Vortex Collapse for the L2-Critical Nonlinear Schr\"odinger Equation

Gideon Simpson; Ian Zwiers

doi:10.48550/arxiv.1010.5864

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$Vortex Collapse for the L2-Critical Nonlinear Schr\"odinger Equation$

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Vortex Collapse for the L2-Critical Nonlinear Schr\"odinger Equation

Gideon Simpson and Ian Zwiers

28 Oct 2010

DOI: https://doi.org/10.48550/arxiv.1010.5864

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Abstract

Mathematics - Analysis of PDEs

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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Title: Vortex Collapse for the L2-Critical Nonlinear Schr\"odinger Equation
Creators: Gideon Simpson
Ian Zwiers
Resource Type: Preprint
Language: English
Academic Unit: Mathematics
Other Identifier: 991019296759304721

Vortex Collapse for the L2-Critical Nonlinear Schr\"odinger Equation

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Abstract

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