Semi-classical analysis for nonlinear Schrödinger equations: WKB analysis, focal points, coherent states

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