Professor of Mathematics
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 RiemannHilbert problems, as well as the paper of Dieng and McLaughlin on \(\overline{\partial}\) methods for RiemannHilbert 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 ZabuskyKruskal experiment and its relation to the FermiPastaUlamTsingou problem is also available (see the entry for Jan. 16).
This week: Presentations by T. Bolles and D. Geroski.
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 Kortewegde 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 ZabuskyKruskal 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.  
Makeup lecture  Thurs, Jan 25 56:30 PM 1866 East Hall 
The Toda lattice. RiemannHilbert 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 RiemannHilbert problem of inverse scattering.  Some lecture notes.  
Week 7  Lecture 11  Tues, Feb 13  The dressing method. Construction of exact solutions from the RiemannHilbert 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  Longtime asymptotics for the defocusing nonlinear Schrödinger equation.  
Lecture 14  Thurs, Feb 22  Longtime asymptotics, continued.  
Week 9  Tues, Feb 27  Winter Break  
Thurs, Mar 1  Winter Break  
Week 10  Lecture 15  Tues, Mar 6  Longtime 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 twosoliton solution of the focusing NLS equation. 

Lecture 18  Thurs, Mar 15  Theory of RiemannHilbert 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 RiemannHilbert 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 dbarsteepest 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. 