Cone Invariance and Homoclinic Standing Waves for Cooperative Finite-Range DNLS Equations

Abstract

We develop a constrained-power variational framework for positive homoclinic standing waves in one-dimensional discrete nonlinear Schrödinger equations with cooperative finite-range interactions. Starting from periodic lattices, we construct constrained energy maximizers and use the associated projected gradient flow to establish positive invariance of the cone of nonnegative, even, and unimodal profiles. A central result is that the monotone ordering of the interaction coefficients, \(a_1\ge a_2\ge\cdots\ge a_M\ge0\), with \(a_1>0\), provides the structural condition ensuring preservation of this cone under finite-range coupling. Under a strict maximum condition above the linear spectral edge, we obtain a uniform frequency gap and period-independent exponential localization. These estimates provide the compactness required for the infinite-period limit and yield positive, even, unimodal, and exponentially localized homoclinic standing waves as strong \(\ell^2(\mathbb Z)\)-limits of periodic constrained maximizers. Numerical computations for several interaction ranges and cooperative kernels illustrate the effects of extended coupling on dispersion and localization and support the periodic-to-homoclinic convergence predicted by the analysis.