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Bright soliton interactions in the variable coefficient Fokas-Lenells equation, Conservation laws, Modulation instability and Soliton tunneling

We present here a study of the bright soliton dynamics in an inhomogeneous fibre by means of variable coefficient Fokas-Lenells equation with time varying dispersion, nonlinearity and gain/loss parameter. At first, we propose our system that governs the propagation of ultrashort pulses in an inhomogeneous fibre. Secondly, under a suitable gauge transformation, we transform the system into a simplified form of variable coefficient Fokas-Lenells equation. The Lax integrability and conservation laws are exhibited. We also study the stability of the generalised plane wave against small amplitude perturbations. Thereafter, by using a nonstandard Hirota bilinearization method with the help of a suitable auxiliary function, we obtain the bright one soliton, two soliton and provide a scheme for obtaining N-bright soliton solutions. The elastic collision dynamics of the two solitons is studied using asymptotic analysis. We also investigate the soliton acceleration/retardation under a suitable choice of dispersion and nonlinearity coefficients. Finally, the dramatic effect of the nonlinear tunnelling of the bright one and two-soliton is also studied under some Gaussian dispersion or nonlinearity.