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Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation. (English) Zbl 0822.35125
Summary: Time-dependent solutions of the two-dimensional Chern-Simons gauged nonlinear Schrödinger equation are investigated in terms of an initial- value problem. We prove that this Cauchy problem is locally well posed in H 2 ( 2 ), and that global solutions exist in H 1 ( 2 ) provided that the initial data are small enough in L 2 ( 2 ). On the other hand, under certain conditions ensuring, for example a negative Hamiltonian, solutions blow up in a finite time which only depends on the initial data. The diverging shape of collapsing structure is finally discussed throughout a self-similar analysis.
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35B40Asymptotic behavior of solutions of PDE
81T13Yang-Mills and other gauge theories
81Q05Closed and approximate solutions to quantum-mechanical equations