The nonlinear Schrödinger (NLS) equation provides a canonical description for the envelope dynamics of a quasi-monochromatic plane wave propagating in a weakly nonlinear dispersive medium when dissipative processes are negligible. It arises in various physical contexts in the description of nonlinear waves such as propagation of a laser beam in a medium whose index of reflection is sensitive to the wave amplitude, water waves at the free surface of an ideal and plasma waves. NLS also appears in the description of the Bose-Einstein Condensate (BEC), a context where it is often called the Gross-Pitaevskii equation. It admits solutions in the form of coherent structures like vortices that define states that can be excited in super helium. The last few years have witnessed a rapid development in research of NLS-related applications such as optical communications, laser surgeries, remote sensing and the BEC. This has created an enormous amount of new mathematical problems for mathematicians.
(Tel-Aviv University, Israel)
Shi Jin (University of Wisconsin)
George Papanicolaou (Stanford University)