Abstract
Spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation are frequently used to model rogue waves and are typically unstable. In this paper we study 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.
Highlights
In one of his foundational studies, Stokes established the existence of traveling nonlinear periodic wave trains in deep water [1]
We find that for short time, the complex double points associated with modes that resonate with the spatially periodic breathers (SPBs) structure split producing disjoint asymmetric bands, while the complex double points associated with nonresonant modes remain closed, substantiating the initial spectral evolutions observed in the numerical experiments
In this paper we investigated the route to stability for even N-mode SPB solutions of the nonlinear Schrödinger (NLS) equation in the framework of a damped higher order NLS (HONLS) equation using the Floquet spectral theory of the NLS equation
Summary
In one of his foundational studies, Stokes established the existence of traveling nonlinear periodic wave trains in deep water [1]. In the present study we observe the onset of novel instabilities as a result of symmetry breaking and the development of critical states in the damped HONLS flow which were nonexistent in the previously examined damped NLS system with even symmetry We determine these instabilities are associated with degenerate complex elements of both the periodic and continuous spectrum, i.e., with both complex “double points” and complex “critical points”, respectively. Variations in the spectrum under the HONLS flow are correlated with deformations of certain NLS solutions to determine the routes to stability for the damped HONLS SPBs. In Section 4, via perturbation analysis, we examine splitting of the complex double points, present in the SPB initial data, under the damped HONLS flow. In this study resonances have a stabilizing effect; the instabilities of nonresonant modes persist on a longer time scale than the instabilities associated with resonant modes
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