Rogue waves in extended Gross-Pitaevskii Models with a Lee-Huang-Yang correction

TL;DR

This work addresses rogue waves in a one-dimensional extended Gross-Pitaevskii equation with Lee-Huang-Yang corrections modeling quantum droplets. A space-time fixed-point method with a doubly periodic basis is used to compute spatiotemporal rogue waves and to explore their continuation in the chemical potential, revealing two RW families: some regimes closely mimic the NLS Peregrine soliton while others exhibit notable deviations due to nonintegrability and competing nonlinearities. Two dynamical generation protocols are demonstrated, namely interfering dam-break flows and gradient catastrophe of localized waveforms, each yielding a spectrum of RW structures including PS-like, HORW, and modulated RW lattices. The results offer insight into RW phenomenology in nonintegrable media with competing interactions and point to feasible experimental observation in ultracold atomic droplets, with implications for nonlinear wave control in related fields.

Abstract

We explore the existence and dynamical generation of rogue waves (RWs) within a one dimensional quantum droplet bearing environment. RWs are computed by deploying a spacetime fixed point scheme to the relevant extended Gross Pitaevskii equation (eGPE). Parametric regions where the ensuing RWs are different from their counterparts in the nonlinear Schroedinger equation are identified. To corroborate the controllable generation, relevant to ultracold atom experiments, of these rogue patterns we exploit two different protocols. The first is based on interfering dam break flows emanating from Riemann initial conditions and the second refers to the gradient catastrophe of a spatially localized waveform. A multitude of possible RWs are found in this system, spanning waveforms reminiscent of the Peregrine soliton, its spatially periodic variants, namely, the Akhmediev breathers, and other higher order RW solutions of the nonlinear Schroedinger equation. Key elements of the shape of the corresponding eGPE RWs traced back to nonintegrability and the presence of competing interactions are discussed. Our results set the stage for probing a multitude of unexplored rogue like waveforms in such mixtures with competing interactions and should be accessible to current ultracold atom experiments.

Figures (7)

• Figure 1: Representative eGPE RW moduli profiles, $|\psi(x,t)|$. (a) Space-time profile of a rogue waveform for $\mu_0\approx -0.12$, and (b) a second-order rogue waveform characterized by $\mu_0\approx-0.02$. (c) [(d)] Snapshot of the eGPE [second-order] rogue waveform in (a) [(b)] at $\tilde{t}=0$ compared to a cubic GPE PS [second-order RW] possessing the same peak amplitude. Parameters used for the PS [HORW] are $L=30$ and $T=190$ [$L=60$ and $T=290$].
• Figure 2: RW families of the eGPE. The $||\phi||_2$ norm of the spatiotemporally identified fixed point $\phi$ starting from (a) an NLS-like PS (first family) and (h) a sech-like waveform (second family) as a function of the chemical potential $\mu_0$. Inset in (h) depicts a magnification of $||\phi||_2$ at $\mu_0\approx-0.1935$, see the green rectangle. Representative waveforms $|\phi(x,t)|$ of the upper (lower) solution branch (i.e., curve) depicted in panel (a) with pink (purple) are shown in panels (b) -(d) ((e)-(g)). Here, panel (b) [(d)] depicts a configuration with $\mu_0\approx-0.1009$ [$\mu_0\approx -0.2$] and panel (e) [(g)] refers to $\mu_0\approx-0.2$ [$\mu_0\approx-0.09$]. We remark that the purple line (hardly visible) near the turning point at $\mu_0\approx -0.2$ practically coincides with the pink. Note the progressive deformation from PS-like structures belonging to the upper branch towards wider RW patterns as $\mu_0$ decreases, and from multi-hump structures (e.g., in (f)) towards PS-like entities (in (g)) in the lower branch as $\mu_0$ increases. In the second RW family, panel (h), we observe a bifurcation from a homogeneous background in (i) to an NLS periodic wave in (j) and gradually a PS-like solution for decreasing $\mu_0$ in (k). Multi-hump RWs occur for further decreasing values of $\mu_0$ (panels (l)-(n)). Vertical dashed lines indicate the values of $\mu_0$ for the solutions depicted in panels (b)-(g) and (i)-(n).
• Figure 3: Spatio-temporal evolution of $|\psi(x,t)|$ of (a) a single-humped (Fig. \ref{['fig:red-family']}(c)) and (b) a multi-humped (Fig. \ref{['fig:red-family']}(l)) RW for $\mu_0 \approx -0.19$ and $\mu_0 \approx -0.1$ respectively. Destabilization of the single-humped RW beyond 2 time-periods is evident, while longevity can be inferred for the multi-hump RW.
• Figure 4: (a) Interfering DBFs leading to modulated RW lattice and PS generation for $\rho_0=0.0125$. (b) Snapshot of $|\psi|$ revealing the formation of an eGPE RW compared to a numerically obtained PS-like wave with $\mu_0\approx -0.0689$. (c) Relevant profile demonstrating a RW lattice compared to a computed eGPE one having $\mu_0\approx -0.085$. The central breather is also compared to a computed eGPE PS at $\mu_0\approx -0.0767$. (d) A structure reminiscent of a higher order solitonic waveform, and (e) a temporally modulated, persistent quantum droplet appears at the center, accompanied by symmetrically emitted matter wave jets, with the central and the lateral structures fitted to the analytical droplet profile with $\mu_0\approx-0.1518$ and $\mu_0\approx -0.087$ respectively. Vertical dashed lines in panel (a) indicate the times used for the profiles shown in panels (b)-(e).
• Figure 5: (a) Interfering DBFs for $\rho_0=0.0875$ leading to RW generation. (b) Snapshot of $|\psi|$ features an eGPE RW compared with the computed rogue waveform shown in Fig. \ref{['fig:red-family']}(a) having $\mu_0\approx -0.1713$. (c) A structure reminiscent of a HORW. (d) Snapshot illustrates a bimodal breathing state, which subsequently merges into (e) a larger amplitude structure centered at $x=0$. Vertical dashed lines in panel (a) mark the time-instants presented in panels (b)-(e).