Semi-classical analysis for nonlinear Schrödinger equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hackensack, NJ [u.a.]
World Scientific
2008
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 243 S. Ill. 24 cm |
| ISBN: | 9812793127 9789812793126 |
Staff View
MARC
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| 100 | 1 | |a Carles, Rémi |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Semi-classical analysis for nonlinear Schrödinger equations |c Rémi Carles |
| 264 | 1 | |a Hackensack, NJ [u.a.] |b World Scientific |c 2008 | |
| 300 | |a XI, 243 S. |b Ill. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Schrödinger equation | |
| 650 | 4 | |a Nonlinear theories | |
| 650 | 0 | 7 | |a Nichtlineare Schrödinger-Gleichung |0 (DE-588)4278277-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a WKB-Methode |0 (DE-588)4190133-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nichtlineare Schrödinger-Gleichung |0 (DE-588)4278277-6 |D s |
| 689 | 0 | 1 | |a WKB-Methode |0 (DE-588)4190133-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017143145&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017143145 | ||
Record in the Search Index
| _version_ | 1804138654095900672 |
|---|---|
| adam_text | Contents
Preface
v
General
Notations
vii
WKB
Analysis
1
1.
Preliminary Analysis
3
1-1
General presentation
..................... 7
1.2
Formal derivation of the equations
............. 9
1.3
Linear
Schrödinger
equation
................. 13
1.3.1
The eikonal equation
................ 13
1.3.2
The transport equations
.............. 18
1-4
Basic results in the nonlinear case
............. 23
1.4.1
Formal properties
.................. 24
1.4.2
Strong solutions
................... 25
1.4.3
Mild solutions
.................... 27
1.4.4
Weak solutions
.................... 29
2.
Weak Nonlinear Geometric Optics
31
2.1
Precised existence results
.................. 32
2.2
Leading order asymptotic analysis
............. 35
2.3
Interpretation
......................... 36
2.4
Higher order asymptotic analysis
.............. 38
2.5
An application: Cauchy problem in Soboiev spaces for non¬
linear
Schrödinger
equations with potential
........ 39
x
Semi-Classical Analysis for Nonlinear
Schrödinger
Equations
3.
Convergence of Quadratic
Observables
via Modulated
Energy Punctionals
45
3.1
Presentation
.......................... 45
3.2
Formal computation
..................... 48
3.3
Justification
.......................... 50
3.3.1
The Cauchy problem for
(3.3)........... 50
3.3.2
Rigorous estimates for the modulated energy
... 51
3.4
Convergence of quadratic
observables
............ 55
4.
Pointwise Description of the Wave Function
59
4.1
Several possible approaches
................. 60
4.2
E. Grenier s idea
....................... 61
4.2.1
Without external potential
............. 62
4.2.2
With an external potential
............. 70
4.2.3
The case 0<q<1
................. 75
4.3
Higher order homogeneous nonlinearities
.......... 78
4.4
On conservation laws
..................... 87
4.5
Focusing nonlinearities
.................... 88
5.
Some Instability Phenomena
91
5.1
Ill-posedness for nonlinear
Schrödinger
equations
..... 91
5.2
Loss of regularity for nonlinear
Schrödinger
equations
. . 97
5.3
Instability at the semi-classical level
............ 100
Caustic Crossing: The Case of Focal Points
109
6.
Caustic Crossing: Formal Analysis 111
6.1
Presentation
..........................
Ill
6.2
The idea of J. Hunter and J. Keller
............. 116
6.3
The case of a focal point
................... 120
6.4
The case of a cusp
...................... 121
7.
Focal Point without External Potential
127
7.1
Presentation
.......................... 127
7.2
Linear propagation, linear caustic
.............. 132
7.3
Nonlinear propagation, linear caustic
............ 140
7.4
Linear propagation, nonlinear caustic
............ 148
Contents xi
7.4.1 Elements
of scattering theory for the nonlinear
Schrödinger
equation
................ 149
7.4.2
Main result
...................... 151
7.4.3
On the propagation of Wigner measures
..... 155
7.5
Nonlinear propagation, nonlinear caustic
.......... 159
7.6
Why initial quadratic oscillations?
