Explicit bright and dark solitons for the variable coefficient Biswas-Milovic equation with competing nonlinearity

This paper is devoted in the study of Biswas-Milovic equation with variable coefficients associated with four different forms of nonlinearity, namely the Kerr law, power law, parabolic law and the dual-power law. We prove that, in presence of additional time dependent damping term, yet there exists different solitary wave solutions under some constraint relations. The generalized form of solitary wave ansatz method in context of doubly periodic Jacobi elliptic functions is carried out to obtain bright and dark soliton solutions of the governing equation. The constraint relations between the model coefficients and the damping coefficient for existence of soliton solutions are derived. In addition, it is shown that for the existence of soliton, the damping coefficient has to be Riemann integrable. The remarkable features of such solitons are demonstrated in several interesting figures.