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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 \(H^ 2 (\mathbb{R}^ 2)\), and that global solutions exist in \(H^ 1 (\mathbb{R}^ 2)\) provided that the initial data are small enough in \(L^ 2 (\mathbb{R}^ 2)\). 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:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
81T13 Yang-Mills and other gauge theories in quantum field theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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