On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice
| Autor: | Penati, T., Sansottera, M., Paleari, S., Koukouloyannis, V., Kevrekidis, P. G. |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.physd.2017.12.012 |
| Popis: | We consider a one-dimensional discrete nonlinear Schr{\"o}dinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-sites vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler variant. Due to the specific choice of lengths of the inter-site interactions, the vortex configurations considered present a degeneracy which causes the standard continuation techniques to be non-applicable. In the present one-dimensional case, the existence of a conserved quantity for the soliton profile (the so-called density current), together with a perturbative construction, leads to the nonexistence of any phase-shift discrete soliton which is at least $C^2$ with respect to the small coupling $\epsilon$, in the limit of vanishing $\epsilon$. If we assume the solution to be only $C^0$ in the same limit of $\epsilon$, nonexistence is instead proved by studying the bifurcation equation of a Lyapunov-Schmidt reduction, expanded to suitably high orders. Specifically, we produce a nonexistence criterion whose efficiency we reveal in the cases of partial and full degeneracy of approximate solutions obtained via a leading order expansion. Comment: 28 pages, slightly changed the title and other details |
| Databáze: | arXiv |
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