Soliton Management for ultrashort pulse: dark and anti-dark solitons of Fokas-Lenells equation with a damping like perturbation and a gauge equivalent spin system

We investigate the propagation of an ultrashort optical pulse using Fokas-Lenells equation (FLE) under varying dispersion, nonlinear effects and perturbation. Such a system can be said to be under soliton management (SM) scheme. At first, under a gauge transformation, followed by shifting of variables, we transform FLE under SM into a simplified form, which is similar to an equation given by Davydova and Lashkin for plasma waves, we refer to this form as DLFLE. Then, we propose a bilinearization for DLFLE in a non-vanishing background by introducing an auxiliary function which transforms DLFLE into three bilinear equations. We solve these equations and obtain dark and anti-dark one-soliton solution (1SS) of DLFLE. From here, by reverse transformation of the solution, we obtain the 1SS of FLE and explore the soliton behavior under different SM schemes. Thereafter, we obtain dark and anti-dark two-soliton solution (2SS) of DLFLE and determine the shift in phase of the individual solitons on interaction through asymptotic analysis. We then, obtain the 2SS of FLE and represent the soliton graph for different SM scheme. Thereafter, we present the procedure to determine N-soliton solution (NSS) of DLFLE and FLE. Later, we introduce a Lax pair for DLFLE and through a gauge transformation we convert the spectral problem of our system into that of an equivalent spin system which is termed as Landau-Lifshitz (LL) system. LL equation (LLE) holds the potential to provide information about various nonlinear structures and properties of the system.