Riccati-based analytical framework for solving the potential Korteweg-de Vries equation

This paper investigates the potential KdV equation by applying the Riccati-based Modified Extended Simple Equation Method (RMESEM). Through a suitable traveling wave transformation, the governing nonlinear partial differential equation is converted into an ordinary differential equation, which is then analyzed using an extended Riccati framework. As a result, several families of exact traveling wave solutions are derived in explicit analytical form. The obtained solutions include solitary wave, periodic, rational, hyperbolic, and trigonometric structures under different parameter conditions. The validity of the derived solutions is confirmed through direct substitution into the original model. In addition, the influence of the involved parameters on the wave behavior is discussed to illustrate the physical significance of the solutions. The present approach provides a systematic and effective technique for constructing exact solutions of nonlinear evolution equations arising in mathematical physics and related applied sciences.