singularity formation

Tuesday, November 1, 2016 - 3:15pm - 4:05pm
Justin Holmer (Brown University)
We consider a version of the nonlinear Schroedinger equation (NLS) with point nonlinearity, which can formulated as the linear Schroedinger equation away from the spatial origin together with a nonlinear jump condition in the derivative across the origin. This model can be viewed as a limiting form of a concentrated nonlinearity and exhibits many of the same properties as the standard nonlinear Schroedinger equation. In fact, in most cases, the analysis is simpler than for standard NLS and thus simpler proofs and/or stronger results are possible.
Tuesday, November 1, 2016 - 10:15am - 11:05am
Gideon Simpson (Drexel University)
Direct numerical simulation of an L2 supercritical variant of the derivative nonlinear Schrödinger equation suggests that there is a finite time singularity. Subsequent exploration with the dynamic rescaling method provided more detail about the blowup and a recent refined asymptotic analysis of the blowup solution gives predictions of the blowup rates. However, due to the mixed hyperbolic-dispersive nature of the equation, these methods have limited the proximity to the blowup time.
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