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Variational and stability properties of coupled NLS equations on the star graph. (English) Zbl 1496.35355

Summary: We consider variational and stability properties of a system of two coupled nonlinear Schrödinger equations on the star graph \(\Gamma\) with the \(\delta\) coupling at the vertex of \(\Gamma\). The first part is devoted to the proof of an existence of the ground state as the minimizer of the constrained energy in the cubic case. This result extends the one obtained recently for the coupled NLS equations on the line.
In the second part, we study stability properties of several families of standing waves in the case of a general power nonlinearity. In particular, we consider one-component standing waves \(e^{i \omega t} (\Phi_1 (x), 0)\) and \(e^{i \omega t} (0, \Phi_2 (x))\). Moreover, we study two-component standing waves \(e^{i \omega t} (\Phi (x), \Phi (x))\) for the case of power nonlinearity depending on a unique power parameter \(p\).
To our knowledge, these are the first results on variational and stability properties of coupled NLS equations on graphs.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
35B35 Stability in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
49M41 PDE constrained optimization (numerical aspects)
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