We provide information on a non-trivial structure of phase space of the cubic nonlinear Schrödinger (NLS) on a three-edge star graph. We prove that, in contrast to the case of the standard NLS on the line, the energy associated with the cubic focusing Schrödinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L 2-norm. We moreover show that the only stationary state with prescribed L 2-norm is indeed a saddle point
Adami, R., Cacciapuoti, C., Finco, D., Noja, D. (2012). On the structure of critical energy levels for the cubic focusing NLS on star graphs. JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL, 45(19) [10.1088/1751-8113/45/19/192001].
On the structure of critical energy levels for the cubic focusing NLS on star graphs
Noja, D
2012
Abstract
We provide information on a non-trivial structure of phase space of the cubic nonlinear Schrödinger (NLS) on a three-edge star graph. We prove that, in contrast to the case of the standard NLS on the line, the energy associated with the cubic focusing Schrödinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L 2-norm. We moreover show that the only stationary state with prescribed L 2-norm is indeed a saddle point
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