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Multi-pulse edge-localized states on quantum graphs

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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.

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Data availability statement

The datasets analysed during the current study are available from the corresponding author on request.

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Acknowledgements

The authors thank Roy Goodman for many suggestions helping us to improve the presentation of this paper and for preparing the illustrating example shown on Fig. 6. A. Kairzhan thanks the Fields Institute for Research in Mathematical Sciences for its support and hospitality during the thematic program on Mathematical Hydrodynamics in July–December, 2020. D.E. Pelinovsky acknowledges the support from the NSERC Discovery grant.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    Adilbek Kairzhan

  2. Department of Mathematics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Dmitry E. Pelinovsky

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  1. Adilbek Kairzhan
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  2. Dmitry E. Pelinovsky
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Correspondence to Dmitry E. Pelinovsky.

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Kairzhan, A., Pelinovsky, D.E. Multi-pulse edge-localized states on quantum graphs. Anal.Math.Phys. 11, 171 (2021). https://doi.org/10.1007/s13324-021-00603-3

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