Dam breaks in the discrete nonlinear Schr\"odinger equation

Abstract

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.