Stationary Nonlinear Schr\"odinger Equation on Simplest Graphs: Boundary conditions and exact solutions

We treat the stationary (cubic) nonlinear Schr\"odinger equation (NSLE) on simplest graphs. Formulation of the problem and exact analytical solutions of NLSE are presented for star graphs consisting of three bonds. It is shown that the method can be extended for the case of arbitrary number of bonds of star graphs and for other simplest topologies such as tree and loop graphs. The case of repulsive and attractive nonlinearities are treated separately

A derivation of boundary conditions for the space-fractional operator with order 0<α<10<\alpha<1 at the vertices of metric graphs

We consider the space-fractional operator with order $0<\alpha<1$ on the metric star graph. The boundary conditions at the vertices of the metric star graph providing the self-adjointness of the operator are derived. The obtained result is extended to the other topologies of the metric graphs