Multi-pulse edge-localized states on quantum graphs

Abstract: Edge-localized stationary states of the focusing nonlinear Schrodinger equation on a general quantum graph are considered in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically to a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. Not only we prove that such states exist in the limit of large mass, but also we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator). In the case of the edge-localized N-soliton states 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.