Registration Deadline: January 18th, 2026

Instructor: Professor Dmitry Pelinovsky, McMaster University

Course Date: January 5th - April 2nd, 2026
Mid-Semester Break: February 16th - 20th, 2026
Lecture Time: TBA
Office Hours: TBA

Registration Fee:

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

Capacity Limit: 25

Format: Online

Course Description

The course consists of two parts. In the first part, we shall cover Fourier transform in Lebesgue spaces, Sobolev spaces and embeddings, and well-posedness theory (existence, uniqueness, and continuous dependence) for linear and nonlinear dispersive partial differential equations. In the second part, we shall review applications of linear operators and constrained energy minimization in the stability theory of nonlinear waves in dispersive partial differential equations.

Topics to be covered:

• Fourier Transform in Lebesgue spaces 
• Fourier transform in L1 (Riemann-Lebesgue lemma)
• Fourier transform in L2 (Plancherel's theorem
• Applications of Fourier transform in L2: equipartition of energy and uncertainty principle
• Fourier transform in Lp (interpolation of operators)
• Applications of Fourier transform in Lp: diffusive and dispersive decays
• Oscillatory Fourier integrals (Van-der-Corput lemma) and applications
• Partial differential equations in Sobolev spaces
• Sobolev spaces and L1 embeddings
• Hs: embeddings and Banach algebra
• Local well-posedness of nonlinear Schrodinger equation
• Local well-posedness of nonlinear diffusion and wave equations
• Sobolev and Gagliardo-Nirenberg inequalities
• Global solutions of nonlinear Schrodinger equation
• Stritcharz inequalities and applications
• Stability of nonlinear waves in Sobolev spaces
• Hamiltonian and gradient systems with symmetries
• Stability of energy minimizers
• Instability of saddle points of energy
• Stationary states in Hamiltonian systems with symmetries
• Orbital stability of energy minimizers with symmetries
• Minimizers and saddle points of energy under constraints
• Orbital stability of constrained energy minimizers
• Hamiltonian systems with Casimir functionals in periodic setting
• Orbital stability of constrained energy minimizers with two constraints

Textbooks:

• F. Linares, G. Ponce, “Introduction to Nonlinear Dispersive Equations” (Springer, Universitext) 2015
• A. Geyer, D.E. Pelinovsky, ``Stability of Nonlinear Waves in Hamiltonian Dynamical Systems” (AMS, Mathematical Surveys and Monographs 288) 2025.

Course Expectations:

Lectures will be delivered online. Attendance is required and contributes to the class participation mark. Auditing is allowed with permission of the instructor. 

Three home assignments will be posted with flexible deadlines. The course is completed by students’ final presentations on the subjects related to the topics covered.

Evaluation method: 

The course mark is composed from three components:

• Three written assignments: 45%
• Class participation: 25%
• Final oral presentations: 30%