发表机构
Xiamen University（厦门大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为MMT波模型、FPUTβ链和DNLS晶格推导出精确非微扰平衡态模式统计，验证其从弱到强非线性均有效，并揭示微扰法在准凝聚附近的失效机制。
AI 中文摘要
我们为三个代表性非线性系统推导了平衡态模式占据数及相关统计量的精确有限尺寸非微扰表示：Majda-McLaughlin-Tabak色散波模型、Fermi-Pasta-Ulam-Tsingou beta非谐晶格以及离散非线性薛定谔（DNLS）晶格场。独立模拟证实了从弱非线性到强非线性范围内的预测。对于DNLS，该理论在弱耦合准凝聚交叉区域仍然准确，在该区域，即使裸非线性系数很小，大的低模式占据数和长程相干性也会放大相互作用效应。有限环DNLS占据数进一步分解为具有不同关联长度的瑞利-金斯（Rayleigh-Jeans）通道的正和，这解释了何时单一瑞利-金斯定律适用以及为何在准凝聚附近失效。在MMT和DNLS中，精确占据数还决定了平均模式频率，即使动力学谱展宽或分裂。非微扰结果允许直接评估两种代表性微扰方法。适当处理平均相互作用可产生准确的低阶近似，包括在强非线性下。然而，在更高阶，修正项不再减小，连续近似以增大的振幅振荡；两种有限阶方法在弱耦合准凝聚附近也失效。因此，低阶一致性或小的裸耦合都不能保证可靠的微扰描述。这些结果为广泛使用的非线性波和晶格模型建立了非微扰平衡态理论，并为非线性光学、色散波、非谐晶格和冷原子系统中能量、粒子和光功率的模式分布提供了定量基础。
英文摘要
We derive exact finite-size nonperturbative representations of equilibrium modal occupations and related statistics for three representative nonlinear systems: the Majda-McLaughlin-Tabak dispersive-wave model, the Fermi-Pasta-Ulam-Tsingou beta anharmonic chain, and the discrete nonlinear Schrodinger lattice field. Independent simulations confirm the predictions from weak to strong nonlinearity. For DNLS, the theory remains accurate across the weak-coupling quasicondensation crossover, where large low-mode occupations and long-range coherence amplify interaction effects even when the bare nonlinear coefficient is small. The finite-ring DNLS occupations are further resolved into a positive sum of Rayleigh-Jeans channels with distinct correlation lengths, explaining when a single Rayleigh-Jeans law applies and why it fails near quasicondensation. In MMT and DNLS, the exact occupations also determine the mean modal frequencies even when the dynamical spectra broaden or split. The nonperturbative results allow a direct assessment of two representative perturbative approaches. Treating the mean interaction appropriately yields accurate low-order approximations, including at strong nonlinearity. At higher orders, however, the corrections cease to decrease and successive approximations oscillate with increasing amplitude; both finite-order approaches also fail near weak-coupling quasicondensation. Thus neither low-order agreement nor a small bare coupling guarantees a reliable perturbative description. The results establish nonperturbative equilibrium theory for widely used nonlinear wave and lattice models and provide a quantitative basis for modal distributions of energy, particles, and optical power in nonlinear optics, dispersive waves, anharmonic lattices, and cold-atom systems.
Comments47 pages, 15 figures
