本书介绍了非线性色散方程理论的最新进展，主要是非线性薛定谔方程。本书适合对偏微分方程及其相关领域感兴趣的研究生和数学研究人员阅读参考。 This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schr?dinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented. Several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research related to dispersive equations and Hamiltonian PDEs.

前辅文
Introduction and summary
I. An overview of results on the Cauchy problem for NLS
  1. Equations
  2. Wellposedness of the Cauchy problen
  3. Scattering results
  4. Estimates on the linear group
  5. Solving the Cauchy problem
  6. Derivative nonlinear Schrödinger equations
II. Further comments
  1. Construction of blowup solutions for conformal NLS from the ground state
  2. Behaviour of higher Sobolev norms
  3. Fourier restriction theory beyond L2
  4. L2-concentration phenomenon
  5. The Schrödinger maximal function
  6. Derivative nonlinear Schrodinger equation
III. 3D H1-critical defocusing NLS
  1. Consider 3D NLS
  2. Fix a time interval I=[0,T]
  3. Sketch of the argument
  4. A concentration property
  5. A version of Morawetz inequality
  6. Construction of an appropriate time interval
  7. Details on the perturbative analysis
  8. A variant of the method
IV. Global wellposedness below energy norm
  1. Description of the method
  2. The example of the NLW
  3. The case of the nonlinear Schrödinger equation
  4. Symplectic capacities and symplectic Hilbert spaces
  5. Global wellposedness of the NLW (4.41)
V. Nonlinear Schrödinger equation with periodic boundary conditions
  1. Introduction
  2. Results on the Cauchy problem
  3. Periodic Strichartz inequalities
  4. Sketch of proof of Theorems 2.1 and 2.7
  5. Invariant Gibbs measures (1D)
  6. Invarinat Gibbs measures (D>1)
  7. Invariant Gibbs measures (unbounded domains)
  8. Quasi-periodic solutions
Appendix 1. Growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential
Appendix 2. Zakharov systems
References
Index