Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

Abstract

Abstract. We study the stability and instability of periodic traveling waves for Korteweg-de Vries type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations, provided that the associated linearized operator enjoys a Jordan block structure. We then discuss when the linearized equation admits solutions exponentially growing in time.

Modulational Instability and Variational Structure

“…Instead, we rely upon a Bloch wave decomposition of the related spectral problem. Lin [28] devised a continuation argument and extended the first index to solitary waves for a class of nonlinear, nonlocal equations; see [33], for instance, for an adaptation in the periodic wave setting. It is standard from Floquet theory (see [34], for instance) that any eigenfunction of (10) takes the form…”

“…Under Assumptions 2 and 5, we shall take the approach in Section 2 and determine its spectral instability near the origin to long wavelengths perturbations. In particular, we shall calculate the modulational instability index , defined in (33), in terms of U , P, M as functions of c and a. (Note that, x 0 ∈ R and T > 0 are arbitrary.)…”

“…To summarize, the modulational instability index, defined in (33), at a periodic traveling wave of (38) is found as the discriminant of the cubic polynomial…”

“…In particular, we follow the approach in Section 2 and Section 3.3 and we calculate the modulational instability index , defined in (33), in terms of inner products between v 1 = u a , w 1…”

New variational characterization of periodic waves in the fractional Korteweg–de Vries equation

“…This criterion corresponds to the criterion for stability of solitary waves [9,26,29,36]. Note that this scalar criterion obtained from the new variational characterization of periodic waves replaces computations of a 2 × 2 matrix needed to establish if the periodic wave is a constrained minimizer of energy subject to fixed momentum and mass as in [24]. In particular, the sharp criterion based on the sign of b ′ (c 0 ) works equally well in the cases when the linearized operator L has one or two negative eigenvalues, see Remark 4.3.…”