Download or read book Inverse Problems and Nonlinear Evolution Equations written by Alexander L. Sakhnovich and published by Walter de Gruyter. This book was released on 2013-07-31 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the method of operator identities and related theory of S-nodes, both developed by Lev Sakhnovich. The notion of the transfer matrix function generated by the S-node plays an essential role. The authors present fundamental solutions of various important systems of differential equations using the transfer matrix function, that is, either directly in the form of the transfer matrix function or via the representation in this form of the corresponding Darboux matrix, when Bäcklund–Darboux transformations and explicit solutions are considered. The transfer matrix function representation of the fundamental solution yields solution of an inverse problem, namely, the problem to recover system from its Weyl function. Weyl theories of selfadjoint and skew-selfadjoint Dirac systems, related canonical systems, discrete Dirac systems, system auxiliary to the N-wave equation and a system rationally depending on the spectral parameter are obtained in this way. The results on direct and inverse problems are applied in turn to the study of the initial-boundary value problems for integrable (nonlinear) wave equations via inverse spectral transformation method. Evolution of the Weyl function and solution of the initial-boundary value problem in a semi-strip are derived for many important nonlinear equations. Some uniqueness and global existence results are also proved in detail using evolution formulas. The reading of the book requires only some basic knowledge of linear algebra, calculus and operator theory from the standard university courses.
Download or read book Inverse Problems for Kinetic and Other Evolution Equations written by I︠U︡riĭ Evgenʹevich Anikonov and published by VSP. This book was released on 2001 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph in the "Inverse and Ill-Posed Problems Series deals with methods of studying multidimensional inverse problems for kinetic and other evolution equations, in particular transfer equations. The methods used are applied to concrete inverse problems, especially multidimensional inverse problems applicable in linear and nonlinear statements.A significant part of the book contains formulas and relations for solving inverse problems, including formulas for the solution and coefficients of kinetic equations, differential-difference equations, nonlinear evolution equations, and second order equations.This monograph will be of value and interest to mathematicians, engineers and other specialists dealing with inverse and ill posed problems.
Download or read book Inverse Problems for Kinetic and Other Evolution Equations written by Yu. E. Anikonov and published by Walter de Gruyter GmbH & Co KG. This book was released on 2014-07-24 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph deals with methods of studying multidimensional inverse problems for kinetic and other evolution equations, in particular transfer equations. The methods used are applied to concrete inverse problems, especially multidimensional inverse problems applicable in linear and nonlinear statements. A significant part of the book contains formulas and relations for solving inverse problems, including formulas for the solution and coefficients of kinetic equations, differential-difference equations, nonlinear evolution equations, and second order equations.
Download or read book Inverse Problems Non Linear Evolution written by Centre national de la recherche scientifique (France). Recherche coopérative sur programme 264 and published by Centre National de la Recherche Scientifique. This book was released on 1980 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Solitons Nonlinear Evolution Equations and Inverse Scattering written by Mark J. Ablowitz and published by Cambridge University Press. This book was released on 1991-12-12 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Nonlinear Evolution Equations and Potential Theory written by J. Kral and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: Preface.- Gottfried Anger: Direct and inverse problems in potential theory.- Viorel Barbu: Regularity results for sane differential equations associated with maximal monotone operators in Hilbert spaces.- Haim Brezis: Classes d'interpolation associées à un opérateur monotone et applications.- Siegfried Dnümmel: On inverse problems for k-dimensional potentials.- Jozef Ka?ur: Application of Rothe's method to nonlinear parabolic boundary value problems.- Josef Král: Potentials and removability of singularities.- Vladimir Lovicar: Theorem of Fréchet and asymptotically almost periodid solutions of.
Download or read book Nonlinear Evolution Equations and Dynamical Systems written by Vladimir G. Makhankov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the 6th International Workshop, 16-26 July 1990, Dubna, USSR
Download or read book Nonlinear Evolution Equations and Dynamical Systems written by Sandra Carillo and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Evolution Equations and Dynamical Systems (NEEDS) provides a presentation of the state of the art. Except for a few review papers, the 40 contributions are intentially brief to give only the gist of the methods, proofs, etc. including references to the relevant litera- ture. This gives a handy overview of current research activities. Hence, the book should be equally useful to the senior resercher as well as the colleague just entering the field. Keypoints treated are: i) integrable systems in multidimensions and associated phenomenology ("dromions"); ii) criteria and tests of integrability (e.g., Painlev test); iii) new developments related to the scattering transform; iv) algebraic approaches to integrable systems and Hamiltonian theory (e.g., connections with Young-Baxter equations and Kac-Moody algebras); v) new developments in mappings and cellular automata, vi) applications to general relativity, condensed matter physics, and oceanography.
