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A unified method for analyzing the dynamics and topological structure associated with a class of Floquet topological insulators is presented. The method is applied to a system that describes the propagation of electromagnetic waves through the bulk of a two-dimensional lattice that is helically driven in the direction of propagation. Tight-binding approximations are employed to derive reduced dynamical systems. Further asymptotic approximations, valid in the high-frequency driving regime, yield a time-averaged system which governs the leading order behavior of the wave. From this follows an analytic calculation of the Berry connection, curvature, and Chern number via analyzing the local behavior of the eigenfunctions near the critical points of the spectrum. Examples include honeycomb, Lieb, kagome, and 1/5-depleted lattices. In the nonlinear regime, novel equations governing slowly varying wave envelopes are derived. For the honeycomb lattice, numerical simulations show that for relatively small nonlinear effects a striking spiral pattern occurs; as nonlinearity increases, localized structures emerge, and for somewhat higher nonlinearity the waves appear to collapse 

Abstract The measured ages of massive, quiescent galaxies atz∼ 3–4 imply that massive galaxies quench as early asz∼ 6. While the number of spectroscopic confirmations of quiescent galaxies atz< 3 has increased over the years, there are only a handful atz> 3.5. We report spectroscopic redshifts of one secure (z= 3.757) and two tentative (z= 3.336 andz= 4.673) massive ( $\log \left(M_{*}/M_{\odot }\right)>10.3$) quiescent galaxies with 11 hr of Keck/MOSFIREK-band observations. Our candidates were selected from the FLAMINGOS-2 Extragalactic Near-InfraredK-band Split (FENIKS) survey, which uses deep Gemini/Flamingos-2K_bK_rimaging optimized for increased sensitivity to the characteristic red colors of galaxies atz> 3 with a strong Balmer/4000 Å break. The rest-frameUVJand (ugi)_scolors of three out of four quiescent candidates are consistent with 1–2 Gyr old stellar populations. This places these galaxies as the oldest objects at these redshifts, and challenges the notion that quiescent galaxies atz> 3 are all recently quenched, post-starburst galaxies. Our spectroscopy shows that the other quiescent-galaxy candidate is a broad-line active galactic nucleus (z= 3.594) with strong, redshifted Hβ+ [OIII] emission with a velocity offset > 1000 km s⁻¹, indicative of a powerful outflow. The star formation history of our highest redshift candidate suggests that its progenitor was already in place byz∼ 7–11, reaching ∼10¹¹M_⊙byz≃ 8. These observations reveal the limit of what is possible with deep near-infrared photometry and targeted spectroscopy from the ground and demonstrate that secure spectroscopic confirmation of quiescent galaxies atz> 4 is feasible only with JWST. 

Abstract Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)‐dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann‐type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one‐dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations. 

Abstract The mathematical description of localized solitons in the presence of large‐scale waves is a fundamental problem in nonlinear science, with applications in fluid dynamics, nonlinear optics, and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg–de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A topical review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton–rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite‐gap description are used to describe soliton–rarefaction wave and soliton–dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the inverse scattering transform. For transmitted solitons, far‐field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave–mean field interaction. 

The semi-classical Korteweg–de Vries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis, Whitham theory is constructed to the higher order. This allows the order one phase and the complete leading-order solution to be obtained; the results are confirmed by extensive numerical calculations. 

Abstract Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose‐Einstein condensates, water waves, and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one‐dimensional NLS equation exhibits significant qualitative and quantitative differences.