Dispersive shock waves in periodic lattices

TL;DR

The paper addresses dispersive shock waves in nonlinear Schrödinger systems with periodic potentials by deploying a Wannier-based tight-binding reduction to a DNLS model, enabling analysis of generalized Riemann problems in deep lattices. Through modulation theory and full-dispersion DNLS reductions, it characterizes one-phase and two-phase wave dynamics, derives DSW edge speeds via DSW-fitting, and connects weakly nonlinear regimes to KdV-type reductions. The study reveals a rich phenomenology, including counterpropagating rarefaction waves, right-propagating DSWs, traveling DSWs, modulational-instability–driven breaks, high-genus oscillations, and breathing structures, with tight-binding models—especially DNLS-2—showing strong fidelity to continuum dynamics when higher-band effects are small. The findings provide a practical, scalable framework for non-convex dispersive hydrodynamics in periodic media with potential experimental relevance in optics and ultracold gases, while outlining directions for vector DNLS extensions to capture interband energy transfer.

Abstract

We introduce and systematically investigate the generation of dispersive shock waves, which arise naturally in physical settings such as optical waveguide arrays and superfluids confined within optical lattices. The underlying physically relevant model is a nonlinear Schrödinger (NLS) equation with a periodic potential. We consider the evolution of piecewise smooth initial data composed of two distinct nonlinear periodic eigenmodes. To begin interpreting the resulting wave dynamics, we employ the tight-binding approximation, reducing the continuous system to a discrete NLS (DNLS) model with piecewise constant initial data (i.e., a Riemann problem), where each constant state represents a discrete Floquet-Bloch mode at the continuum model level. The resulting tight-binding approximation is shown to display higher-fidelity for {deeper} periodic potentials. This reduced DNLS model effectively models the dynamics at the minima of the periodic potential of the original continuum NLS. Within such a single-band DNLS framework, we apply tools from Whitham modulation theory and long-wave quasi-continuum reductions to uncover and analyze a rich spectrum of non-convex, discrete dispersive hydrodynamic phenomena, comparing the resulting phenomenology with that of the periodic-potential-bearing continuum model.

Figures (8)

• Figure 1: (a) The band structure for the linear Schrödinger equation with potential $12\sin^2(x)$, displaying the first three bands shown in the first Brillouin zone, (b) Zoom on the lowest eigenvalue band showing the change in curvature (diffraction sign) between the band edge at $k=0$ and the edge points $k=\pm1$.
• Figure 2: The (left) amplitude and (right) wavemean variations of the nonlinear eigenmodes of Eq. \ref{['eq: Nonlinear steady-state problem']} as a function of the eigenvalue $\mu$, for $V_0=12$.
• Figure 3: The periodic eigenmode ($u(x)$) with eigenvalue $\mu=3.17$ constructed by superposing first-band Wannier functions for $V_0=12$. The periodic state is shown with blue solid color, while the scaled and shifted Wannier basis functions are shown in red-dashed line (A). (B) A single Wannier basis function centered at $n=0$. For $V_0=12\gg1$, the basis function is nearly Gaussian-like Alfimov_2002.
• Figure 4: The wavepattern at $t=1000$ arising from a Riemann problem ($\psi(x,0)=\sqrt{2},\;x<0$ and $\psi(x,0)=1,\;x>0$) posed to the continuum, bulk NLS (Eq. \ref{['eq: Periodic dNLS']} with $V(x)\equiv 0$). Here, a counter-propagating rarefaction wave and a DSW structure develop across a continuously expanding plane-wave background, facilitating the equilibration of hydrodynamic pressure across the initial step in wave amplitude $|\psi|$.
• Figure 5: A catalog of dam-break results at $t=1000$, for fixed $\mu_{+}$, while varying $\mu_{-}<3.19$, illustrating the generation of a radiating, right-propagating DSW for $V_0=12$. The dam break at $t=0$ (location of the initial step) occurs at $x_n=nL=301\pi$. Within each of the 4 panels, there are sub-panels showing the dynamics at the continuum level (left) and discrete level, (i.e. at the locations coinciding with the potential minima) (right). The results from the continuum NLS simulations in both sub-panels (Eq. \ref{['eq: Periodic dNLS']}) have been shown in blue solid line, with its tight-binding approximation (DNLS, Eq. \ref{['system-wanner']}) in red dashed line. As $\mu_{-}$ increases, there is an eventual disagreement, particularly at the solitonic edge (B,C), which approaches the vacuum point ($|\psi|=0$). Moreover, in panel (D), while the NLS and its tight-binding approximation show reasonable agreement, the DSW exhibits prominent leftward radiation, possibly due to significant higher-order dispersive effects.