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1. Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

   https://doi.org/10.1111/sapm.12651  

   Biondini, Gino; Chernyavsky, Alexander (February 2024, Studies in Applied Mathematics)

   Abstract We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability. 

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2. Local well-posedness of dispersive equations with quasi-periodic initial data

   https://doi.org/10.1090/proc/17110  

   Papenburg, Hagen (July 2025, Proceedings of the American Mathematical Society)

   We prove unconditional local well-posedness in a space of quasi-periodic functions for dispersive equations of the form ${\mathrm{\partial <\#comment/>}}_{t}u+Lu+{\mathrm{\partial <\#comment/>}}_x\left(u^{p+1}\right)=0,$ where $L$ is a multiplier operator with purely imaginary symbol which grows at most exponentially. The class of equations to which our method applies includes the generalized Korteweg-de Vries equation, the generalized Benjamin-Ono equation, and the derivative nonlinear Schrödinger equation. We also discuss well-posedness of some dispersive models which do not have a problematic derivative in the nonlinearity, namely, the nonlinear Schrödinger equation and the generalized Benjamin-Bona-Mahony equation, with quasi-periodic initial data. In this way, we recover and improve upon results from Damanik and Goldstein [J. Amer. Math. Soc. 29 (2016), pp. 825–856], Damanik et al. [J. Math Pures Appl. (9) 186 (2024), pp. 251–302], and Damanik et al. [J. Funct. Anal. 286 (2024), No. 110238] by shorter arguments. 

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3. 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. 

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4. Whitham modulation equations for the regularized Boussinesq equation with cubic nonlinearity

   https://doi.org/10.1017/jnw.2026.10042  

   Hoefer, Mark A; Vainchtein, Anna (January 2026, Journal of Nonlinear Waves)

   A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi–Pasta–Ulam lattice with a general cubic interaction force. Explicit periodic travelling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic travelling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic velocities is shown to lead to modulational instability of periodic travelling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic travelling waves and initial value problems that also reveal additional short-wavelength instabilities. 

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5. 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. 

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