On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice

PHYSICA D-NONLINEAR PHENOMENA

卷:370

Article

Penati, T.¹  Sansottera, M.¹  Paleari, S.¹  Koukouloyannis, V.²  Kevrekidis, P. G.³ 

[1] Milano Univ, Dept Math F Enriques, Via Saldini 50, I-20133 Milan, Italy

[2] Qatar Univ, Coll Arts & Sci, Dept Math Stat & Phys, POB 2713, Doha, Qatar

[3] Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA

关键词: Discrete non-linear Schrodinger;    Discrete solitons;    Discrete vortex;    Current conservation;    Perturbation theory;    Lyapunov-Schmidt decomposition;   

DOI  :  10.1016/j.physd.2017.12.012

来源: Elsevier

【 摘 要 】

We consider a one-dimensional discrete nonlinear Schrodinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-site 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. (C) 2017 Elsevier B.V. All rights reserved.

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