A Class of Solutions to the 3D Cubic Nonlinear Schrodinger Equation that Blows Up on a Circle | Zendy

Justin Holmer | Zendy; Svetlana Roudenko | Zendy

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A Class of Solutions to the 3D Cubic Nonlinear Schrodinger Equation that Blows Up on a Circle

Author(s) -

Justin Holmer,

Svetlana Roudenko

Publication year - 2010

Publication title -

applied mathematics research express

Language(s) - English

Resource type - Journals

eISSN - 1687-1200

pISSN - 1687-1197

DOI - 10.1093/amrx/abq016

Subject(s) - quintic function , axial symmetry , plane (geometry) , nonlinear system , class (philosophy) , nonlinear schrödinger equation , mathematics , mathematical analysis , schrödinger equation , mathematical physics , physics , quantum mechanics , geometry , computer science , artificial intelligence

We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS) i\partial_t u + \Delta u + |u|^2 u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Rapha\"el's construction \cite{R} of a family of solutions to the 2d quintic focusing NLS, i\partial_t u + \Delta u + |u|^4 u=0, that blow-up on a circle.

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