UNIVERSITY OF BRITISH COLUMBIA
MATH 557
SPRING 2010
LINEAR AND NONLINEAR WAVES
Description: Our goal is to understand the qualitative behaviour of solutions of
linear and nonlinear wave equations which are important in areas such as physics,
applied math, and geometry. Our examples will mostly be equations of dispersive
type, such as wave, Klein-Gordan, Schrödinger, and KdV type equations, and wave
and Schrödinger maps (geometric equations). Some of the fundamental questions we
consider: Can solutions be defined locally in time? Do solutions exist for all time, or
do they “blow up” at finite time? If the former, how do they “look” after a long time?
Do they become trivial? Do they settle down to some interesting configuration?
Outline:
1. Linear equations: solution formulas, group velocity, stationary phase, decay and
Strichartz estimates.
2. Nonlinear equations: Hamiltonian structure, conserved quantities, and solitary
waves.
3. Local wellposedness: Existence and uniqueness of time local solutions, continuous dependence on data.
4. Global wellposedness vs. finite time singularity.
5. Scattering theory.
6. Orbital stability of solitary waves.
7. Introduction to some geometric equations.
References:
1. Thierry Cazenave: Semilinear Schrödinger equations, Courant Lecture Notes in
Mathematics 10. 2003.
2. Jalal Shatah and Michael Struwe: Geometric wave equations, Courant Lecture
Notes in Mathematics 2. 1998.
3. Walter A. Strauss: Nonlinear wave equations, CBMS Regional Conference Series
in Mathematics, 73. American Mathematical Society, Providence, RI, 1989.
Prerequisites: Basic properties of Sobolev spaces and Fourier transform will be
needed throughout the course, and will be reviewed in the first lecture.
Course homepage: http://www.math.ubc.ca/∼ttsai/math557/
Evaluation: Biweekly assignments.
Instructor: Tai-Peng Tsai, Math 109, phone 604-822-2591, ttsai@math.ubc.ca.