Prepared by Gino Biondini and Barbara Prinari
A small research group has been meeting at the American Institute of Mathematics (AIM) in Palo Alto, CA, during the week of Feb. 18-22 to work on integrable systems of nonlinear Schroedinger type, a special class of nonlinear partial differential equations (PDEs).
Nonlinear Schroedinger (NLS) equations are the simplest models that describe the evolution of weakly nonlinear dispersive wave trains. As such, they have been studied as models for many important natural phenomena, such as deep water waves, ion-acoustic waves in plasmas, and propagation of laser pulses in optical fibers. Indeed, the NLS equation has provided an invaluable tool for the study of optical fiber telecommunication systems over the last forty years. An emerging application that has also attracted great scientific attention in the last 10 years is related to Bose-Einstein condensates (BECs). A BEC is a state of matter of a dilute gas of boson atoms cooled to temperatures very close to absolute zero. Under such conditions, a large fraction of the atoms occupy the lowest quantum state and quantum effects become evident even on a macroscopic scale. The state of the BEC is then described by the wavefunction of the condensate as a whole, which obeys a nonlinear equation known as the Gross-Pitaevskii equation. The existence of BECs was theorized in the 1920′s by Bose and Einstein, but only observed experimentally in 1995. Since then, the field has exploded. In a one-dimensional approximation (cigar-shaped traps), the model equation is then precisely the NLS equation. Recent studies also suggest that nonlinear solitary wave interactions described by the Kadomtsev-Petviashvili equation (another integrable system) could help to explain the generation of localized large amplitude waves such as those generated by the interaction of two wave stems in the Tohoku Japanese tsunami of 2012.
One of the most interesting features of nonlinear integrable systems from a mathematical point of view is the fact that they possess a surprisingly rich and beautiful structure, and powerful analytical and asymptotic mathematical techniques are available to investigate the properties of these systems and solutions.
Nonlinear integrable systems have been extensively studied, and a large body of knowledge has been accumulated on them. At the same time, new solutions and new properties continue to be found, and many fundamental questions are still open. Among them, initial-value problems with non-trivial boundary conditions and boundary value problems. This SQuaRE (Structured Quartet Research Ensemble) aims at addressing and resolving some of these issues.