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.

我们在大质量极限下的一般量子图上构建非线性薛定谔方程的边局域定态。与之前的工作相比，我们包括任意多脉冲正态，它们渐近地由N 个孤子组成，每个孤子都位于有界（悬垂、循环或内部）边缘。我们给出了图的边长的充分条件，在该条件下，这种状态存在于大质量的极限中。此外，我们计算这些多脉冲状态的精确莫尔斯指数（相应线性化算子中负特征值的数量）。如果边缘局部状态的N 个孤子驻留在悬垂和循环边缘上，我们证明莫尔斯指数正好是N. 这项工作的技术新颖性是通过避免椭圆函数（和相关的指数级小缩放）并根据给定图的相关部分的 Dirichlet-to-Neumann 映射关闭存在论点来实现的。我们用花、哑铃和单区间图的三个例子来说明一般结果。