The instability of the Hocking–Stewartson pulse and its geometric phase in the Hopf bundle

Abstract

This work demonstrates an innovative numerical method for counting and locating eigenvalues with the Evans function. Utilizing the geometric phase in the Hopf bundle, the technique calculates the winding of the Evans function about a contour in the spectral plane, describing the eigenvalues enclosed by the contour for the Hocking–Stewartson pulse of the complex Ginzburg–Landau equation. Locating eigenvalues with the geometric phase in the Hopf bundle was proposed by Way (2009), and proven by Grudzien et al. (2016). Way demonstrated his proposed method for the Hocking–Stewartson pulse, and this manuscript redevelops this example as in the proof of the method in Grudzien et al. (2016), modifying his numerical shooting argument, and introduces new numerical results concerning the phase transition.