Description
We consider the cubic nonlinear Schrödinger (NLS) equation with a linear damping on the one dimensional torus and we investigate the stability of some solitary wave profiles within the dissipative dynamics. The undamped cubic NLS equation is well known to admit a family of periodic waves given by Jacobi elliptic functions of cnoidal type. We show that the family of cnoidal waves is orbitally stable. More precisely, by considering a sufficiently small perturbation of a given cnoidal wave at initial time, the evolution will always remain close (up to symmetries of the equation) to the cnoidal wave whose mass is modulated according to the dissipative dynamics.
Since cnoidal waves are not exact solutions to the damped NLS, the perturbation is forced away from the family of solitary wave profiles. To achieve our result and control this secular growth, we construct an approximated solitary wave profile that embodies the first order effects of the damping. The perturbation around the approximated profile is controlled by means of a Lyapunov functional, for which we derive suitable decay estimates.
