Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation. (English) Zbl 0822.35125

Bergé, L.; de Bouard, A.; Saut, J. C.

Nonlinearity 8, No. 2, 235-253 (1995).

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

Keywords:

Cauchy problem; blow up; self-similar analysis