Coherent Structures, Pattern Formation and Solitons
Lectures: MWF 10:30-11:20pm (Zoom: https://washington.zoom.us/j/94971840180)
Prereqs: Amath 569 or Instructor Permission
Instructor: Bernard Deconinck
Office Hours: M4-5pm, T9-11am (Zoom: https://washington.zoom.us/j/94971840180)
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.
Here's a dropbox folder with all lecture videos and whiteboard pdfs.
- Logistics and class overview. Zoom link.
- Introduction. mp4, whiteboard.
- Solitons. Linear dispersion. Zoom link, whiteboard.
- Stationary Phase. Zoom link, whiteboard.
- Pattern formation. Zoom link, whiteboard.
- Multiple Scales 1. Zoom link, whiteboard.
- Multiple Scales 2. Zoom link, whiteboard.
- Multiple Scales 3. Zoom link, whiteboard.
- Special Solutions: Potential Energy Method. Zoom link, whiteboard.
- Special Solutions: Phase Plane. (no recording available:-(), whiteboard.
- Similarity Solutions. Zoom link, whiteboard.
- Lie symmetries. Recording link, whiteboard.
- Hamiltonian systems. Zoom link, whiteboard.
- Calculus of Variations. Zoom link, whiteboard.
- Infinite-dimensional Hamiltonian systems. Zoom link, whiteboard.
- Conserved Quantities. Zoom link, whiteboard.
- The KdV hierarchy. Zoom link, whiteboard.
- Lax pairs. Zoom link, whiteboard.
- Inverse Scattering 1. Big Picture. Zoom link, whiteboard.
- Inverse Scattering 2. Forward. Zoom link, whiteboard.
- Inverse Scattering 3. Time dependence. Zoom link, whiteboard.
- Inverse Scattering 4. Inverse 1. Zoom link, whiteboard.
- Inverse Scattering 5. Reconstruction. Zoom link, whiteboard.
- Inverse Scattering 6. Zeros of a(k). Zoom link, whiteboard.
- Inverse Scattering 7. Solitons. Zoom link, whiteboard.
- Inverse Scattering 8. Analyticity of M(x,k). Zoom link, whiteboard.
- Inverse Scattering 9. Boundedness of M(x,k) for real k. Zoom link, whiteboard.
- The Painleve equations. Zoom link, whiteboard.
- P1, the ARS conjecture. Zoom link, whiteboard.
- Painleve for PDEs. Zoom link, whiteboard.
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.
In this time of Covid: Zoom, etc.
I've decided to lecture synchronously, so you can attend the lectures at the same time that they were supposed to take place would we have been so lucky as to be able to be on campus, together in a classroom. These lectures will be recorded, so if you miss a lecture, or if the lecture schedule presents conflicts for you, you can watch them asynchronously as well (Note: if after a few lectures, I find myself synchronously lecturing for a minute audience, I reserve the right to switch to asynchronous). I hope the synchronous delivery will allow for some good interactions, questions and hopefully answers.
Zoom link (for lectures and office hours): https://washington.zoom.us/j/94971840180
BTW: there's something to be said for old-fashioned note taking, whether you're watching the lectures synchronously or asynchronously.
UW's student conduct policy: https://www.washington.edu/studentconduct/ (Links to an external site.)
Disability Resources: https://depts.washington.edu/uwdrs/ (Links to an external site.)
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/)..