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

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.