Introduction to nonlinear dispersive equations
著者
書誌事項
Introduction to nonlinear dispersive equations
（Universitext）
Springer, c2015
2nd ed
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注記
Includes bibliographical references (p. 279-298) and index
内容説明・目次
内容説明
This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg-de Vries equation and the nonlinear Schroedinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schroedinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schroedinger equation and the generalized Korteweg-de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schroedinger equation, taking the reader to the forefront of recent research.
The second edition of Introduction to Nonlinear Dispersive Equations builds upon the success of the first edition by the addition of updated material on the main topics, an expanded bibliography, and new exercises. Assuming only basic knowledge of complex analysis and integration theory, this book will enable graduate students and researchers to enter this actively developing field.
目次
1. The Fourier Transform.- 2. Interpolation of Operators.- 3. Sobolev Spaces and Pseudo-Differential Operators.- 4. The Linear Schrodinger Equation.- 5. The Non-Linear Schrodinger Equation.- 6. Asymptotic Behavior for NLS Equation.- 7. Korteweg-de Vries Equation.- 8. Asymptotic Behavior for k-gKdV Equations.- 9. Other Nonlinear Dispersive Models.- 10. General Quasilinear Schrodinger Equation.- Proof of Theorem 2.8.- Proof of Lemma 4.2.- References.- Index.
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