New exact solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method

By means of modified extended direct algebraic method (MEDA) the multiple exact complex solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. New complex solutions for nonlinear equations such as the variant of the RLW equation, the variant of the PHI-four equation and the variant Boussinesq equations are obtained.

💡 Research Summary

The paper presents a systematic procedure for obtaining multiple exact complex solutions of several nonlinear partial differential equations (NPDEs) by means of a newly formulated Modified Extended Direct Algebraic (MEDA) method. The authors first motivate the need for more flexible analytic techniques beyond traditional direct algebraic, inverse scattering, and perturbation methods, especially when dealing with nonlinear wave equations that possess rich solution structures.

The MEDA framework consists of three main steps. First, a traveling‑wave transformation (\xi = x - vt) (or a similar linear combination of space and time variables) reduces the original NPDE to an ordinary differential equation (ODE) in the single variable (\xi). Second, a balance between the highest order derivative term and the nonlinear term determines the polynomial degree (N) of the ansatz. Third, the solution is assumed in the form
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