离散非线性薛定谔模型中有限背景下时间准周期解的存在性、稳定性及时空动力学
Existence, stability and spatio-temporal dynamics of time-quasiperiodic solutions on a finite background in discrete nonlinear Schrödinger models
AI总结：
该研究探究离散非线性薛定谔模型在有限背景下的空间局域时间准周期解，证明其反连续极限下的存在性，通过数值延拓分析有限耦合下的存续性与谱稳定性，结合分岔图与数值模拟揭示其演化规律与动力学特性。
摘要：
在本研究中，我们探究了离散非线性薛定谔（DNLS，discrete nonlinear Schrödinger）型模型在有限背景之上支撑空间局域、时间准周期解的可能性。我们严格证明了这类解在模型的反连续、零耦合极限附近是存在的。随后我们采用数值延拓方法，说明这类解在有限耦合条件下仍能存续，并探究了其谱稳定性。我们得到了一幅复杂的分岔图，显示这类解从具有单点和双点激发的更简单形式，逐步发展为更复杂的多点激发形式，且自聚焦和自散焦非线性区域的解分支存在直接关联。我们进一步探究了在两种非线性符号下，所得解在频率趋于零极限时的变化情况。我们还通过直接数值模拟探究了解的动力学行为，对分析进行了补充。
英文摘要：
In the present work we explore the potential of models of the discrete nonlinear Schrödinger (DNLS) type to support spatially localized and temporally quasiperiodic solutions on top of a finite background. Such solutions are rigorously shown to exist in the vicinity of the anti-continuum, vanishing coupling limit of the model. We then use numerical continuation to illustrate their persistence for finite coupling, as well as to explore their spectral stability. We obtain an intricate bifurcation diagram showing a progression of such solutions from simpler ones bearing single- and two-site excitations to more complex, multi-site ones with a direct connection of the branches of the self-focusing and self-defocusing nonlinear regime. We further probe the variation of the solutions obtained towards the limit of vanishing frequency for both signs of the nonlinearity. Our analysis is complemented by exploring the dynamics of the solutions via direct numerical simulations.
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