On $$\varvec{}$$-mixed-type soliton propagation in dispersive nonautonomous long waves with waveguides

In this paper N-soliton propagations for the Calogero–Bogoyavlenskii–Schiff (CBS) equation in an inhomogeneous media which describes the long nonautonomous waves are obtained. Here attention is focused to study the effect of the dispersion coefficient on the propagation solitons waves. It is found that N-bright-dark solitons are produced by periodic or coupled periodic and pulses waves. Solitons waves are propagated for two and three pulses with periodic oscillating. Further, the double-periodic and solitary waves are dispersive to broken-solitons waves for the graded-index with oscillating reflection components. These results are useful for the application for long-distance telecommunication and optical fiber.