The effect of a small loss or gain in the periodic NLS anomalous wave dynamics. I

The focusing Nonlinear Schr"odinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. Using the finite gap method, two of us (PGG and PMS) have recently solved, to leading order and in terms of elementary functions of the initial data, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number of unstable modes. In this paper, concentrating on the simplest case of a single unstable mode, we study the periodic Cauchy problem of the AWs for the NLS equation perturbed by a linear loss or gain term. Using the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic model describing quantitatively how the solution evolves, after a suitable transient, into slowly varying lower dimensional patterns (attractors) in the $(x,t)$ plane, characterized by $Delta X=L/2$ in the case of loss, and by $Delta X=0$ in the case of gain, where $Delta X$ is the $x$-shift of the position of the AW during the recurrence, and $L$ is the period. This process is described, to leading order, in terms of elementary functions of the initial data. Since dissipation can hardly be avoided in all natural phenomena involving AWs, and since a small dissipation induces $O(1)$ effects on the periodic AW dynamics, generating the slowly varying loss/gain attractors analytically described in this paper, we expect that these attractors, together with their generalizations corresponding to more unstable modes, will play a basic role in the theory of periodic AWs in nature.