We consider quasilinear generalizations of the Korteweg-de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. In particular, our framework includes hydrodynamic formulation of the nonlinear Schrödinger equations. The periodic waves we study exhibit on each period two pulses, one converging to a bright soliton and one converging to a dark soliton, when wavelength goes to infinity. We show that such waves, for sufficiently large periods, are spectrally unstable. To do so, we combine two approaches. The first one is to calculate the asymptotic expansion of the Hessian matrix of the action integral and concludes using arXiv:1505.01382 as in arXiv:1710.03936 . This shows instability when both limiting solitary waves are stable. The second approach studies the convergence of the spectrum when the period goes to infinity and is applied in remaining cases, when one of the solitary waves is unstable. To carry out the latter, we prove the convergence of an appropriate renormalization of the periodic Evans function as in arXiv:1802.02830 .
Because of the relatively large dimension of family of periodic traveling waves such stability/instability criteria typically fail to set a neat dichotomy. This is in strong contrast with what happens for solitary waves where for a large class of equations the family of waves with a fixed endstate is one-dimensional, parametrized by velocity, and the sign of the second-order derivative of the Boussinesq momentum determines the stability/instability. Yet, as proved in [BMR20], following [BMR16], one may expect a reduction of the complexity of periodic waves stability in some asymptotic regimes.
In [BMR16] the authors show some stability/instability criteria for periodic traveling waves of large classes of Hamiltonian PDEs. The goal of [BMR20] was to see what happened for those criteria in two different limits. The first one, the harmonic limit, considers periodic waves with small amplitude around a constant state. The second one, the soliton limit, studies periodic waves with large period that converge to a solitary wave. The outcome is that in the harmonic limit, periodic waves were always orbitally stable, and in the soliton limit, periodic waves share the same stability properties as their limiting soliton.
Here we are interested in periodic waves of large periods, asymptotically described on each periodic cell by the superposition of two distinct solitary waves sharing the same endstate. We will call such waves two-pulse periodic waves. In the end we show that they are always spectrally unstable.
For the classes of equations that we consider, wave profiles are in correspondence with trajectories of a two-dimensional Hamiltonian differential equation, hence with level sets of a two-dimensional function. As a consequence the three asymptotic regimes mentioned above, the two studied in [BMR20] and the one studied here, contain all the generic ways in which a family of periodic waves may end.
The instability we prove is at the spectral and linear level, and holds both under co-periodic and localized perturbations. To our knowledge, the conversion of spectral instability into nonlinear instability for periodic traveling waves is a largely open question for quasilinear dispersive PDEs like the ones we are considering. In the semilinear case however we expect nonlinear orbital instability under localized perturbations to follow from the analysis of [JLL19]. Yet we mention that based on studies for parabolic systems [JNRZ14], space-modulated stability and not orbital stability is expected to be the sharp notion of stability for periodic waves subject to localized perturbations. Incidentally we also stress that, even for the semilinear case, nonlinear stability with respect to localized perturbations for periodic waves of Hamiltonian systems is a largely open question. For a recent significant step in this direction see [BGdR25].
Half of our result is obtained by adapting to our present asymptotic regime the calculations of [BMR20]. In this way, we show that if the two solitons are orbitally stable then the corresponding two-pulse waves of sufficiently large period are spectrally unstable. Actually this part of the proof shows that this is the case when the sum of the second-order derivatives of the Boussinesq momenta is positive, which holds in particular when both are positive. At the basis of the result lies a co-periodic instability result [BMR16].
Let us mention that the other half of the instability result cannot be obtained by studying modulational instability. For an introduction to the latter we refer to [BHJ16]. The idea is quite natural and the authors of [BMR20] have also studied in their companion paper [BMR21] what happens in the same regimes to the modulational criteria derived in [BNR14a]. Nevertheless, as we develop in Remark 3.4.2, in our regime we show that when the arguments mentioned do not conclude to instability the modulational system is hyperbolic, hence the absence of modulational instability.
Instead, we complete our study with a direct spectral perturbation analysis. When doing so we draw inspiration from single-pulse soliton analyses in [Gar97,SS01,YZ19]. More precisely, adapting [YZ19] to our current regime, we show that a suitable rescaled version of the periodic Evans functions converges, at an exponential rate, to the product of the Evans functions of the two solitons. With this result in hands, we conclude that if at least one of the solitary waves is unstable then the corresponding two-pulse waves of sufficiently large period are spectrally unstable.
The situation analyzed in the latter part of the argument is probably easier to guess: the two solitary waves being superposed far away from each other, there is some room for solitary-wave instabilities to develop. The former situation, when both solitary waves are stable, is intuitively less clear. A possible heuristic scenario is that perturbations could change velocities of each solitary wave somewhat independently hence
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