[PDF] The Direct Scattering Study Of The Parametrically Driven Nonlinear Schrodinger Equation eBook

Author : Tuncay Aktosun
Publisher : Springer Nature
Page : 631 pages
File Size : 15,98 MB
Release : 2020-05-19
Category : Mathematics
ISBN : 3030384314

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Authored by two experts in the field who have been long-time collaborators, this monograph treats the scattering and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. The existence, uniqueness, construction, and characterization aspects are treated with mathematical rigor, and physical insight is provided to make the material accessible to mathematicians, physicists, engineers, and applied scientists with an interest in scattering and inverse scattering. The material presented is expected to be useful to beginners as well as experts in the field. The subject matter covered is expected to be interesting to a wide range of researchers including those working in quantum graphs and scattering on graphs. The theory presented is illustrated with various explicit examples to improve the understanding of scattering and inverse scattering problems. The monograph introduces a specific class of input data sets consisting of a potential and a boundary condition and a specific class of scattering data sets consisting of a scattering matrix and bound-state information. The important problem of the characterization is solved by establishing a one-to-one correspondence between the two aforementioned classes. The characterization result is formulated in various equivalent forms, providing insight and allowing a comparison of different techniques used to solve the inverse scattering problem. The past literature treated the type of boundary condition as a part of the scattering data used as input to recover the potential. This monograph provides a proper formulation of the inverse scattering problem where the type of boundary condition is no longer a part of the scattering data set, but rather both the potential and the type of boundary condition are recovered from the scattering data set.

Author : Nicolas Burq
Publisher : American Mathematical Society
Page : 102 pages
File Size : 35,99 MB
Release : 2024-05-15
Category : Mathematics
ISBN : 1470469790

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Author : Peter E. Zhidkov
Publisher : Springer
Page : 154 pages
File Size : 13,33 MB
Release : 2001-04-24
Category : Mathematics
ISBN : 9783540418337

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- of nonlinear the of solitons the the last 30 theory partial theory During years - has into solutions of a kind a differential special equations (PDEs) possessing grown and in view the attention of both mathematicians field that attracts physicists large and of the of the problems of its novelty problems. Physical important applications for in the under consideration are mo- to the observed, example, equations leading mathematical discoveries is the Makhankov One of the related V.G. by [60]. graph from this field methods that of certain nonlinear by equations possibility studying inverse these to the problem; equations were analyze quantum scattering developed this method of the inverse called solvable the scattering problem (on subject, are by known nonlinear At the the class of for same time, currently example [89,94]). see, the other there is solvable this method is narrow on hand, PDEs sufficiently and, by of differential The latter called the another qualitative theory equations. approach, the of various in includes on pr- investigations well-posedness approach particular solutions such or lems for these the behavior of as stability blowing-up, equations, these and this of approach dynamical systems generated by equations, etc., properties in wider class of a makes it to an problems (maybe possible investigate essentially more general study).

Author : Donlapark Pornnopparath
Publisher :
Page : 131 pages
File Size : 45,22 MB
Release : 2018
Category :
ISBN :

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We consider the initial value problem for various type of nonlinear Schrödinger equations with derivative nonlinearity which cannot be treated by normal perturbative arguments because of the loss in derivative from the nonlinearity. The first part of the study involves finding the well-posedness in low regularity Sobolev spaces for different types of nonlinearities. The key idea is to capture a part of the solution that resembles the linear Schrödinger dynamic while keeping the remaining part spatial and frequency localized. With this, we can study the interactions between the truncations of the solution at different frequencies and obtain a meaningful perturbative analysis. In the second part, we study the dynamic of the cubic nonlinear Schrödinger equation in the energy critical Sobolev space by projecting the solution onto different wave packets which are frequency and spatial localized at all time. As a result, we obtain the asymptotic behavior, modified scattering profile and asymptotic completeness of the solution without relying on the integrable structure of the equation.

Author : Mark J. Ablowitz
Publisher : SIAM
Page : 433 pages
File Size : 17,52 MB
Release : 2006-05-15
Category : Mathematics
ISBN : 089871477X

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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.