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.

Standing waves on quantum graphs

“…the tadpole graph [9,44,59,102,103], the dumbbell graph [77,95], and the flower graph [85]. More general results were obtained in the limit of large mass in [12,33,88], where the existence of single-pulse states and multi-pulse states was proven for a general graph, as we will describe in section 6.…”

“…The period function T + (p, q) defined in (5.15) satisfies certain monotonicity properties which are useful in the analysis of pulses on edges of a metric graph G. These results were obtained in [85,88] for cubic nonlinearity, which are reviewed here. Recall that A(u) := u 2 − u 4 (for p = 1) attains a minimum at u = ±p * , where p * = 1 √ 2 .…”

“…Finally, the asymptotic representation T + (p, q) ∼ − ln(p + q), as p, q → 0 proven in [33] (see also [88]) suggests the following property, which is important for the study of pulses in the limit of large in ω = − 2 .…”

“…Another method can be used to overcome this difficulty in the limit of large negative ω, which in the case of subcritical nonlinearities transfers to the limit of large mass M(Φ) = μ in the variational problem (3.2). The positive edge-localized states were considered in the limit of large mass in [12,33,57,88,91].…”

“…Multi-pulse states were classified for the stationary NLS equation with the cubic nonlinearity in the limit of large mass under some consistency assumptions in [88] based on the properties of the DtN mappings proven in [33].…”