2017/07/31 by T. Penati, Tiziano Penati, M. Sansottera +5 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Bifurcation #Degeneracy (biology) #Degenerate energy levels #Lattice (music) #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Dynamics and Pattern Formation #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum mechanics #Soliton #nlin.PS
paper · pdf · doi:10.1016/j.physd.2017.12.012
28 pages, slightly changed the title and other details
openalex publication_date 2017/12/27 · openalex created_date 2018/01/05 · arxiv created 2018/03/08 · arxiv updated 2018/03/09 · openalex updated_date 2026/08/05
We consider a one-dimensional discrete nonlinear Schrödinger (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 C2 with respect to the small coupling ε, in the limit of vanishing ε. If we assume the solution to be only C0 in the same limit of ε, 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.