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

Abstract

We consider a one-dimensional discrete nonlinear Schr{ö}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.

28 pages, slightly changed the title and other details

References in corpus (3)

• Asymmetric vortex solitons in nonlinear periodic lattices
• On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice
• An extensive resonant normal form for an arbitrary large Klein-Gordon model

Cited by in corpus (8)

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• Discrete breathers in ${\varphi }^4$ and related models
• Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms
• Spatiotemporal dynamics in a twisted, circular waveguide array
• On the nonexistence of degenerate phase-shift multibreathers in a zigzag Klein-Gordon model