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The current paper recovers soliton solutions in magneto-optic waveguides having generalized form of Kudryashov’s self-phase modulation structure. The retrieval of soliton solutions is achieved by the well-known and primitive G′/G-expansion scheme. The intermediary functions that made this retrieval possible are Jacobi’s elliptic functions and Weierstrass’ elliptic functions. The parameter constraints for the solitons to exist are also presented.

This article studies optical dromions with spatio–temporal dispersion having parabolic law of nonlinear refractive index change in presence of white noise. Two integration approaches reveal the dromion solutions. They are the extended auxiliary equation approach and the generalized projective Riccati equation scheme. A wide spectrum of dromion solution structures have been revealed from such algorithms. The parameter constraints are also enlisted for these dromions to exist.

In this article, we propose a new method to construct many new exact solutions with parameters for the generalized KdV–mKdV (GKdV–mKdV) equation with higher-order nonlinear terms. The proposed method is a generalization of the well-known G′G,1G-expansion method in the case of the hyperbolic function solutions. Also, we use a direct algebraic method based on the generalized Liénard equation to find other diffrent new exact solutions of the above GKdV–mKdV equation. Soliton solutions, periodic solutions, rational functions solutions, hyperbolic functions solutions and symmetrical hyperbolic Lucas functions solutions are obtained. Comparing our new results obtained in this article with the well-known results are given. The generalized G′G,1G-expansion method presented in this article is straightforward, concise and it can also be applied to other nonlinear partial differential equations in mathematical physics.