Universität Wien

250167 VO Select. Top. in PDE (Schröd. and Klein-Gordon Eq.) (2005W)

Dispersion and Attractors for Linear and Nonlinear Schrödinger and Klein-Gordon Equations

0.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Vorbesprechung: October 6 and 7, 10.00 - 12.00 and 16.00 - 18.00, Nordbergstrasse 15, room A 109.

Details

Language: German

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 12.10. 17:00 - 19:00 Seminarraum
  • Friday 14.10. 14:00 - 16:00 Seminarraum
  • Wednesday 19.10. 17:00 - 19:00 Seminarraum
  • Friday 21.10. 14:00 - 16:00 Seminarraum
  • Friday 28.10. 14:00 - 16:00 Seminarraum
  • Friday 04.11. 14:00 - 16:00 Seminarraum
  • Wednesday 09.11. 17:00 - 19:00 Seminarraum
  • Friday 11.11. 14:00 - 16:00 Seminarraum
  • Wednesday 16.11. 17:00 - 19:00 Seminarraum
  • Friday 18.11. 14:00 - 16:00 Seminarraum
  • Wednesday 23.11. 17:00 - 19:00 Seminarraum
  • Friday 25.11. 14:00 - 16:00 Seminarraum
  • Wednesday 30.11. 17:00 - 19:00 Seminarraum
  • Friday 02.12. 14:00 - 16:00 Seminarraum
  • Wednesday 07.12. 17:00 - 19:00 Seminarraum
  • Friday 09.12. 14:00 - 16:00 Seminarraum
  • Wednesday 14.12. 17:00 - 19:00 Seminarraum
  • Friday 16.12. 14:00 - 16:00 Seminarraum
  • Wednesday 11.01. 17:00 - 19:00 Seminarraum
  • Friday 13.01. 14:00 - 16:00 Seminarraum
  • Wednesday 18.01. 17:00 - 19:00 Seminarraum
  • Friday 20.01. 14:00 - 16:00 Seminarraum
  • Wednesday 25.01. 17:00 - 19:00 Seminarraum
  • Friday 27.01. 14:00 - 16:00 Seminarraum

Information

Aims, contents and method of the course

I. Asymptotic Methods:
i) Method of stationary phase.
ii) WKB approximation: the Hamilton-Jacobi equation, the rays and caustics, the transport equations, the Stokes phenomenon.
iii) Applications to the acoustic, Schrödinger and Klein-Gordon equations:
local energy decay, geometric optics, wave packets, group velocity, energy propagation.
Ref: [1-5].
II. Global Attractors of Hyperbolic Nonlinear PDEs
i) 1D Klein-Gordon field coupled to a nonlinear oscillator:
a) splitting onto dispersive and bound components,
b) local energy decay of the dispersive component.
c) quasimeasures and multiplicators,
d) compactness of the trajectory,
e) nonlinear spectral analysis of omega-limiting trajectories:
limiting equation and the Titchmarsh Convolution Theorem.
Ref: [6], [7] (Chapter 3).

ii) 3D wave equation coupled to a relativistic particle in presence of external confining potential.

a) Li\'enard-Wiechert integral representation,
b) energy flow to infinity,
c) convolution representation and the Wiener Tauberian Theorem: radiative damping,
d) omega-limiting states.
Ref: [7] (pp 37-41), [8].

Assessment and permitted materials

Minimum requirements and assessment criteria

I. Asymptotic Methods:

i) Method of stationary phase.

ii) WKB approximation: the Hamilton-Jacobi equation, the rays and
caustics,
the transport equations, the Stokes phenomenon.

iii) Applications to the acoustic, Schr\"odinger and Klein-Gordon
equations:
local energy decay, geometric optics, wave packets, group velocity,
energy propagation.

Ref: [1-5].

II. Global Attractors of Hyperbolic Nonlinear PDEs

i) 1D Klein-Gordon field coupled to a nonlinear oscillator:

a) splitting onto dispersive and bound components,

b) local energy decay of the dispersive component.

c) quasimeasures and multiplicators,

d) compactness of the trajectory,

e) nonlinear spectral analysis of omega-limiting trajectories:
limiting equation and the Titchmarsh Convolution Theorem.

Ref: [6], [7] (Chapter 3).

ii) 3D wave equation coupled to a relativistic particle in presence
of external confining potential.

a) Li\'enard-Wiechert integral representation,

b) energy flow to infinity,

c) convolution representation and the Wiener Tauberian Theorem:
radiative damping,

d) omega-limiting states.

Ref: [7] (pp 37-41), [8].

Examination topics

Method of stationary phase, the Hamilton-Jacobi equation, the Wiener-Paley theory, Quasimeasures, the Titchmarsh Convolution Theorem, Energy propagation, the Wiener Tauberian Theorem.

Reading list

[1] M.V.Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations, Springer, Berlin, 1993.

[2] M.Fedoryuk, Partial Differential Equations V. Asymptotic Methods for Partial Differential Equations,
Encyclopaedia of Mathematical Sciences Vol. 34, Springer, Berlin, 1999.

[3] Komech, Linear Partial Differential Equations with Constant Coefficients, p.127-260 in: Yu.V.Egorov, A.I.Komech, M.A.Shubin, Elements of the Modern Theory of Partial Differential Equations, Springer, Berlin, 1999.

[4] I.M.Gel'fand, G.E.Shilov, Generalized Functions. Vol. I:
Properties and Operations, Academic Press, New York, 1964.

[5] A.Komech, Lectures on Quantum Mechanics (nonlinear PDE point of view), preprint of Max-Planck Institute for Mathematics in the Sciences, No. 25/2005, 2005.
http://www.mis.mpg.de/preprints/ln/lecturenote-2505-abstr.html,
http://arxiv.org/abs/math-ph/0505059.

[6] A.Komech, On attractor of a singular nonlinear U(1)-invariant Klein-Gordon equation, p. 599-611 in: Proceedings of the 3rd ISAAC Congress, Freie Universität Berlin, Berlin, 2003.

[7] A.Komech, On Global Attractors of Hamilton Nonlinear Wave Equations, preprint of Max-Planck Institute for Mathematics in the Sciences, No. 24/2005, 2005.
http://www.mis.mpg.de/preprints/ln/lecturenote-2405-abstr.html

[8] A.Komech, H.Spohn, M.Kunze, Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Diff. Eqns., 22 (1997), no. 1/2, 307-335.

Association in the course directory

Last modified: Mo 07.09.2020 15:40