Effect of small loss or gain on the anomalous wave recurrence

Anomalous (rogue waves, freak waves) are extreme waves of anomalously large amplitude with respect to the surrounding waves, arising apparently from nowhere and disappearing without leaving any trace. Focusing Nonlinear Schrodinger equation is used as a simple basic model for generation of anomalous waves due to modulation instability in nonlinear systems. In the spatially-periodic setting anomalous waves are described by special finite-gap solutions corresponding to almost degenerate Riemann surfaces, therefore it is possible to construct explicit approximate solutions of the initial value problem. We also obtained analytic formulas describing the effect of small loss/gain in case of one unstable mode and show that also very small loss/gain essentially changes the recurrence law. These formulas explains some phenomena observed earlier in water tank experiments and numerical simulations.