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Register a new userStability of periodic waves in Hamiltonian PDEs
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Sylvie Benzoni-Gavage
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Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability -in fact, emph{orbital} stability, since we are dealing with space-invariant problems-, is far from being straightforward when the best spectral stability we can expect is a emph{neutral} one. Indeed, because of the Hamiltonian structure, the spectrum of the linearized equations cannot be bounded away from the imaginary axis, even if we manage to deal with the point zero, which is always present because of space invariance. Some other means make a crucial use of the underlying structure. This is clearly the case for the variational approach, which basically uses the Hamiltonian -or more precisely, a constrained functional associated with the Hamiltonian and with other conserved quantities- as a Lyapunov function. When it works, it is very powerful, since it gives a straight path to orbital stability. An alternative is the modulational approach, following the ideas developed by Whitham almost fifty years ago. The main purpose here is to point out a few results, for KdV-like equations and systems, that make the connection between these three approaches: spectral, variational, and modulational.
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Stability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature of the Hessian matrix of action integrals associated with those waves. In a previous paper (Nonlinearity 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes 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. They derived a sufficient condition for orbital stability with respect to co-periodic perturbations, and a necessary condition for spectral stability, both in terms of the negative signature - or Morse index - of the Hessian matrix of the action integral. Here the asymptotic behavior of this matrix is investigated in two asymptotic regimes, namely for small amplitude waves and for waves approaching a solitary wave as their wavelength goes to infinity. The special structure of the matrices involved in the expansions makes possible to actually compute the negative signature of the action Hessian both in the harmonic limit and in the soliton limit. As a consequence, it is found that nondegenerate small amplitude waves are orbitally stable with respect to co-periodic perturbations in this framework. For waves of long wavelength, the negative signature of the action Hessian is found to be exactly governed by the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave.
Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction-diffusion systems were first considered by Doelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], and viscous systems of conservation laws have been addressed by Johnson, Noble, Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipative models are considered, and a most basic question is investigated, namely the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg--de Vries equation, and the less known Euler--Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water waves theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre [Comm. Partial Differential Equations 2005], Oh and Zumbrun [Arch. Ration. Mech. Anal. 2003], and Johnson, Zumbrun and Bronski [Phys. D 2010]. In addition, reduced necessary conditions are obtained in the small amplitude limit. Then numerical investigations are carried out for the modulated equations of the Euler--Korteweg system with two types of pressure laws, namely the quadratic law of shallow water equations, and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.
We consider stability of non-rotating gaseous stars modeled by the Euler-Poisson system. Under general assumptions on the equation of states, we proved a turning point principle (TPP) that the stability of the stars is entirely determined by the mass-radius curve parameterized by the center density. In particular, the stability can only change at extrema (i.e. local maximum or minimum points) of the total mass. For very general equation of states, TPP implies that for increasing center density the stars are stable up to the first mass maximum and unstable beyond this point until next mass extremum (a minimum). Moreover, we get a precise counting of unstable modes and exponential trichotomy estimates for the linearized Euler-Poisson system. To prove these results, we develop a general framework of separable Hamiltonian PDEs. The general approach is flexible and can be used for many other problems including stability of rotating and magnetic stars, relativistic stars and galaxies.
Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from sub-critical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram-- specifically, determination of rigorous Eckhaus-type stability conditions-- remains an interesting open problem.
We study the phenomenon of revivals for the linear Schrodinger and Airy equations over a finite interval, by considering several types of non-periodic boundary conditions. In contrast with the case of the linear Schrodinger equation examined recently (which we develop further), we prove that, remarkably, the Airy equation does not generally exhibit revivals even for boundary conditions very close to periodic. We also describe a new, weaker form of revival phenomena, present in the case of certain Robin-type boundary conditions for the linear Schrodinger equation. In this weak revival, the dichotomy between the behaviour of the solution at rational and irrational times persists, but in contrast with the classical periodic case, the solution is not given by a finite superposition of copies of the initial condition.
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