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Apr 17 2018:

The slides from the student presentations can be downloaded below. Thanks to all of the students for interesting talks.

There are new lecture notes available on Topic 9, the theory of Riemann-Hilbert problems, as well as the paper of Dieng and McLaughlin on \(\overline{\partial}\) methods for Riemann-Hilbert problems.

A Mathematica notebook illustrating some properties of various equations of the form \(u_{tt}-u_{xx}+V'(u)=0\) can be downloaded below (see the entry in the schedule for Jan. 4).

A second Mathematica notebook illustrating the Zabusky-Kruskal experiment and its relation to the Fermi-Pasta-Ulam-Tsingou problem is also available (see the entry for Jan. 16).

This week: Presentations by T. Bolles and D. Geroski.

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Home Page for Math 651

Class Meetings:

Tuesdays and Thursdays 11:40-1:00 PM in 1866 East Hall.

Office hours:

Tuesdays, Wednesdays, and Thursdays, 2-3 PM in 5826 East Hall, or by appointment.

Grading and Course Policies:

Students will be evaluated on the basis of Soliton Picture

The KdV soliton reproduced at the Scott-Russell Aquaduct, 1995

Schedule
Week Meeting Date In Class Resources Homework
Week 1 Lecture 1 Thurs, Jan 4 Principles of wave propagation. Dispersion and nonlinearity. Solitary waves and solitons. 1. Mathematica notebook for understanding solitons (enable dynamic content).

2. Some lecture notes.

Week 2 Lecture 2 Tues, Jan 9 The Korteweg-de Vries equation and its properties.
Thurs, Jan 11 No Class (to be rescheduled later)
Week 3 Lecture 3 Tues, Jan 16 KdV continued.

1. Mathematica notebook for reproducing the Zabusky-Kruskal experiment (enable dynamic content).

2. Some lecture notes.

Lecture 4 Thurs, Jan 18 Lax pairs and other integrable equations. Some lecture notes.  
Week 4 Lecture 5 Tues, Jan 23 The Toda lattice.
Lecture 6 Thurs, Jan 25 The Toda lattice. The inverse problem via orthogonal polynomials and QR factorization.
Make-up lecture

Thurs, Jan 25

5-6:30 PM

1866 East Hall

The Toda lattice. Riemann-Hilbert formulation of the inverse problem. Some lecture notes. HW 1 pdf. Due Thursday, February 8. Here is a copy of the 1975 paper on the Toda lattice by Moser.
Week 5 Lecture 7 Tues, Jan 30 The inverse scattering transform for the defocusing nonlinear Schrödinger equation. Direct scattering and Jost solutions.
Lecture 8 Thurs, Feb 1 The inverse scattering transform for the defocusing nonlinear Schrödinger equation. Time evolution of scattering data.
Week 6 Lecture 9 Tues, Feb 6 Further properties of Jost solutions.
Lecture 10 Thurs, Feb 8 The Riemann-Hilbert problem of inverse scattering. Some lecture notes.
Week 7 Lecture 11 Tues, Feb 13 The dressing method. Construction of exact solutions from the Riemann-Hilbert problem.
Lecture 12 Thurs, Feb 15 Dressing, continued. Some lecture notes. HW 2 pdf. Due Tuesday, March 6.
Week 8 Lecture 13 Tues, Feb 20 Long-time asymptotics for the defocusing nonlinear Schrödinger equation.
Lecture 14 Thurs, Feb 22 Long-time asymptotics, continued.
Week 9 Tues, Feb 27 Winter Break
Thurs, Mar 1 Winter Break
Week 10 Lecture 15 Tues, Mar 6 Long-time asymptotics, continued. Some lecture notes.
Lecture 16 Thurs, Mar 8 The focusing nonlinear Schrödinger equation.
Week 11 Lecture 17 Tues, Mar 13 The focusing nonlinear Schrödinger equation, contd. Solitons.

1. Some lecture notes.

2. A Mathematica notebook for exploring the two-soliton solution of the focusing NLS equation.

Lecture 18 Thurs, Mar 15 Theory of Riemann-Hilbert problems.
Week 12 Lecture 19 Tues, Mar 20 Theory, contd.
Lecture 20 Thurs, Mar 22 Theory, contd.
Week 13 Lecture 21 Tues, Mar 27 Theory, contd.
Lecture 22 Thurs, Mar 29 Theory, contd.
Week 14 Lecture 23 Tues, Apr 3 Fredholm theory of Riemann-Hilbert problems.
Lecture 24 Thurs, Apr 5 Application to the focusing and defocusing nonlinear Schrödinger equation. Some lecture notes.
Week 15 Lecture 25 Tues, Apr 10 The dbar-steepest descent method. The paper of Dieng and McLaughlin discussed in today's lecture.
Lecture 26 Thurs, Apr 12

Student presentations I:

1. A. Georgakopoulos: Integrable structure of conformal mappings.

2. Y. Liao: Integrable operators.

Alex's slides.

Yuchen's slides.

Week 16 Lecture 27 Tuesday, Apr 17

Student presentations II:

3. T. Bolles: Squared eigenfunctions and solution of linearized integrable equations.

4. D. Geroski: Hirota's direct method in soliton theory.

Tyler's slides.

David's slides.