Multi-pulse edge-localized states on quantum graphs

TL;DR

This paper investigates the existence and stability of multi-pulse edge-localized states of the focusing nonlinear Schrödinger equation on quantum graphs, providing explicit Morse index calculations without elliptic functions.

Contribution

It introduces a new method using Dirichlet-to-Neumann maps to prove the existence of multi-pulse states and computes their Morse index explicitly.

Findings

Existence of multi-pulse states on quantum graphs for large mass

Morse index of N-soliton states is exactly N

Avoids elliptic functions in the analysis

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.