合作有限范围DNLS方程的锥不变性与同宿驻波
Cone Invariance and Homoclinic Standing Waves for Cooperative Finite-Range DNLS Equations
AI总结：
该研究针对合作有限范围DNLS方程，构建受限功率变分框架，利用投影梯度流建立特定构型锥的不变性，结合严格极大条件得到指数局域化同宿驻波，数值计算验证了相关收敛性。
摘要：
针对具有合作有限范围相互作用的一维离散非线性薛定谔方程（DNLS）中的正同宿驻波，我们构建了受限功率变分框架。从周期晶格出发，我们构造了受限能量极大值点，并利用相关的投影梯度流建立了非负、偶对称且单峰构型的锥的正不变性。核心结果是，相互作用系数的单调排序\na₁≥a₂≥…≥a_M≥0（其中a₁>0）提供了确保该锥在有限范围耦合下保持不变的结构条件。在线性谱边缘之上的严格极大条件下，我们得到了一致的频率间隙和与周期无关的指数局域化。这些估计为无限周期极限提供了所需的紧性，并产生了正、偶对称、单峰且指数局域化的同宿驻波，它们是周期受限极大值点的强ℓ²(ℤ)极限。对多种相互作用范围和合作核的数值计算，阐明了扩展耦合对色散和局域化的影响，并验证了分析所预测的周期到同宿的收敛性。
英文摘要：
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.
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