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Institutes
Institute for Analysis
Workgroup Nonlinear Partial Differential Equations
Nonlinear Schrödinger equations - stationary aspects

Nonlinear Schrödinger equations - stationary aspects (Winter Semester 2010/11)

Course starts: October 28th


This course is a continuation of the lecture of Prof. Schnaubelt in the previous summer
semester on "Nonlinear Schrödinger Equations - dynamical aspects". However, the contents
and methods are independent.

In this course I will study solitary waves of the nonlinear Schrödinger equation (NLS). They
are solutions of

$-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u \mbox{ in } I\!\!R^n$

I will mainly discuss variational methods for proving existence of solutions. In the case of
constant coefficients we will also study qualitative properties of positive solutions.

Prerequisites

Familiarity with variational methods is helpful. Some of the basic facts of the calculus of
variations will be reviewed. Knowledge in Lebesgue integral, Sobolev spaces, and functional
analytical concepts like weak convergence is essential.

Schedule
Lecture: Thursday 11:30-13:00 1C-04 Begin: 28.10.2010
Lecturers
Lecturer Prof. Dr. Wolfgang Reichel
Office hours: Monday, 11:30-13:00 Before you e-mail: call or come!
Room 3.035 Kollegiengebäude Mathematik (20.30)
Email: Wolfgang.Reichel@kit.edu

Contents

A preliminary list of topics:

  1. Motivation and examples
  2. Constant coefficient case
  3. Asymptotically constant coefficients
  4. Periodic coefficients

References

  1. H. Berestycki, P.L. Lions: Nonlinear scalar fi eld equations I. Arch. Rational Mech. Anal. 82, 313-345 (1983).
  2. B. Gidas, Wei-Ming Ni, L. Nirenberg: Symmetry of positive solutions of nonlinear elliptic equations in Rn. Math. Anal. Appl., Part A, 369-402. Adv. in Math. Suppl. Stud. 7a (1981).
  3. A. Pankov: Periodic nonlinear Schroedinger equation with application to photonic crystals. Milan J. Math. 73, 259-287 (2005).
  4. W. Strauss: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149-162 (1977).
  5. M. Struwe: Variational Methods. Springer Verlag.
  6. C. A. Stuart: A variational approach to bifurcation in LP on an unbounded symmetrical domain. Math. Ann. 263, 51-59 (1983).
  7. M. Willem: Minimax Theorems. Birkhäuser Verlag.