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We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the L2(R)-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number is greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization. 

In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial translations) of smooth traveling wave solu- tions, and their stability criteria are expressed in terms of Jacobians of the conserved quantities with respect to these parameters. The stability criteria utilizes a general Hamiltonian structure which exists for every b > 1, and hence applies outside of the completely integrable cases (b = 2 and b = 3). 

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.