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

本文的目的是证明，对于一大类称为标量粘性平衡定律的非线性演化方程，周期性行波解的谱（线性）不稳定性条件意味着它们在适当的周期性 Sobolev 空间中的轨道（非线性）不稳定性。该分析基于适定性理论、数据解图的平滑性以及非线性迭代下平衡不稳定性的抽象结果。由此产生的不稳定性准则应用于两个周期波族。第一族由具有有限基本周期的小振幅波组成，这些波从速度临界值附近的局部 Hopf 分岔中出现。第二类包括任意大的周期波，这些周期波由同宿（全局）分岔产生，并且当其基本周期趋于无穷大时趋向于限制行进脉冲。对于这两个族，应用该准则得出它们在与波的基本周期相同的周期Sobolev空间中的非线性粘性平衡定律流动下的轨道不稳定性。