2020/08/23 by Enrique Alarcón Álvarez, Alvarez, Enrique, Ramón G. Plaza +1
Mathematics · #35B10 #35B35 #35C07 #35K55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2008.10120
openalex publication_date 2020/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper studies both existence and spectral stability properties of\nbounded spatially periodic traveling wave solutions to a large class of scalar\nviscous balance laws in one space dimension with a reaction function of\nmonostable or Fisher-KPP type. Under suitable structural assumptions, it is\nshown that this class of equations underlies two families of periodic waves.\nThe first family consists of small amplitude waves with finite fundamental\nperiod which emerge from a Hopf bifurcation around a critical value of the wave\nspeed. The second family pertains to arbitrarily large period waves which arise\nfrom a homoclinic bifurcation and tend to a limiting traveling (homoclinic)\npulse when their fundamental period tends to infinity. For both families, it is\nshown that the Floquet (continuous) spectrum of the linearization around the\nperiodic waves intersects the unstable half plane of complex values with\npositive real part, a property known as spectral instability. For that purpose,\nin the case of small-amplitude waves it is proved that the spectrum of the\nlinearized operator around the wave can be approximated by that of a constant\ncoefficient operator around the zero solution and determined by a dispersion\nrelation which intersects the unstable complex half plane. In the case of large\nperiod waves, we verify that the family satisfies the assumptions of the\nseminal result by Gardner (1997, J. Reine Angew. Math. 491, pp. 149-181) of\nconvergence of periodic spectra in the infinite-period limit to that of the\nunderlying homoclinic wave, which is unstable. A few examples are discussed.\n