Registration Deadline: TBA

Instructor: Professor Dmitry Pelinovsky, McMaster University

Course Date: TBA
Mid-Semester Break: TBA
Lecture Time: TBA
Office Hours: TBA

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD$500

Capacity Limit: TBA

Format: TBA

Course Description

Nonlinear dispersive wave equations arise as reduced mathematical models from governing equations of mathematical physics. These reduced models combine the leading-order balance between nonlinear and dispersive effects present in wave propagation. Questions concerning the well-posedness of the time evolution, the existence and stability of coherent structures such as traveling, standing, or time-periodic solutions, and the long-time dynamics near these coherent structures are of paramount importance for applications.

This graduate course is intended for students in analysis and applied mathematics. It will include two main parts:

1. well-posedness of nonlinear partial differential equations in Sobolev spaces,
2. stability of nonlinear waves in Hamiltonian systems with symmetries.

The first part is based on the book of F. Linares and G. Ponce, "Introduction to Nonlinear Dispersive Equations" (Springer, Universitext), 2015. It will cover introduction to Sobolev spaces: embeddings, Banach algebra, Sobolev and Gagliardo-Nirenberg inequalities, Stritcharz inqualities, and applications to local and global well-posedness of nonlinear PDEs.

The second part is based on the book of A. Geyer and D. E. Pelinovsky "Stability of nonlinear waves in Hamiltonian dynamical systems", AMS Monograph Series (2025, to be published). It will cover conserved quantities and symmetries, stability of energy minimizers, instability of saddle points, orbital stability of constrained minimizers, Casmir functionals, and applications to stability of spatially periodic and spatial decaying solutions of nonlinear PDEs.