Dynamical analysis of solitons, breathers and periodic rogue waves for the variable-coefficient fourth-order nonlinear Schrödinger equation

This work focuses on higher-order localized waves for coupled variable-coefficient fourth-order nonlinear Schrödinger equation that describe the simultaneous propagation of optical pulses in an inhomogeneous optical fiber. Based on the Lax pair, N-th Darboux transformation is constructed and iterative expression of localized wave solutions are obtained. Three-dimensional animations are used to reveal the dynamical characteristics and evolution of nonautonomous localized waves solutions. Profiles and contour plots distributions of nonautonomous solitons, breathers, and rogue waves are shown. The variable coefficients functions \({\gamma _1}\left( t \right) \) ${\gamma }_1\left(t\right)$ and \(\sigma \left( t \right) \) $\sigma \left(t\right)$ are discussed to reveal how the magnitude, shape, and position of background waves are affected. It is hopeful that results may provide theoretical basis for the research on the localized waves in an inhomogeneous optical fiber in the future.