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We present a multiparameter family of a soliton on a background solution to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the nonlinear Schrödinger equation limit and the limit of rogue wave solutions.

We study the infinite integrable nonlinear Schrödinger equation hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, Akhmediev breathers, Kuznetsov-Ma breathers, periodic solutions, and rogue wave solutions for this infinite-order hierarchy. We find that "even- order" equations in the set affect phase and "stretching factors" in the solutions, while "odd-order" equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are always complex.

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We show theoretically, that the detuning between the resonance frequencies of differently pumped DFB sections gives rise to two different pulsation mechanisms, 1) dispersive self Q-switching of a single-mode and 2) beating oscillations between two modes of nearly equal threshold gain. Our analysis is based on the dynamic coupled wave equations accomplished with carrier rate equations. We demonstrate the existence of certain isolated values of the detuning between both sections, at which two longitudinal eigenmodes become degenerate. In the degeneration point, the longitudinal excess factor of spontaneous emission has a singularity and the system of eigenmodes becomes incomplete. We derive reduced equations governing the dynamics in the vicinity of degeneration points. For an example device, the numerical integration of these equations clearly demonstrates the two different self pulsations with repetition rates of more than 100 GHz.