Semi-Classical Analysis For Nonlinear Schrodinger Equations

Rémi Carles9812793127, 9789812793126, 9789812793133

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.
Contents: WKB Analysis: Preliminary Analysis; Weak Nonlinear Geometric Optics; Convergence of Quadratic Observables via Modulated Energy Functionals; Pointwise Description of the Wave Function; Some Instability Phenomena; Caustic Crossing: The Case of Focal Points: Caustic Crossing: Formal Analysis; Focal Point without External Potential; Focal Point in the Presence of an External Potential; Some Ideas for Supercritical Cases.

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Table of contents :
Contents……Page 10
Preface……Page 6
General Notations……Page 8
WKB Analysis……Page 14
1. Preliminary Analysis……Page 16
1.1 General presentation……Page 20
1.2 Formal derivation of the equations……Page 22
1.3.1 The eikonal equation……Page 26
1.3.2 The transport equations……Page 31
1.4 Basic results in the nonlinear case……Page 36
1.4.1 Formal properties……Page 37
1.4.2 Strong solutions……Page 38
1.4.3 Mild solutions……Page 40
1.4.4 Weak solutions……Page 42
2. Weak Nonlinear Geometric Optics……Page 44
2.1 Precised existence results……Page 45
2.2 Leading order asymptotic analysis……Page 48
2.3 Interpretation……Page 49
2.4 Higher order asymptotic analysis……Page 51
2.5 An application: Cauchy problem in Sobolev spaces for nonlinear Schr odinger equations with potential……Page 52
3.1 Presentation……Page 58
3.2 Formal computation……Page 61
3.3.1 The Cauchy problem for (3.3)……Page 63
3.3.2 Rigorous estimates for the modulated energy……Page 64
3.4 Convergence of quadratic observables……Page 68
4. Pointwise Description of the Wave Function……Page 72
4.1 Several possible approaches……Page 73
4.2 E. Grenier’s idea……Page 74
4.2.1 Without external potential……Page 75
4.2.2 With an external potential……Page 83
4.2.3 The case 0 < < 1……Page 88
4.3 Higher order homogeneous nonlinearities……Page 91
4.4 On conservation laws……Page 100
4.5 Focusing nonlinearities……Page 101
5.1 Ill-posedness for nonlinear Schrodinger equations……Page 104
5.2 Loss of regularity for nonlinear Schrodinger equations……Page 110
5.3 Instability at the semi-classical level……Page 113
Caustic Crossing: The Case of Focal Points……Page 122
6.1 Presentation……Page 124
6.2 The idea of J. Hunter and J. Keller……Page 129
6.3 The case of a focal point……Page 133
6.4 The case of a cusp……Page 134
7.1 Presentation……Page 140
7.2 Linear propagation, linear caustic……Page 145
7.3 Nonlinear propagation, linear caustic……Page 153
7.4 Linear propagation, nonlinear caustic……Page 161
7.4.1 Elements of scattering theory for the nonlinear Schrodinger equation……Page 162
7.4.2 Main result……Page 164
7.4.3 On the propagation of Wigner measures……Page 168
7.5 Nonlinear propagation, nonlinear caustic……Page 172
7.6.1 Notion of linearizability……Page 180
7.6.2 The L2-supercritical case: > 2/n……Page 186
7.6.3 The L2-critical case: = 2/n……Page 191
7.6.4 Nonlinear superposition……Page 194
7.7 Focusing on a line……Page 195
8.1 Isotropic harmonic potential……Page 198
8.2 General quadratic potentials……Page 211
8.3 About general subquadratic potentials……Page 221
9. Some Ideas for Supercritical Cases……Page 226
9.1 Cascade of phase shifts……Page 229
9.1.1 A formal computation……Page 230
9.1.2 A rigorous computation……Page 235
9.1.3 Why do the results disagree?……Page 239
9.2 And beyond?……Page 242
Bibliography……Page 246
Index……Page 256