Download or read book Methods of Inverse Problems in Physics written by Dilip N. Ghosh Roy and published by CRC Press. This book was released on 1991-03-14 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: This interesting volume focuses on the second of the two broad categories into which problems of physical sciences fall-direct (or forward) and inverse (or backward) problems. It emphasizes one-dimensional problems because of their mathematical clarity. The unique feature of the monograph is its rigorous presentation of inverse problems (from quantum scattering to vibrational systems), transmission lines, and imaging sciences in a single volume. It includes exhaustive discussions on spectral function, inverse scattering integral equations of Gel'fand-Levitan and Marcenko, Povzner-Levitan and Levin transforms, Møller wave operators and Krein's functionals, S-matrix and scattering data, and inverse scattering transform for solving nonlinear evolution equations via inverse solving of a linear, isospectral Schrodinger equation and multisoliton solutions of the K-dV equation, which are of special interest to quantum physicists and mathematicians. The book also gives an exhaustive account of inverse problems in discrete systems, including inverting a Jacobi and a Toeplitz matrix, which can be applied to geophysics, electrical engineering, applied mechanics, and mathematics. A rigorous inverse problem for a continuous transmission line developed by Brown and Wilcox is included. The book concludes with inverse problems in integral geometry, specifically Radon's transform and its inversion, which is of particular interest to imaging scientists. This fascinating volume will interest anyone involved with quantum scattering, theoretical physics, linear and nonlinear optics, geosciences, mechanical, biomedical, and electrical engineering, and imaging research.
Download or read book Nonlinear Evolution Equations written by Antonio Degasperis and published by . This book was released on 1990 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Direct and Inverse Methods in Nonlinear Evolution Equations written by Robert M. Conte and published by Springer Science & Business Media. This book was released on 2003-10-21 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities à la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.
Download or read book Direct and Inverse Methods in Nonlinear Evolution Equations written by Robert M. Conte and published by Springer. This book was released on 2014-03-12 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many physical phenomena are described by nonlinear evolution equation. Those that are integrable provide various mathematical methods, presented by experts in this tutorial book, to find special analytic solutions to both integrable and partially integrable equations. The direct method to build solutions includes the analysis of singularities à la Painlevé, Lie symmetries leaving the equation invariant, extension of the Hirota method, construction of the nonlinear superposition formula. The main inverse method described here relies on the bi-hamiltonian structure of integrable equations. The book also presents some extension to equations with discrete independent and dependent variables. The different chapters face from different points of view the theory of exact solutions and of the complete integrability of nonlinear evolution equations. Several examples and applications to concrete problems allow the reader to experience directly the power of the different machineries involved.
Download or read book Lectures on Nonlinear Evolution Equations written by Reinhard Racke and published by Birkhäuser. This book was released on 2015-08-31 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered.
Download or read book Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations written by Pham Loi Vu and published by CRC Press. This book was released on 2023-05-15 with total page 453 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inverse Scattering Problems and Their Applications to Nonlinear Integrable Equations, Second Edition is devoted to inverse scattering problems (ISPs) for differential equations and their applications to nonlinear evolution equations (NLEEs). The book is suitable for anyone who has a mathematical background and interest in functional analysis, differential equations, and equations of mathematical physics. This book is intended for a wide community working with ISPs and their applications. There is an especially strong traditional community in mathematical physics. In this monograph, the problems are presented step-by-step, and detailed proofs are given for considered problems to make the topics more accessible for students who are approaching them for the first time. New to the Second Edition All new chapter dealing with the Bäcklund transformations between a common solution of both linear equations in the Lax pair and the solution of the associated IBVP for NLEEs on the half-line Updated references and concluding remarks Features Solving the direct and ISP, then solving the associated initial value problem (IVP) or initial-boundary value problem (IBVP) for NLEEs are carried out step-by-step. The unknown boundary values are calculated with the help of the Lax (generalized) equations, then the time-dependent scattering data (SD) are expressed in terms of preassigned initial and boundary conditions. Thereby, the potential functions are recovered uniquely in terms of the given initial and calculated boundary conditions. The unique solvability of the ISP is proved and the SD of the scattering problem is described completely. The considered ISPs are well-solved. The ISPs are set up appropriately for constructing the Bӓckhund transformations (BTs) for solutions of associated IBVPs or IVPs for NLEEs. The procedure for finding a BT for the IBVP for NLEEs on the half-line differs from the one used for obtaining a BT for non-linear differential equations defined in the whole space. The interrelations between the ISPs and the constructed BTs are established to become new powerful unified transformations (UTs) for solving IBVPs or IVPs for NLEEs, that can be used in different areas of physics and mechanics. The application of the UTs is consistent and efficiently embedded in the scheme of the associated ISP.
Download or read book Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations written by Pham Loi Vu and published by CRC Press. This book was released on 2019-11-11 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Nonlinear Evolution Equations written by Songmu Zheng and published by CRC Press. This book was released on 2004-07-08 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear evolution equations arise in many fields of sciences including physics, mechanics, and material science. This book introduces some important methods for dealing with these equations and explains clearly and concisely a wide range of relevant theories and techniques. These include the semigroup method, the compactness and monotone operator
Download or read book Some Topics on Inverse Problems written by Centre national de la recherche scientifique (France). Recherche coopérative sur programme 264 and published by . This book was released on 1988 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt:
