Abstract: The cubic nonlinear Schrödinger equation is a simple model for a wide range of interesting physical phenomena, including Bose—Einstein condensation, rogue water waves, and nonlinear optics. Mathematically, this equation is a fundamental example in the field of nonlinear dispersive PDE, an active research area that progressed rapidly in the last 20 or so years.
In this talk, we will discuss several fundamental mathematical questions about the cubic NLS, including the existence of solutions, the behavior of solutions (for example, blow-up or scattering), and the presence of certain special `solitary wave’ solutions. These questions will quickly lead us into the territory of current research in dispersive equations.
This talk is intended for undergraduate students who may be interested in pursuing graduate study in mathematics. The talk should be accessible to any student who is familiar with calculus and a bit of analysis/ODEs. Some exposure to Fourier analysis would be helpful, but this will not be assumed.