Coloquio de Matemáticas Aplicadas
In this talk, I will address both the existence and the stability properties of bounded spatially periodic traveling wave solutions to a large class of scalar viscous balance laws in one space dimension with a reaction function of monostable type. It is shown that, under suitable assumptions, this class of equations underlies two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a Hopf bifurcation around a critical value of the wave speed. The second family pertains to arbitrarily large period waves which arise from a homoclinic bifurcation and tend to a limiting traveling (homoclinic) pulse when their fundamental period tends to infinity. For both families, it is shown that the Floquet (continuous) spectrum of the linearization around the periodic waves intersects the unstable half plane of complex values with positive real part, a property known as spectral instability. Furthermore, this spectral information is applied to prove that these periodic waves are orbitally (nonlinearly) unstable under the flow of the viscous balance law in periodic Sobolev spaces with the same period as the fundamental period of the wave. A few examples will be discussed. This is joint work with E. Álvarez (Math. Physics Dept., IIMAS-UNAM) and J. Angulo-Pava (Univ. of Sao Paulo).