1 research outputs found
On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
In this paper, we complement recent results of Bronski and Johnson and of
Johnson and Zumbrun concerning the modulational stability of spatially periodic
traveling wave solutions of the generalized Korteweg-de Vries equation. In this
previous work it was shown by rigorous Evans function calculations that the
formal slow modulation approximation resulting in the Whitham system accurately
describes the spectral stability to long wavelength perturbations. Here, we
reproduce this result without reference to the Evans function by using direct
Bloch-expansion methods and spectral perturbation analysis. This approach has
the advantage of applying also in the more general multi-periodic setting where
no conveniently computable Evans function is yet devised. In particular, we
complement the picture of modulational stability described by Bronski and
Johnson by analyzing the projectors onto the total eigenspace bifurcating from
the origin in a neighborhood of the origin and zero Floquet parameter. We show
the resulting linear system is equivalent, to leading order and up to
conjugation, to the Whitham system and that, consequently, the characteristic
polynomial of this system agrees (to leading order) with the linearized
dispersion relation derived through Evans function calculation.Comment: 19 pages