Stationary Solitons in discrete NLS with non-nearest neighbour interactions

The paper’s “Main numerical result” reports homoclinic orbits for the 4D map (8) derived from the stationary DNLS with next-nearest-neighbor coupling for ε ∈ (0,1], A ∈ [−0.145, −0.115], based on high-order parametrizations of invariant manifolds and a numerical intersection test; it does not furnish a rigorous existence proof and explicitly frames the result as numerical evidence , with map (8) defined from the steady-state reduction of (5) and transversality checked numerically . By contrast, the model’s solution gives a standard variational/mountain-pass construction yielding a rigorous existence result; a small but easily repaired gap is the need for a uniform upper bound on the mountain-pass levels c_m (obtainable along the one-site path tδ_0, which yields sup_t(−S_m(tδ_0)) = ε^2, uniform in m). With that repair, the argument is complete and, in fact, applies on the larger parameter set A ∈ (−1/4,0), uniformly in ε ∈ (0,1].