Slow Modulations of Periodic Waves in Hamiltonian PDEs, with Application to Capillary Fluids

Abstract

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 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.

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

“…The parameterν > 0 is the viscosity and D(ρ, v) is a second order differential operator. We mention that the inviscid version of system (2.1) for certain classes of the operators [D(ρ, v)] x is sometimes called the Euler-Korteweg system [6,7]. The linear dispersion relation for system (2.1) has the form The key property of the dispersion relation we need is determined by the longwave expansion of ω 0 (k, ρ 0 ), which we assume generically has the form…”

Co-periodic stability of periodic waves in some Hamiltonian PDEs

“…For (qKdV), this matrix reads Remark 10. This testing against explicit formulas incidentally enabled us to find out that the numerical computations displayed in [BGNR14] were, to some extent, corrupted, because of inconsistent choices for the finite difference step ∆ν and for the integration step size ∆ω -in fact, ∆ν was taken too small.…”

“…As a matter of fact, the theory of linear stability of periodic waves under 'localized' perturbations -that is, perturbations going to zero at infinity -is still in its infancy (see for instance [BD09,GH07b,BDN11] as regards spectral stability for KdV and the cubic NLS, and [Rod15] for asymptotic linear stability of KdV waves), and the nonlinear stability under such perturbations is an open problem. In [BGNR14,BGNR13], the authors have contributed to the field by exhibiting several necessary conditions for the spectral stability of periodic waves in Hamiltonian PDEs. In particular, they have proved in a rather general setting that the hyperbolicity of the modulated equations 'à la Whitham' is necessary for the spectral stability of the underlying wave.…”

The Korteweg capillarity system. Integrable reduction via gauge and reciprocal links

“…[57,58] and literature cited therein). It may be set in the more general context of the isothermal capillarity system derived 'ab initio' by Antanovskii in [2], viz…”