More Like this

1. Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves

   https://doi.org/10.4171/aihpc/65  

   Haragus, Mariana; Johnson, Mathew A; Perkins, Wesley R; de_Rijk, Björn (July 2023, Annales de l'Institut Henri Poincaré C, Analyse non linéaire)

   We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato–Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinearoptics. So far, nonlinear stability of such solutions has only been established against co-periodicperturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbationswhich are localized, i.e., integrable on the line. Such localized perturbations naturally yieldthe absence of a spectral gap, so we must rely on a substantially different method with origins inthe stability analysis of periodic waves in reaction–diffusion systems. The relevant linear estimateshave been obtained in recent work by the first three authors through a delicate decomposition of theassociated linearized solution operator. Since its most critical part just decays diffusively, the nonlineariteration can only be closed if one allows for a spatio-temporal phase modulation. However,the modulated perturbation satisfies a quasilinear equation yielding an apparent loss of regularity.To overcome this obstacle, we incorporate tame estimates on the unmodulated perturbation, whichsatisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to closean independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLEagainst localized perturbations with precisely the same decay rates as predicted by the linear theory. 

   more » « less
2. Near invariance of quasi-energy spectrum of Floquet Hamiltonians

   https://doi.org/10.4171/jst/603  

   Sagiv, Amir; Weinstein, Michael I (April 2026, Journal of Spectral Theory)

   The spectral analysis of the unitary monodromy operator, associated with a time-periodically (parametrically) forced Schrödinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form H^{\varepsilon}(t)=H^{0} + \varepsilon^{a} W(\varepsilon^{a} t, -i\nabla) , whereH^{0}is an unperturbed autonomous Hamiltonian,a\geq 1, andW(T,\cdot)has a period ofT_{\mathrm{per}}>0. In particular, in the small\varepsilon>0regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with theH^{\varepsilon}(t)dynamics, and that of the autonomous (unforced) flow,\exp[-iH^{0} T_{\mathrm{per}}\varepsilon^{-a}]. We considerH^{0}which is spatially periodic on\R^{n}with respect to a lattice. Using the decomposition ofH^{0}andH^{\varepsilon}(t)into their actions on spaces (Floquet–Bloch fibers) of pseudo-periodic functions, we establish a spectral near-invariance property for the monodromy operator, when acting on data which are\varepsilon-localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by -emphband-limited (Floquet–Bloch) wavepackets; (iii) the envelope dynamics of such wavepackets is well approximated by an effective (homogenized) PDE; and (iii) an exact invariance property for band-limited Floquet–Bloch wavepackets, which follows from the effective dynamics. We apply our general results to a number of periodic Hamiltonians,H^{0}, of interest in the study of photonic and quantum materials. 

   more » « less
3. Experimental investigations of linear and nonlinear periodic travelling waves in a viscous fluid conduit

   https://doi.org/10.1017/jfm.2022.993  

   Mao, Yifeng; Hoefer, Mark A. (January 2023, Journal of Fluid Mechanics)

   Conduits generated by the buoyant dynamics between two miscible Stokes fluids with high viscosity contrast, a type of core–annular flow, exhibit a rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small- and large-amplitude periodic travelling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic travelling wave profiles. Measurements are favourably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber–frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is explained by incorporating a weak recirculating flow into the full two-Stokes flow model. When the boundary condition corresponds to the temporal profile of a nonlinear periodic travelling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics. 

   more » « less
4. Large-amplitude modulation of periodic traveling waves

   https://doi.org/10.3934/dcdss.2022070  

   Métivier, Guy; Zumbrun, Kevin (January 2022, Discrete and Continuous Dynamical Systems - S)

   We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit. 

   more » « less
5. Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field

   https://doi.org/10.1007/s00220-023-04771-0  

   Costin, Ovidiu; Costin, Rodica; Jauslin, Ian; Lebowitz, Joel L (September 2023, Communications in Mathematical Physics)

   Springer (Ed.)

   We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space x 0, the Schrödinger equation of the system is i∂t ψ = − 2 1 ∂x 2 ψ + (x)(U − E x cos ωt)ψ, t > 0, x ∈ R, where (x) is the Heaviside function and U > 0 is the effective confining potential (we choose units so that m = e = = 1). The amplitude E of the external electric field and the frequency ω are arbitrary. We prove existence and uniqueness of classical solutions of the Schrödinger equation for general initial conditions ψ(x, 0) = f (x), x ∈ R. When 2the initial condition is in L the evolution is unitary and the wave function goes to zero at any fixed x as t → ∞. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-L 2 initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all t > 0. For these initial conditions we show that the solution approaches in the large t limit a periodic state that satisfies an infinite set of equations formally derived, under the assumption that the solution is periodic by Faisal et al. (Phys Rev A 72:023412, 2005). Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior, as seen both analytically and numerically. It shows a steep increase in the current as the frequency passes a threshold value ω = ωc , with ωc depending on the strength of the electric field. For small E, ωc represents the threshold in the classical photoelectric effect, as described by Einstein’s theory. 

   more » « less