2015/04/29 by Colin Grudzien, Grudzien, Colin
Computer Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Laser-Matter Interactions and Applications #Nonlinear Dynamics and Pattern Formation #Quantum optics and atomic interactions #math.DS
paper · pdf · doi:10.48550/arxiv.1504.07896
openalex publication_date 2015/04/29 · arxiv created 2016/06/29 · arxiv updated 2016/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, and proven by Grudzien, Bridges & Jones. Way demonstrated his proposed method for the Hocking-Stewartson pulse, and this manuscript redevelops this example as in the proof of the method, modifying his numerical shooting argument, and introduces new numerical results concerning the phase transition.