New analytical wave solutions for the gardner equation via the Riccati-modified extended simple equation method

The Gardner equation is a well-known example of a nonlinear evolution equation that intersects the classical Korteweg-de Vries (KdV) and modified KdV equations, which occur in the real world in internal waves of stratified fluid, ion-acoustic waves in plasma, and in the propagation of signals in nonlinear media. In this paper, we use the Riccati-based Modified Extended Simple Equation Method (RM-ESEM) to build a comprehensive and coherent system of precise analytical solutions of the Gardner equation. The resultant solutions are of trigonometric, hyperbolic, rational, exponential, and mixed forms, which give a thorough insight into the nonlinear dynamics of the underlying waves. These solutions are known to be stable, periodical and transient, which are proven by graphical analysis and their physical relevance. The originality of the work is that it brings together several families of solutions in a single analytic framework, greatly enlarging the familiar solution space and allowing one to see interesting nonlinear phenomena, such as soliton interactions and parameter-dependent wave dynamics. These findings have found theoretical and useful applications in fluid physics, nonlinear optical systems, and plasma physics.