Coherent Structures, Pattern Formation and Solitons

SLN 22775 Lectures: MWF 3:30-4:50pm (Note: most weeks, lectures will be on MW. The Friday lecture time may be used to make up a lecture, as needed) Prereqs: Amath 569 or Instructor Permission Instructor: Bernard Deconinck deconinc@uw.edu Office Hours: M10-11am, T9-11am

Course Description

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.

Textbook

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.

Message Board

We're using Piazza for the class message board. 

Syllabus (subject to relatively minor changes)

1. Introduction. Background and motivation: The FPU problem, Poincare's work on King Oscar II's problem.
2. Quick overview of Linear dispersive partial differential equations using Fourier transforms.
3. Handwavy derivation of the Korteweg-de Vries equation and the Nonlinear Schrodinger equation. Reference materials: About John Scott Russell, John Scott Russell's original soliton recreated.
4. Exact solutions of partial differential equations: stationary solutions. Solitary waves and solitons, elliptic solutions (depending on class interest)
5. Infinite-dimensional Hamiltonian and Lagrangian systems. Conserved quantities. Poisson brackets. Liouville integrability. 
6. Conserved quantities. Infinite number of conserved quantities for KdV. The Miura transform, Modified KdV. the KdV hierarchy. Integrable equations, hierarchies.
7. Concepts of stability.
8. Lax Pairs. Principles of the inverse scattering method.
9. Testing for integrability: Painleve methods. 
10. Additional topics, as time permits.

Grading

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. When a homework set is assigned, we have already covered all the class material needed for you to do every single problem on the set. Don't wait until the end of the submission period to start! 

Since your project count for 40% of your grade, you should expect to put as much work into it as in 3.5 or 4 homework sets. You will know the topic of your project around the halfway point of the quarter (you pick, I approve; I will make suggestions available as well), so you will have plenty of time to work on it. 

Various

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Religious Accommodation Policy: Washington state law requires that UW develop a policy for accommodation of student absences or significant hardship due to reasons of faith or conscience, or for organized religious activities. The UW’s policy, including more information about how to request an accommodation, is available at Religious Accommodations Policy (https://registrar.washington.edu/staffandfaculty/religious-accommodations-policy/). Accommodations must be requested within the first two weeks of this course using the Religious Accommodations Request form (https://registrar.washington.edu/students/religious-accommodations-request/)..