Multi-pulse edge-localized states on quantum graphs

Abstract

We construct the edge-localized stationary states of the nonlinear Schrödinger equation on a general quantum graph in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. We give sufficient conditions on the edge lengths of the graph under which such states exist in the limit of large mass. In addition, we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator) for these multi-pulse states. If N solitons of the edge-localized state reside on the pendant and looping edges, we prove that the Morse index is exactly N. The technical novelty of this work is achieved by avoiding elliptic functions (and related exponentially small scalings) and closing the existence arguments in terms of the Dirichlet-to-Neumann maps for relevant parts of the given graph. We illustrate the general results with three examples of the flower, dumbbell, and single-interval graphs.