Coherent Structures, Pattern Formation and Solitons
DEN 210, MWF 10:30-11:20pm
Prereqs: Amath 569 or Instructor Permission
Instructor: Bernard Deconinck
Office Hours: M4-5pm, T9-11am
Methods for integrable and near-integrable nonlinear partial differential equations such as the Korteweg-de Vries equation and the Nonlinear Schrodinger equation; symmetry reductions and solitons; soliton interactions; infinite-dimensional Hamiltonian systems; Lax pairs and inverse scattering; Painleve analysis.
There is no required textbook for this course as I don't think a suitable one exists. I also didn't formally recommend any books for this course, so the bookstore doesn't have anything on the shelves for Amath573.
My typed-up lecture notes are available.
We're using Piazza for the class message board.
Syllabus (subject to relatively minor changes)
- Introduction. Context. Some history. Reference materials: The FPU problem, Poincare's work on King Oscar II's problem.
- Quick overview of Linear dispersive partial differential equations using Fourier transforms.
- Handwaving derivation of the Korteweg-de Vries equation and the Nonlinear Schrodinger equation. Reference materials:About John Scott Russell (Links to an external site.), John Scott Russell's original soliton recreated (Links to an external site.).
- Exact solutions of partial differential equations as obtained through symmetry reduction. Simplest case: stationary solutions. Solitary waves and solitons.
- Infinite-dimensional Hamiltonian and Lagrangian systems. Conserved quantities. Noether's theorem. Poisson brackets. Liouville integrability. If time permits: Bihamiltonian structures.
- Conserved quantities. Infinite number of conserved quantities for KdV. The Miura transform, Modified KdV. the KdV hierarchy. Integrable equations, hierarchies.
- Two soliton solutions and their interactions. Brief mentioning of Hirota's method and Backlund transformations.
- Lax Pairs. Principles of the inverse scattering method. Trace formulae.
- Testing for integrability I: prolongation methods.
- Testing for integrability II: Painleve methods.
As time permits: extra topics from (a) periodic solutions, (b) higher-dimensional problems, (c) lattice problems, (d) Whitham modulation theory, etc.
In addition to homework, each of you will present their findings on a class-related project. We will set some days outside of regular class time aside for the presentation of these projects. You are expected to be present for the presentations of your colleagues. Your course grade will be calculated by weighing your homework and project work in the proportions 60% and 40%, respectively.
Homework sets are assigned biweekly. Homework is due at the beginning of class on its due date. Late homework is not accepted. Every homework set you hand in should have a header containing your name, student number, due date, course, and the homework number as a title. Your homework should be neat and readable. Your homework score may reflect the presentation of your homework set.