Modulational stability of periodic waves

We discuss the stability of one-dimensional stationary periodic solutions of partial differential equations. A particularity of this problem is that stability results strongly depend upon the class of allowed perturbations. These perturbations may be periodic, with period which is either the same or an integer multiple of the period of the stationary solution, or they may be localized. While we briefly outline the differences and techniques for spectral and linear stability, we focus on the nonlinear stability of these periodic solutions. As an example, we analyze the stability of periodic solutions of the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics.