## The Nonlinear Schroedinger Equation Working Group

### Arizona based Group members:

Momar Dieng

Jesus Adrian Espinola-Rocha

Bob Jenkins

Ken McLaughlin

Jason Newport

#### Lectures and publications

### Research Topics

#### Introduction to integrable nonlinear partial differential equations

In the 2005-2006 academic year, graduate students Jesus Adrian Espinola-Rocha, Bob Jenkins, and Jason Newport learned and then gave a complete and self-contained presentation of the spectral and inverse spectral theory associated to the nonlinear Schroedinger hierarchy of partial differential equations

#### d-bar analysis and long-time behavior

Dieng and McLaughlin are developing techniques for the asymptotic analysis of d-bar problems. A proving ground: long-time asymptotics for the defocusing NLS equation. This has been the main activity of the working group for the Fall of 2006.

#### semiclassical spectral analysis

Goal: describe the asymptotic behavior of point spectrum and reflection coefficient for the Zakharov-Shabat operator associated to the focusing NLS equation.#### Singular limits for special initial data

Jeff DiFranco and I carried out a detailed asymptotic analysis of the inverse problem associated to the defocusing NLS equation, with square well initial data. The punch line is that there is a continuous analog of the Gibbs phenomenon which describes the manner in which the NLS equation regularizes jump discontinuities.

Gibbs phenomena as time decreases to zero for step initial data. This is actually an exact solution of the linear equation, but Jeff DiFranco's thesis work (and our article, above), show that the leading order behavior of the defocusing nonlinear Schrodinger equation is the same.

Jason Newport is currently studying a singular limit of the inverse problem associated to Dirac mass inital data for the defocusing NLS equation.

*Some of the material on this page is based on work supported in part by the National Science Foundation under grant numbers DMS-0200749, and DMS-0451495 (McLaughlin). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.*