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The paper studies the optical solitons with coupled nonlinear Schrödinger system (CNLSS) that describes the propagation of waves in birefringence polarization-preserving fibers with four-wave mixing effect. The system is discussed with the effects of group velocity dispersion, self-phase and cross-phase modulations. The mechanism that is used to investigate is the extended Fan-sub equation method. It is observed that this system exhibits a collection of rich solitons (single and combined) for different values of parameters. The 3D figures are also plotted to show the physical behavior of obtained solutions under different constraint conditions.

For describing the propagation of ultrashort pulses in the birefringent or two-mode fiber, we consider the coupled nonlinear Schrödinger equations with higher-order effects. According to the Ablowitz–Kaup–Newell–Segur system, 3$\times$3 Lax pair for such equations is derived. Based on the Lax pair, we construct the conservation laws and Darboux transformation (DT). One- and two-soliton solutions are obtained via the DT, and we graphically present the one soliton and bound-state two solitons.

In this paper, we study a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber for two orthogonally polarized optical pulses. With respect to the slowly-varying envelopes of two interacting optical modes and based on the existing binary Darboux transformation, we obtain four types of the bound-state solitons: degenerate-I, degenerate-II, degenerate–non-degenerate, and non-degenerate–non-degenerate bound-state solitons. We graphically analyze the interactions between the degenerate or non-degenerate solitons and four types of the bound-state solitons. When the degenerate solitons interact with the bound-state solitons, amplitudes and widths of the degenerate solitons remain unchanged. When the non-degenerate solitons interact with the bound-state solitons, amplitudes and widths of the bound-state solitons remain unchanged.

Temporal birefringent effects in the fibers change the crosstalk behaviors inside and between the fiber cores in the linear and non-linear optical power areas. This paper studies a non-linear Schrödinger system with the four-wave mixing term, which describes the optical solitons in a birefringent fiber. We construct the generalized Darboux transformation, and acquire the higher-order semirational solutions consisting of the second- and third-order semirational solutions, which represent the complex amplitudes of the electric fields in the two orthogonal polarizations. We acquire the interactions between/among the two/three solitons. Such interactions are elastic and generate the rogue waves around the interacting regions. We obtain the interactions among the second-/third-order rogue waves and two/three solitons, respectively. When $|dc-|b{|}^2|$ decreases, amplitude of the second-order rogue wave increases, with $d$ and $c$ accounting for the self-phase modulation and cross-phase modulation, respectively, while $b$ representing the four-wave mixing effect. With $|dc-|b{|}^2|$ kept invariant, when $c$ increases and ${\sigma }_1=0$, amplitudes of the second-order rogue wave and two bright solitons increase, while when $d$ increases and ${\sigma }_2=0$, amplitudes of the second-order rogue wave and two dark solitons increase, with ${\sigma }_1$ and ${\sigma }_2$ being the constants.

We have considered soliton pulse compression in a dispersion decreasing ideal elliptical birefringent fiber with phase modulation. We have shown that there is exact balancing between the effective gain and the effective phase modulation and as a result arrive at the fundamental soliton solution. For the various dispersion decreasing profiles we have considered, we have shown that the hyperbolic profile is the best suitable choice for the dispersion decreasing profile as it provides fairly good compression even after compensating for the fiber loss.