AMATH 573 B: Coherent Structures, Pattern Formation and Solitons

Coherent Structures, Pattern Formation and Solitons

SLN 10240

deconinc@uw.edu

Office Hours: W4-5pm, Th9-11am

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. Context. Some history. Reference materials: 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 as obtained through symmetry reduction. Simplest case: stationary solutions. Solitary waves and solitons.
5. Infinite-dimensional Hamiltonian and Lagrangian systems. Conserved quantities. Noether's theorem. Poisson brackets. Liouville integrability. If time permits: Bihamiltonian structures.
6. Conserved quantities. Infinite number of conserved quantities for KdV. The Miura transform, Modified KdV. the KdV hierarchy. Integrable equations, hierarchies.
7. Two soliton solutions and their interactions. Brief mentioning of Hirota's method and Backlund transformations.
8. Lax Pairs. Principles of the inverse scattering method. Trace formulae.
9. 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.

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.

Teacher expectations

Since this is an advanced course (think of it as a PhD elective), you are expected to be fairly independent and resourceful. That doesn't mean you shouldn't come to office hours. It does mean that you should try to solve problems before you want hints and pointers. The homework sets will be hard (and presumably instructional) and occasionally tedious. Do not wait until the last minute to do them. You have all the tools to do them when they are assigned, 14 days before they are due. Although it is not required, the use of a computer algebra system (Maple, Mathematica, Sage, etc) will be very helpful. If you use this, please upload your notebooks together with the write up of your homework sets. 

The project should be substantial (40% of your grade!) and your presentation should be polished and professional. You will know what your project is on (you pick, I sign off on your choice) around the middle of the quarter. Given the percentages, you should expect to spend as much time on it as on three homework sets. 

Ultimately, this is a fun course that will make you go "Ooh, Aah!" numerous times. But it will require work, from you and from me.

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/)..