On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

Abstract

We consider stationary waves on nonlinear quantum star graphs, i.e. solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e. the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting such as the classic Sturm oscillation theorem.

Standing waves on quantum graphs

“…Similar classifications of bound states were obtained from explicit analysis of the stationary NLS equations with cubic nonlinearity on the dumbbell graph in [77,95], the truncated star graph [28], and the periodic graph [107]. Positive bound states on the tadpole graph in the case of quintic nonlinearity were considered with the period function in [102].…”

Standing waves on quantum graphs

“…These excited states are usually not interesting in applications because they are typically unstable in the time evolution of the NLS equation [100]. Similar classifications of bound states were obtained from explicit analysis of the stationary NLS equations with cubic nonlinearity on the dumbell graph in [74,92], the truncated star graph [27], and the periodic graph [104]. Positive bound states on the tadpole graph in the case of quintic nonlinearity were considered with the period function in [99].…”

Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

“…The boundary control method, the leaf-peeling method, and the distributed-parameter system have been used to investigate problems of controllability, observability, and stability for the wave equation, but the method of generalized functions has not been used. As we mentioned before, different ODE and PDE problems on star graphs were considered in [12,13,[29][30][31]. Now, let us explain how our work differs from the works mentioned above.…”