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

TL;DR

This paper proves the nonexistence of certain phase-shift discrete solitons in a one-dimensional nonlocal dNLS lattice due to degeneracy issues, using perturbative and bifurcation analysis techniques.

Contribution

It introduces a nonexistence criterion for phase-shift discrete solitons in a nonlocal dNLS model with degeneracy, extending the understanding of soliton solutions in such lattices.

Findings

No $C^2$ phase-shift solitons exist at small coupling $psilon$.

Nonexistence also holds for solutions only $C^0$ in the same limit.

The criterion is effective in cases of partial and full degeneracy of approximate solutions.

Abstract

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.