............. 167
7.6.1
Notion of linearizability
............... 167
7.6.2
The ^-supercritical case:
σ
>
2/n
........ 173
7.6.3
The ^-critical case:
σ
=
2/n
........... 178
7.6.4
Nonlinear superposition
............... 181
7.7
Focusing on a line
...................... 182
8.
Focal Point in the Presence of an External Potential
185
8.1 Isotropie
harmonic potential
................. 185
8.2
General quadratic potentials
................. 198
8.3
About general subquadratic potentials
........... 208
9.
Some Ideas for Supercritical Cases
213
9.1
Cascade of phase shifts
.................... 216
9.1.1
A formal computation
................ 217
9.1.2
A rigorous computation
............... 222
9.1.3
Why do the results disagree?
............ 226
9.2
And beyond?
......................... 229
Bibliography
233
Index
243
|
| any_adam_object | 1 |
| author | Carles, Rémi |
| author_facet | Carles, Rémi |
| author_role | aut |
| author_sort | Carles, Rémi |
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| callnumber-raw | QC174.26.W28 |
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| callnumber-sort | QC 3174.26 W28 |
| callnumber-subject | QC - Physics |
| classification_rvk | SK 540 SK 920 |
| ctrlnum | (OCoLC)198759348 (DE-599)BVBBV035338822 |
| dewey-full | 530.12/4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.12/4 |
| dewey-search | 530.12/4 |
| dewey-sort | 3530.12 14 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV035338822 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:31:37Z |
| institution | BVB |
| isbn | 9812793127 9789812793126 |
| language | English |
| lccn | 2008297932 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017143145 |
| oclc_num | 198759348 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 DE-384 |
| owner_facet | DE-355 DE-BY-UBR DE-11 DE-384 |
| physical | XI, 243 S. Ill. 24 cm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Carles, Rémi Verfasser aut Semi-classical analysis for nonlinear Schrödinger equations Rémi Carles Hackensack, NJ [u.a.] World Scientific 2008 XI, 243 S. Ill. 24 cm txt rdacontent n rdamedia nc rdacarrier Schrödinger equation Nonlinear theories Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd rswk-swf WKB-Methode (DE-588)4190133-2 gnd rswk-swf Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 s WKB-Methode (DE-588)4190133-2 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017143145&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Carles, Rémi Semi-classical analysis for nonlinear Schrödinger equations Schrödinger equation Nonlinear theories Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd WKB-Methode (DE-588)4190133-2 gnd |
| subject_GND | (DE-588)4278277-6 (DE-588)4190133-2 |
| title | Semi-classical analysis for nonlinear Schrödinger equations |
| title_auth | Semi-classical analysis for nonlinear Schrödinger equations |
| title_exact_search | Semi-classical analysis for nonlinear Schrödinger equations |
| title_full | Semi-classical analysis for nonlinear Schrödinger equations Rémi Carles |
| title_fullStr | Semi-classical analysis for nonlinear Schrödinger equations Rémi Carles |
| title_full_unstemmed | Semi-classical analysis for nonlinear Schrödinger equations Rémi Carles |
| title_short | Semi-classical analysis for nonlinear Schrödinger equations |
| title_sort | semi classical analysis for nonlinear schrodinger equations |
| topic | Schrödinger equation Nonlinear theories Nichtlineare Schrödinger-Gleichung (DE-588)4278277-6 gnd WKB-Methode (DE-588)4190133-2 gnd |
| topic_facet | Schrödinger equation Nonlinear theories Nichtlineare Schrödinger-Gleichung WKB-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017143145&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT carlesremi semiclassicalanalysisfornonlinearschrodingerequations |