Lax pair, conservation laws, Darboux transformation and localized waves of a variable-coefficient coupled Hirota system in an inhomogeneous optical fiber

Optical fiber communication plays an important role in the modern communication. In this paper, we investigate a variable-coefficient coupled Hirota system which describes the vector optical pulses in an inhomogeneous optical fiber. With respect to the complex wave envelopes, we construct a Lax pair and a Darboux transformation, both different from the existing ones. Infinitely-many conservation laws are derived based on our Lax pair. We obtain the one/two-fold bright-bright soliton solutions, one/two-fold bright-dark soliton solutions and one/two-fold breather solutions via our Darboux transformation. When α(t),β(t) and δ(t) are the trigonometric functions, we present the bright-bright soliton, bright-dark soliton and breather which are all periodic along the propagation direction, where α(t),β(t) and δ(t) represent the group velocity dispersion, third-order dispersion and nonlinear terms of the self-phase modulation and cross-phase modulation. Interactions between the two bright-bright soliton, two bright-dark solitons and two breathers are presented. Bound state of the two bright-bright solitons is formed. Widths and velocities of the two bright-bright solitons do not change but their amplitudes change after their interaction via the asymptotic analysis. Periods of the bright-dark solitons decrease when the periods of the trigonometric α(t),β(t) and δ(t) decrease.