标题： 离散非线性薛定谔方程中的坝体破裂 显示英文标题

标题： Dam breaks in the discrete nonlinear Schrödinger equation

摘要： 在本工作中，我们研究了色散冲击波（DSW）在{焦散}离散非线性薛定谔方程（DNLS）中的成核现象，这是一个在非线性光学和原子凝聚体中具有广泛相关性的模型。 在这里，我们研究了所谓的水坝破裂问题的动力学，其初始数据由两个参数表征，其中一个对应于晶格间距，另一个是右边的流体力学背景。 我们的分析连接了耦合强度趋近于零的反连续极限与已建立的连续可积模型。 为了阐明极端极限之间的过渡，我们理论部署了 Whitham 调制理论、DNLS 的各种准连续渐近约化以及存在性和稳定性分析，并将我们的发现与系统的数值计算联系起来。 我们的工作揭示了在离散化中存在一个尖锐的阈值，在此阈值下观察到了来自水坝破裂的定性连续动力学。 此外，我们在小耦合极限下观察到了丰富的波图案，包括非定常（和定常）的 Whitham 冲击波、行进的 DSW、离散 NLS 孤子和暗孤子波等。 此外，我们发现了 DSW 破裂现象以及随后多相波列的形成，这是由于\textit{两相}波列的广义调制不稳定性引起的。 我们认为这项工作是深入研究在不同维度和不同非线性情况下 DNLS 模型中明显丰富的 DSW 现象的一个起点。 显示英文摘要

摘要： In the present work we study the nucleation of Dispersive shock waves (DSW) in the {defocusing}, discrete nonlinear Schr{\"o}dinger equation (DNLS), a model of wide relevance to nonlinear optics and atomic condensates. Here, we study the dynamics of so-called dam break problems with step-initial data characterized by two-parameters, one of which corresponds to the lattice spacing, while the other being the right hydrodynamic background. Our analysis bridges the anti-continuum limit of vanishing coupling strength with the well-established continuum integrable one. To shed light on the transition between the extreme limits, we theoretically deploy Whitham modulation theory, various quasi-continuum asymptotic reductions of the DNLS and existence and stability analysis and connect our findings with systematic numerical computations. Our work unveils a sharp threshold in the discretization across which qualitatively continuum dynamics from the dam breaks are observed. Furthermore, we observe a rich multitude of wave patterns in the small coupling limit including unsteady (and stationary) Whitham shocks, traveling DSWs, discrete NLS kinks and dark solitary waves, among others. Besides, we uncover the phenomena of DSW breakdown and the subsequent formation of multi-phase wavetrains, due to generalized modulational instability of \textit{two-phase} wavetrains. We envision this work as a starting point towards a deeper dive into the apparently rich DSW phenomenology in a wide class of DNLS models across different dimensions and for different nonlinearities.