Inverse Problems and Nonlinear Evolution Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Inverse Problems and Nonlinear Evolution Equations write by Alexander L. Sakhnovich. This book was released on 2013-07-31. Inverse Problems and Nonlinear Evolution Equations available in PDF, EPUB and Kindle. 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.
Solitons, Nonlinear Evolution Equations and Inverse Scattering
Solitons, Nonlinear Evolution Equations and Inverse Scattering - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Solitons, Nonlinear Evolution Equations and Inverse Scattering write by Mark J. Ablowitz. This book was released on 1991-12-12. Solitons, Nonlinear Evolution Equations and Inverse Scattering available in PDF, EPUB and Kindle. 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.
Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations
Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations write by Pham Loi Vu. This book was released on 2023-05-15. Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations available in PDF, EPUB and Kindle. 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.
Evolution Equations and Approximations
Evolution Equations and Approximations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Evolution Equations and Approximations write by Kazufumi Ito. This book was released on 2002. Evolution Equations and Approximations available in PDF, EPUB and Kindle. Annotation Ito (North Carolina State U.) and Kappel (U. of Graz, Austria) offer a unified presentation of the general approach for well-posedness results using abstract evolution equations, drawing from and modifying the work of K. and Y. Kobayashi and S. Oharu. They also explore abstract approximation results for evolution equations. Their work is not a textbook, but they explain how instructors can use various sections, or combinations of them, as a foundation for a range of courses. Annotation copyrighted by Book News, Inc., Portland, OR
Direct and Inverse Methods in Nonlinear Evolution Equations
Direct and Inverse Methods in Nonlinear Evolution Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Direct and Inverse Methods in Nonlinear Evolution Equations write by Robert M. Conte. This book was released on 2003-10-21. Direct and Inverse Methods in Nonlinear Evolution Equations available in PDF, EPUB and Kindle. 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.
