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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 ${H}^{2}\left({ℝ}^{2}\right)$, and that global solutions exist in ${H}^{1}\left({ℝ}^{2}\right)$ provided that the initial data are small enough in ${L}^{2}\left({ℝ}^{2}\right)$. 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:
 35Q55 NLS-like (nonlinear Schrödinger) equations 35B40 Asymptotic behavior of solutions of PDE 81T13 Yang-Mills and other gauge theories 81Q05 Closed and approximate solutions to quantum-mechanical equations
##### Keywords:
Cauchy problem; blow up; self-similar analysis