Course Description
Wave propagation subject to nonlinear effects occurs in physical
systems as diverse as the atmosphere, ocean surface waves, lasers,
magnets, and quantum particles, and such systems are modelled by PDE
known collectively as "nonlinear wave equations". Qualitatively,
behaviour in these systems is often characterized by a competition
between dispersive (spreadingout, wavelike) and nonlinear
(concentrating, enhancing) effects. Mathematically, the study of
these PDE has become one of the dominant research areas in analysis,
combining as it does functional and harmonic analysis, and
concepts from mathematical physics, Hamiltonian systems, and ODE
theory.
This course develops the mathematical tools used to address the
most important questions concerning nonlinear wave equations as
models of physical phenomena:
Can solutions be uniquely defined locally
in time? Do solutions exist for all time, or do they "blow up" in
finite time? If a solution exists for all time, how does it look like after a long time?
Does it become trivial? Does it settle down to some interesting
configuration?
Topics
Here is the
tentative outline. It can be adjusted according to audience background
and interests.
 Mathematical background: Lebesgue integral, Lp and Sobolev spaces,
Fourier transform.

Linear wave equations: dispersion relations, solution formulas, group velocity, stationary phase,
decay and Strichartz estimates.
 Nonlinear wave equations: Examples (shallow water, Boussinesq,
KdV, nonlinear Schrödinger and Wave equations),
symmetries, conservation laws,
Hamiltonian formulation, scaling and criticality,
solitary waves.
 Local in time existence and uniqueness of solutions, continuous
dependence on data, Duhamel formulation, contraction mapping principle.
 Global behavior: finite time singularity, scattering theory,
orbital stability of solitary waves.
References

T. Cazenave: Semilinear Schrödinger equations
(2003)
 full gory details of analysis of NLS

T. Tao, Nonlinear Dispersive Equations (2006)
 a thorough introduction to the modern mathematical theory of
nonlinear waves. Chapters 13 are most relevant to us.

G. Whitham, Linear and Nonlinear Waves (1974)
 a classical applied text. Part II covers many of the basic
notions for linear and nonlinear dipsersive PDE.
 W. A. Strauss: Nonlinear wave equations (1989)
 a short, efficient overview of the mathematical stateoftheart at the time.

C. Sulem, P.L. Sulem, The Nonlinear Schroedinger Equation (1999)

G. Fibich, The Nonlinear Schroedinger Equation (2015)
 F. Linares and G. Ponce,
Introduction to Nonlinear Dispersive Equations (2015)
 ssd: old lecture notes by myself
Files of some references will be available in a public
owncloud folder, whose link will be given.
Prerequisites
Basic properties of Sobolev spaces and Fourier
transform will be needed throughout the course, and will be reviewed
in the beginning of the course.
Evaluation
The evaluation is based on homework assignments and class participation.
Instructor and lectures
Instructor:
Dr. TaiPeng Tsai, Math building
room 109, phone 6048222591, ttsai at math.ubc.ca.
Lectures:
MWF, 13:00  13:50, Math Annex MATX 1102
Office hours:
TBA, and by appointment
(Tsai's schedule).