On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions

TL;DR

This paper demonstrates that Bloch-expansion methods can effectively analyze the modulational stability of periodic solutions in the generalized KdV equation, aligning with previous Evans function results but applicable in more general settings.

Contribution

It introduces a Bloch-expansion approach to study spectral stability, extending analysis to multi-periodic cases where Evans functions are not available.

Findings

Bloch-expansion methods reproduce Evans function stability results.

The linearized system near the origin matches the Whitham system.

Characteristic polynomial aligns with the linearized dispersion relation.

Abstract

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.