On the stabilization of breather-type solutions of the damped higher order nonlinear Schrödinger equation

Spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger  (NLS) equation are frequently used to model rogue waves and are typically unstable.  In this talk we examine the effects of dissipation and higher order nonlinearities on the stabilization of N-mode SPBs, 1 ≤ N ≤ 3, in the framework of a damped higher order NLS (HONLS) equation.  We observe the onset of  novel  instabilities  associated with the development of critical  states resulting from symmetry breaking  in the damped HONLS system. We develop a broadened Floquet characterization of  instabilities of solutions of  the NLS equation by showing that instabilities are  associated with degenerate complex elements of not only the discrete, but also the continuous Floquet spectrum. As a result, the Floquet criteria for the stabilization of a solution of the damped HONLS centers around the elimination of all complex degenerate elements of the spectrum. For  a given  initial N-mode SPB,  a short-time perturbation analysis shows that the  complex double points associated with resonant modes split under the damped HONLS while those associated with  nonresonant modes remain closed. The corresponding  damped HONLS numerical experiments corroborate that instabilities associated with  nonresonant modes persist  on a longer time scale than the instabilities associated with resonant modes.