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The purpose of this paper is to prove that, for a large class of nonlinearevolution equations known as scalar viscous balance laws, the spectral (linear)instability condition of periodic traveling wave solutions implies theirorbital (nonlinear) instability in appropriate periodic Sobolev spaces. Theanalysis is based on the well-posedness theory, the smoothness of thedata-solution map, and an abstract result of instability of equilibria undernonlinear iterations. The resulting instability criterion is applied to twofamilies of periodic waves. The first family consists of small amplitude waveswith finite fundamental period which emerge from a local Hopf bifurcationaround a critical value of the velocity. The second family comprisesarbitrarily large period waves which arise from a homoclinic (global)bifurcation and tend to a limiting traveling pulse when their fundamentalperiod tends to infinity. In the case of both families, the criterion isapplied to conclude their orbital instability under the flow of the nonlinearviscous balance law in periodic Sobolev spaces with same period as thefundamental period of the wave.

Orbital instability of periodic waves for scalar viscous balance laws