Abstract In this paper, we derived new explicit complex hyperbolic and complex trigonometric function solutions, especially dark, bright, combined dark–bright, singular, combined singular soliton and other soliton solutions from the conformable fractional Biswas–Milovic equation with Kerr law and Parabolic law nonlinearity that describes the long-distance optical communications and optical fibres for the solitons propagation. The reported results might play an important role in a nonlinear medium for explaining the physical meaning of the studied model. Therefore, it is also shown that the applied method provides a more powerful mathematical tool for investigating exact solitary wave solutions for many other nonlinear evolution equations in nonlinear medium.