Contents
Preface
WKB Analysis
1. Preliminary Analysis
1.1 General presentation
1.2 Formal derivation of the equations
1.3 Linear Schrödinger equation
1.3.1 The eikonal equation
1.3.2 The transport equations
1.4 Basic results in the nonlinear case
1.4.1 Formal properties
1.4.2 Strong solutions
1.4.3 Mild solutions
1.4.4 Weak solutions
2. Weakly Nonlinear Geometric Optics
2.1 Improved existence results
2.2 Leading order asymptotic analysis
2.3 Interpretation
2.4 Higher order asymptotic analysis
2.5 An application: Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential
2.6 Multiphase expansions
2.6.1 Resonant phases
2.6.2 The set of transport equations
2.6.3 Functional setting
2.6.4 Error estimate
3. Convergence of Quadratic Observables via Modulated Energy Functionals
3.1 Presentation
3.2 Formal computation
3.3 Justification
3.3.1 The Cauchy problem for (3.3)
3.3.2 Rigorous estimates for the modulated energy
3.4 Convergence of quadratic observables
4. Pointwise Description of the Wave Function
4.1 Several possible approaches
4.2 E. Grenier’s idea
4.2.1 Without external potential
4.2.2 With an external potential
4.2.3 The case 0 < α < 1
4.3 Higher order homogeneous nonlinearities
4.4 On conservation laws
4.5 Focusing nonlinearities
5. Some Instability Phenomena
5.1 Ill-posedness for nonlinear Schrödinger equations
5.2 Loss of regularity for nonlinear Schrödinger equations
5.3 Instability at the semi-classical level
5.4 Negative order and infinite loss of regularity
Caustic Crossing: the Case of Focal Points
6. Caustic Crossing: Formal Analysis
6.1 Presentation
6.2 The idea of J. Hunter and J. Keller
6.3 The case of a focal point
6.4 The case of a cusp
7. Focal Point without External Potential
7.1 Presentation
7.2 Linear propagation, linear caustic
7.3 Nonlinear propagation, linear caustic
7.4 Linear propagation, nonlinear caustic
7.4.1 Elements of scattering theory for the nonlinear Schrödinger equation
7.4.2 Main result
7.4.3 On the propagation of Wigner measures
7.5 Nonlinear propagation, nonlinear caustic
7.6 Why initial quadratic oscillations?
7.6.1 Notion of linearizability
7.6.2 The L2-supercritical case: σ > 2/d
7.6.3 The L2-critical case: σ = 2/d
7.6.4 Nonlinear superposition
7.7 Focusing on a line
8. Focal Point in the Presence of an External Potential
8.1 Isotropic harmonic potential
8.2 General quadratic potentials
8.3 About general subquadratic potentials
9. Some Ideas for Supercritical Cases
9.1 Cascade of phase shifts
9.1.1 A formal computation
9.1.2 A rigorous computation
9.1.3 Why do the results disagree?
9.2 And beyond?
Coherent States
10. The Linear Case
10.1 On the uncertainty principle
10.2 Propagation of coherent states
10.3 The Gaussian case
10.4 Error estimate and Ehrenfest time
10.5 Optimality of Lemma 10.4
10.6 Wave packet transform
11. Nonlinear Coherent States: Main Tools
11.1 Notion of criticality
11.2 NLS with a time-dependent potential
11.2.1 Some algebraic miracles
11.2.2 Strichartz estimates
11.2.3 Growth of Sobolev norms and momenta
12. Power-like Nonlinearity
12.1 Subcritical case
12.2 Critical case
12.3 A nonlinear superposition principle in the critical case
12.4 Comments and further results
13. Hartree-Type Nonlinearity
13.1 Critical case
13.2 Supercritical case
13.3 The case of two initial coherent states
13.3.1 General considerations
13.3.2 The approximate solution
13.3.3 Estimating the cross terms
13.3.4 Error estimate in the critical case
13.3.5 Supercritical case
Bibliography
Index