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Semi-classical Analysis For Nonlinear Schrodinger Equations: Wkb Analysis, Focal Points, Coherent States (Second Edition)

Author: Remi Carles

Publisher: World Scientific

Published: 2020-10-05

Total Pages: 367

ISBN-13: 9811227926

The second edition of this book consists of three parts. The first one is dedicated to the WKB methods and the semi-classical limit before the formation of caustics. The second part treats the semi-classical limit in the presence of caustics, in the special geometric case where the caustic is reduced to a point (or to several isolated points). The third part is new in this edition, and addresses the nonlinear propagation of coherent states. The three parts are essentially independent.Compared with the first edition, the first part is enriched by a new section on multiphase expansions in the case of weakly nonlinear geometric optics, and an application related to this study, concerning instability results for nonlinear Schrödinger equations in negative order Sobolev spaces.The third part is an overview of results concerning nonlinear effects in the propagation of coherent states, in the case of a power nonlinearity, and in the richer case of Hartree-like nonlinearities. It includes explicit formulas of an independent interest, such as generalized Mehler's formula, generalized lens transform.

Semi-classical Analysis For Nonlinear Schrodinger Equations

Author: Remi Carles

Publisher: World Scientific

Published: 2008-03-04

Total Pages: 256

ISBN-13: 9814471747

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These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger 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 Schrödinger 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.

Semi-classical Analysis for Nonlinear Schrödinger Equations

Author: Rémi Carles

Publisher: World Scientific Publishing Company Incorporated

Published: 2008

Total Pages: 243

ISBN-13: 9812793127

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Semi-classical Analysis for Nonlinear Schrödinger Equations

Author: Rémi Carles

Publisher: World Scientific Publishing Company

Published: 2020-09-29

Total Pages: 0

ISBN-13: 9789811227905

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The second edition of this book consists of three parts. The first one is dedicated to the WKB methods and the semi-classical limit before the formation of caustics. The second part treats the semi-classical limit in the presence of caustics, in the special geometric case where the caustic is reduced to a point (or to several isolated points). The third part is new in this edition, and addresses the nonlinear propagation of coherent states. The three parts are essentially independent. Compared with the first edition, the first part is enriched by a new section on multiphase expansions in the case of weakly nonlinear geometric optics, and an application related to this study, concerning instability results for nonlinear Schrdinger equations in negative order Sobolev spaces. The third part is an overview of results concerning nonlinear effects in the propagation of coherent states, in the case of a power nonlinearity, and in the richer case of Hartree-like nonlinearities. It includes explicit formulas of an independent interest, such as generalized Mehler's formula, generalized lens transform.

Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation (AM-154)

Author: Spyridon Kamvissis

Publisher: Princeton University Press

Published: 2003-09-07

Total Pages: 281

ISBN-13: 069111482X

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Providing an asymptotic analysis via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic regime, this text exploits complete integrability to establish pointwise asymptotics for this problem's solution.

Semiclassical Analysis for Diffusions and Stochastic Processes

Author: Vassili N. Kolokoltsov

Publisher: Springer

Published: 2007-12-03

Total Pages: 360

ISBN-13: 3540465871

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The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.

Wigner Measure and Semiclassical Limits of Nonlinear Schrödinger Equations

Author: Ping Zhang

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 197

ISBN-13: 9780821847015

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This book is based on a course entitled ``Wigner measures and semiclassical limits of nonlinear Schrodinger equations,'' which the author taught at the Courant Institute of Mathematical Sciences at New York University in the spring of 2007. The author's main purpose is to apply the theory of semiclassical pseudodifferential operators to the study of various high-frequency limits of equations from quantum mechanics. In particular, the focus of attention is on Wigner measure and recent progress on how to use it as a tool to study various problems arising from semiclassical limits of Schrodinger-type equations. At the end of each chapter, the reader will find references and remarks about recent progress on related problems. The book is self-contained and is suitable for an advanced graduate course on the topic.

Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation

Author: Spyridon Kamvissis

Publisher:

Published: 2002

Total Pages: 367

ISBN-13:

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Semilinear Schrodinger Equations

Author: Thierry Cazenave

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 346

ISBN-13: 0821833995

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The nonlinear Schrodinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. This book presents various mathematical aspects of the nonlinear Schrodinger equation. It studies both problems of local nature and problems of global nature.

Wigner Measure and Semiclassical Limits of Nonlinear Schrödinger Equations

Author: Ping Zhang

Publisher: American Mathematical Soc.

Published:

Total Pages: 212

ISBN-13: 9780821883563

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"This book is based on a course entitled "Wigner measures and semiclassical limits of nonlinear Schrodinger equations," which the author taught at the Courant Institute of Mathematical Sciences at New York University in the spring of 2007. The author's main purpose is to apply the theory of semiclassical pseudodifferential operators to the study of various high-frequency limits of equations from quantum mechanics. In particular, the focus of attention is on Wigner measure and recent progress on how to use it as a tool to study various problems arising from semiclassical limits of Schrodinger-type equations." "At the end of each chapter, the reader will find references and remarks about recent progress on related problems. The book is self-contained and is suitable for an advanced graduate course on the topic."--BOOK JACKET.