Abstract
Spatially periodic breather solutions (SPBs) of the nonlinear Schrodinger (NLS) equation, i.e., the heteroclinic orbits of unstable Stokes waves, are typically unstable. In this paper, we study the effects of dissipation on single-mode and multi-mode SPBs using a linearly damped NLS equation. The number of instabilities the background Stokes wave possesses and the damping strength are varied. Viewing the damped dynamics as near integrable, the perturbed flow is analyzed by appealing to the spectral theory of the NLS equation. A broad categorization of the routes to stability of the SPBs and how the route depends on the mode structure of the SPBs and the instabilities of the Stokes wave is obtained as well as the distinguishing features of the damped flow.
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