Friday, November 4, 2016 - 11:30am - 12:20pm
Pavel Lushnikov (University of New Mexico)
Many nonlinear systems of partial differential equations have a striking phenomenon of spontaneous formation of singularities in a finite time (blow up). Blow up is often accompanied by a dramatic contraction of the spatial extent of solution, which is called by collapse. A collapse in a nonlinear Schrodinger equation (NLSE) describes the self-focusing of the intense laser beam in the nonlinear Kerr medium (like usual glass) with the propagation distance z playing the role of time.
Thursday, November 3, 2016 - 3:15pm - 4:05pm
Svetlana Roudenko (George Washington University)
We discuss the focusing nonlinear Klein-Gordon equation starting with the cubic nonlinearity in 3 dimensions. Inspired by the paper of Donninger-Schlag on this equation, we further investigate the blow up and scattering behavior of its solutions. We extend the theoretical boundaries of the blow up regions and discuss the behavior of solutions there, for example, formation of a singularity away from the origin, and behavior near the ground and excited states. We also show extensions to other dimensions and nonlinearities.
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