Ground state on the dumbbell graph (Q2800167)

The paper is concerned with standing waves of the focusing nonlinear Schrödinger equation \(i\Psi_t=\Psi_{xx}+ 2| \Psi|^2\Psi\) on the dumbbell graph which is represented by two circles of equal length joined by a line segment. At the two junctions Kirchhoff boundary conditions are applied. A ground state is a standing wave \(\Psi(t,x)=e^{i\Lambda t}\Phi (x)\) where \(\Phi\) minimizes the energy under the constraint \(Q(\Psi)=\int| \Psi|^2dx=Q_0\).NEWLINENEWLINENEWLINEThe main results are as follows. For small \(Q_0\) the ground state is given by \(\Phi\equiv \mathrm{const}\). If \(Q_0\) increases a pitchfork bifurcation occurs at some \(Q_0^\ast >0\). The bifurcating asymmetric standing waves are ground states for \(Q_0\gtrsim Q^\ast_0\). A second bifurcation of symmetric standing waves from \(\Phi\equiv \mathrm{const}\) appears at \(Q_0^{\ast\ast}>Q_0^\ast\).NEWLINENEWLINENEWLINEThe paper contains also a complete characterization of the linear spectrum of the Laplacian operator on the dumbbell graph as well as numerical approximations of the ground states.