A. Kairzhan and D.E. Pelinovsky
Multi-pulse edge-localized states on quantum graphs
Analysis and Mathematical Physics 11 (2021) 171 (26 pages)
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.
Keywords:
Nonlinear Schrodinger equation; quantum graphs,
existence of stationary states, Morse index, period function.