion book dedicated to discrete integrable systems. Both volumes address primarily graduate students and nonspecialist researchers but will also benefit lecturers looking for suitable material for advanced courses and researchers interested in specific topics.
Book Synopsis Integrability of Nonlinear Systems by : Yvette Kosmann-Schwarzbach
Download or read book Integrability of Nonlinear Systems written by Yvette Kosmann-Schwarzbach and published by Springer Science & Business Media. This book was released on 2004-02-17 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: The lectures that comprise this volume constitute a comprehensive survey of the many and various aspects of integrable dynamical systems. The present edition is a streamlined, revised and updated version of a 1997 set of notes that was published as Lecture Notes in Physics, Volume 495. This volume will be complemented by a companion book dedicated to discrete integrable systems. Both volumes address primarily graduate students and nonspecialist researchers but will also benefit lecturers looking for suitable material for advanced courses and researchers interested in specific topics.
We consider a two-dimensional (2D) generalization of a recently proposed model [Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual "extended" unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). As a result, typical scenarios of instability development are exhibited through direct simulations.
Book Synopsis Solitons and Vortices in Two-dimensional Discrete Nonlinear Schrödinger Systems with Spatially Modulated Nonlinearity by :
Download or read book Solitons and Vortices in Two-dimensional Discrete Nonlinear Schrödinger Systems with Spatially Modulated Nonlinearity written by and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider a two-dimensional (2D) generalization of a recently proposed model [Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual "extended" unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). As a result, typical scenarios of instability development are exhibited through direct simulations.
The volume contains the proceedings of the OTAMP 2006 (Operator Theory, Analysis and Mathematical Physics) conference held at Lund University in June 2006. The conference was devoted to the methods of analysis and operator theory in modern mathematical physics. The following special sessions were organized: Spectral analysis of Schrödinger operators; Jacobi and CMV matrices and orthogonal polynomials; Quasi-periodic and random Schrödinger operators; Quantum graphs.
Book Synopsis Methods of Spectral Analysis in Mathematical Physics by : Jan Janas
Download or read book Methods of Spectral Analysis in Mathematical Physics written by Jan Janas and published by Springer Science & Business Media. This book was released on 2008-12-16 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume contains the proceedings of the OTAMP 2006 (Operator Theory, Analysis and Mathematical Physics) conference held at Lund University in June 2006. The conference was devoted to the methods of analysis and operator theory in modern mathematical physics. The following special sessions were organized: Spectral analysis of Schrödinger operators; Jacobi and CMV matrices and orthogonal polynomials; Quasi-periodic and random Schrödinger operators; Quantum graphs.
