The focusing nonlinear Schroedinger (NLS) equation: Rigorous semiclassical and long time asymptotics
by
Stephanos Venakides
Duke University
Coauthors: Alexander Tovbis (UCF), Xin Zhou (Duke), Robert Buckingham (U. Michighan)

In collaboration with Alexander Tovbis (UCF) and Xin Zhou (Duke), we study the semiclassical focusing nonlinear Schrödinger equation

ieqt+e2 qxx+|q|2q=0,      e→ 0

with a one-parameter-family (m > 0) of initial data

q(x, 0) = A(x) e [ iS(x)/(e)],     A = sech (x),     S' = -mtanh(x).

We derived and proved formulae for the asymptotic solution q(x, t, e) when the semiclassical parameter e tends to 0. The derived formulae describe the first breaking curve ( nonlinear caustic) for all positive values of m and are valid globally in time when m ≥ 2; when 0 < m < 2, the initial data acquire a solitonic content and a second break occurs. We then extended the asymptotic analysis to the long time regime (1 << t << ln( 1 / e)) where these formulae become more tractable. We have extended many of our results to a more general class of initial data, shedding some light in the nature of the modulational instability.

In collaboration with Robert Buckingham (U. Michighan), we solve the so called "shock problem", that describes the long time evolution of two colliding truncated plane waves. We find that different waveform structures are separated in space by fronts traveling with constant speeds and we calculate the speeds and the waveforms.

The NLS equation is generically encountered in wave propagation through nonlinear media. One of its most important aspects is its modulational instability: Regular wavetrains, such as fully nonlinear periodic or quasi-periodic NLS solutions (they are the nonlinear analogs of linear combinations of plane waves) are unstable to modulation and may break up to more complicated wave structures. We will outline the recent mathematical techniques that allow the above rigorous calculation of the dominant contribution to the waveform, showing how analyticity in the initial spectral data allows modulated waves to persist in space-time up to boundaries (breaking curves or nonlinear caustics) across which the waves undergo a phase transition. The emphasis will be presenting the core ideas and staying away from the (abundant) technical issues.

Date received: July 10, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-27.