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Author:
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ISBN:
Category :
Languages : en
Pages : 10
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Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 10
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Author: F.A. Berezin
Publisher: Springer Science & Business Media
ISBN: 9780792312185
Category : Mathematics
Languages : en
Pages : 590
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Book Description
Author: Jan Chabrowski
Publisher: World Scientific
ISBN: 9789810240769
Category : Mathematics
Languages : en
Pages : 256
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Book Description
This book deals with nonlinear boundary value problems for semilinear elliptic equations on unbounded domains with nonlinearities involving the subcritical Sobolev exponent. The variational problems investigated in the book originate in many branches of applied science. A typical example is the nonlinear Schrdinger equation which appears in mathematical modeling phenomena arising in nonlinear optics and plasma physics. Solutions to these problems are found as critical points of variational functionals. The main difficulty in examining the compactness of Palais-Smale sequences arises from the fact that the Sobolev compact embedding theorems are no longer true on unbounded domains. In this book we develop the concentration-compactness principle at infinity, which is used to obtain the relative compactness of minimizing sequences. This tool, combined with some basic methods from the Lusternik-Schnirelman theory of critical points, is to investigate the existence of positive, symmetric and nodal solutions. The book also emphasizes the effect of the graph topology of coefficients on the existence of multiple solutions.
Author: Masao Nagasawa
Publisher: Springer Science & Business Media
ISBN: 3034805608
Category : Mathematics
Languages : en
Pages : 333
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Book Description
Schrödinger Equations and Diffusion Theory addresses the question “What is the Schrödinger equation?” in terms of diffusion processes, and shows that the Schrödinger equation and diffusion equations in duality are equivalent. In turn, Schrödinger’s conjecture of 1931 is solved. The theory of diffusion processes for the Schrödinger equation tells us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrödinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrödinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level. --- This book is a self-contained, very well-organized monograph recommended to researchers and graduate students in the field of probability theory, functional analysis and quantum dynamics. (...) what is written in this book may be regarded as an introduction to the theory of diffusion processes and applications written with the physicists in mind. Interesting topics present themselves as the chapters proceed. (...) this book is an excellent addition to the literature of mathematical sciences with a flavour different from an ordinary textbook in probability theory because of the author’s great contributions in this direction. Readers will certainly enjoy the topics and appreciate the profound mathematical properties of diffusion processes. (Mathematical Reviews)
Author: Rmi Carles
Publisher: World Scientific
ISBN: 9812793135
Category : Mathematics
Languages : en
Pages : 256
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Book Description
These lecture notes review recent results on the high-frequency analysis of nonlinear SchrAdinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear SchrAdinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated. These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided. Sample Chapter(s). Chapter 1: Preliminary Analysis (277 KB). Contents: WKB Analysis: Preliminary Analysis: Weak Nonlinear Geometric Optics; Convergence of Quadratic Observables via Modulated Energy Functionals; Pointwise Description of the Wave Function; Some Instability Phenomena; Caustic Crossing: The Case of Focal Points: Caustic Crossing: Formal Analysis; Focal Point without External Potential; Focal Point in the Presence of an External Potential; Some Ideas for Supercritical Cases. Readership: Pure and applied mathematicians; physicists."
Author:
Publisher:
ISBN:
Category :
Languages : sv
Pages : 17
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Book Description
Author: Peter E. Zhidkov
Publisher: Springer Science & Business Media
ISBN: 3540418334
Category : Mathematics
Languages : en
Pages : 153
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Book Description
The emphasis of this book is on questions typical of nonlinear analysis and qualitative theory of PDEs. The selection of the material is related to the author's attempt to illuminate those particularly interesting questions not yet covered in other monographs though they have been the subject of published articles. One chapter, for example, is devoted to the construction of invariant measures for dynamical systems generated by certain equations and a result from a recent paper on basic properties of a system of eigenfunctions of a stationary problem. Also considered is an application of the method of qualitative theory of ODes to proving the existence of radial solutions of stationary problems and stability of solutions of NLSE nonvanishing as the spatial variable tends to infinity. Finally a recent result on the existence of an infinite sequence of invariant measures for the inegrable KdV equation is presented.
Author: Antonio Ambrosetti
Publisher: Springer Science & Business Media
ISBN: 3764373962
Category : Mathematics
Languages : en
Pages : 187
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Book Description
Several important problems arising in Physics, Di?erential Geometry and other n topics lead to consider semilinear variational elliptic equations on R and a great deal of work has been devoted to their study. From the mathematical point of view, the main interest relies on the fact that the tools of Nonlinear Functional Analysis, based on compactness arguments, in general cannot be used, at least in a straightforward way, and some new techniques have to be developed. n On the other hand, there are several elliptic problems on R which are p- turbative in nature. In some cases there is a natural perturbation parameter, like inthe bifurcationfromthe essentialspectrum orinsingularlyperturbed equations or in the study of semiclassical standing waves for NLS. In some other circ- stances, one studies perturbations either because this is the ?rst step to obtain global results or else because it often provides a correct perspective for further global studies. For these perturbation problems a speci?c approach,that takes advantage of such a perturbative setting, seems the most appropriate. These abstract tools are provided by perturbation methods in critical point theory. Actually, it turns out that such a framework can be used to handle a large variety of equations, usually considered di?erent in nature. Theaimofthismonographistodiscusstheseabstractmethodstogetherwith their applications to several perturbation problems, whose common feature is to n involve semilinear Elliptic Partial Di?erential Equations on R with a variational structure.
Author: Thierry Cazenave
Publisher:
ISBN: 9781470417604
Category :
Languages : en
Pages : 323
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Book Description
Author: Andrzej Szulkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 15
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