Vortex Collapse for the L2-Critical Nonlinear Schrödinger Equation

Details Vortex Collapse for the L2-Critical Nonlinear Schrödinger Equation Vortex Collapse for the L2-Critical Nonlinear Schrödinger Equation

2010-10-28

arxiv.org/abs/1010.5864v1

Mathematics

Analysis of PDEs

36 pages, 10 figures

Scientific paper

The focusing cubic nonlinear Schr\"odinger equation in two dimensions admits vortex solitons, standing wave solutions with spatial structure, Qm(r,theta) = e^{i m theta} Rm(r). In the case of spin m = 1, we prove there exists a class of data that collapse with the vortex soliton profile at the log-log rate. This extends the work of Merle and Rapha\"el, (the case m = 0,) and suggests that the L2 mass that may be concentrated at a point during generic collapse may be unbounded. Difficulties with m >= 2 or when breaking the spin symmetry are discussed.

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