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". They include the classical wave
equation, Schrödinger equation and KdV type 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?
Prerequisites
Basic properties of
Fourier transform, Lebesgue integral, Lp and Sobolev spaces
will be needed throughout the course, and will be reviewed
in the beginning of the course.
Topics
Here is the
tentative outline. It can be adjusted according to audience background
and interests.
 Introduction and mathematical background:
Fourier transform, Lebesgue integral, Lp and Sobolev spaces.
(1 week)

Linear wave equations: dispersion relations, solution formulas, group velocity, stationary phase,
decay and Strichartz estimates. (3 weeks)
 Nonlinear wave equations: Examples (shallow water, Boussinesq,
KdV, nonlinear Schrödinger and Wave equations),
symmetries, conservation laws,
Hamiltonian formulation, scaling and criticality,
solitary waves. (2 weeks)
 Local in time existence and uniqueness of solutions, continuous
dependence on data, Duhamel formulation, contraction mapping principle. (3 weeks)
 Global behavior: finite time singularity, scattering theory,
orbital stability of solitary waves. (3 weeks)
References

T. Cazenave: Semilinear Schrödinger equations
(2003)
 full glory 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)
 B. Erdogan and N. Tzirakis, Dispersive Partial Differential Equations, Wellposedness and
Applications, 2016
 Scattering and Solion Dynamics (ssd.pdf): old lecture notes by myself
Owncloud webfolder
Assignments and their solutions will be posted in an owncloud
webfolder hosted by the math department. The link of the webfolder
will be given to the audience.
Also in the webfolder you will find my lecture notes and files of
some references.
Evaluation
The evaluation is based on five (5) 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  13:50pm.
Office hours:
Tue Thu 1111:50am, Wed 33:50pm, and by appointments
(Tsai's schedule).