Dam breaks in the discrete nonlinear Schrödinger equation

TL;DR

The paper addresses dispersive shock waves in the defocusing DNLS under dam-break initial data, using Whitham modulation theory and quasi-continuum reductions to connect anti-continuum and continuum limits. It reveals a sharp discretization threshold that separates continuum-like shock dynamics from highly discrete phenomena, and identifies a rich spectrum of wave patterns, including traveling DSWs, kinks, dark solitons, DSW breakdown, and two-phase modulational instabilities, all supported by extensive numerical simulations. The results are framed by reductions to KdV, KdV-5, and Kawahara equations, which illuminate how dispersion order and lattice spacing shape the shock structure and edge speeds, and they highlight two-phase resonances as a mechanism for DSW breakdown and multi-phase wavetrains. The work provides a foundation for deeper exploration of discrete dispersive hydrodynamics across DNLS models and dimensions, with potential applications in nonlinear optics and atomic condensates, and points to future directions on short-time instability dynamics and higher-genus wave states.

Abstract

In the present work we study the nucleation of Dispersive shock waves (DSW) in the {defocusing}, discrete nonlinear Schr{ö}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.

Figures (24)

• Figure 1: For $\beta=1.5>1/(2E^2(\pi/4,2))$ (Small-to-large coupling threshold), we observe continuum NLS-like-wave patterns, as shown in the panels (A) a rarefaction wave emanates from the dam break with $u_+=0$ and (B) a (left) right propagating (rarefaction) DSW emerges for $u_+=0.5$ respectively. Below the small-to-large coupling threshold (i.e., for $\beta=1$), we show snapshots of DNLS simulations. In the vacuum case ($u_+=0$), (C) besides the propagating rarefaction wave, a novel, right propagating traveling wave feature is seen to emerge (zoomed view in the inset). On the other hand, when $u_+=0.5$, (D) higher-order dispersive effects are witnessed which alter the (left) right propagating (rarefaction) DSW patterns. In all the panels, the DNLS simulations at $t=1000$ are shown in the blue dashed curve, with the $r^{(1)}$-simple wave of the one-phase modulation system overlaid in a red dashed line.
• Figure 2: Zero contour map (for various $\kappa$) of the coefficient of the leading-order dispersion ($\tilde{\alpha}_3(A,h;\kappa)$=0). Evidently, for a fixed $\kappa$ and sufficiently large $Ah\gg 1$, a negative leading order dispersion coefficient can be expected.
• Figure 3: Snapshots of small amplitude DNLS DSW emitted at $t=1000$ for $u_{+}=0.9$, as $\beta$ varies from $0.1$ to $0.6$. Here, in (a) $\beta=0.1$, and we obtain DSW in a regime with negative dispersion curvature $\tilde{\alpha}_3<0$. Thereafter, as $\beta$ is increased, in steps of $0.1$, we observe traveling DSW in (B) and (C) which correspond to $\beta=0.2$, $0.3$ respectively and crossover DSW for (D), corresponding to $\beta=0.4$ and finally DSW in a regime with positive dispersion curvature in (E), (F) [$\beta=0.5$, $0.6$ respectively], whose waveforms are significantly influenced by higher-order dispersive effects.
• Figure 4: (A) The dispersion coefficients in the long-wave expansion are evaluated at intermediate hydrodynamic background states ($|u_m|, \kappa_m$), obtained from a sequence of dam-break problems by fixing $u_{-}=1$, $u_{+}=0.9$ and varying the coupling strength $\beta$. For $\beta=0.1$, the third-order dispersion coefficient is larger in magnitude than the fifth-order term, indicating a regime dominated by negative third-order dispersion. However, near $\beta\approx0.2$, the third-order dispersion coefficient is close to zero signaling the onset of competing dispersive effects at third and fifth orders therein. Eventually for $\beta=0.6$, the third-order term dominates again—this time within a regime of positive dispersion. (B) The (appropriately scaled) third- and fifth-order dispersion coefficients.
• Figure 5: We compare the right propagating shock structures of the DNLS for $\beta=0.1,0.2,0.6$, $u_{+}=0.9$ (snapshot shown for $t=1000$) with corresponding counterparts of the quasi-continuum Kawahara/KdV5/KdV reduction depending on their asymptotic validity. The quasi-continuum reductions were derived by examining disturbances propagating on the intermediate background $u_m$ in each case: (A) the DNLS result compared to the Kawahara DSW (magenta) and KdV DSW (black) for $\beta=0.1$, the Kawahara one providing the better approximation, (B) the DNLS result corresponding to $\beta=0.2$ compared to the KdV5 (green) Kawahara traveling DSW (magenta), with the Kawahara providing the slightly closer approximation, (C) the DNLS result for $\beta=0.3$ compared to its Kawahara reduction (red). Here, the disagreements associated with the linear edge speed (and thus the velocity width) are more pronounced, (D) the DNLS result for $\beta=0.6$ compared to its corresponding KdV reduction. Besides the trailing radiation, good agreement is evident.