On the new sine-Gordon solitons of the generalized Korteweg-de Vries and modified Korteweg-de Vries models via beta operator

In this paper, we applied the sine-Gordon expansion method (SGEM) and the rational sine-Gordon expansion method (RSGEM) for obtaining some new analytical solutions of the (2+1)- dimensional generalized Korteweg-de Vries (gKdV) and modified Korteweg-de Vries (mKdV) equations with a beta operator. The sine-Gordon expansion method (SGEM) has recently been extended to a rational form, referred to as the rational sine-Gordon expansion method (RSGEM). By applying a specific transformation, the equations are reduced to a nonlinear ordinary differential equation (NODE), allowing for the derivation of analytical solutions in various forms, including complex, hyperbolic, rational, and exponential. All these solutions are expressed through periodic functions using SGEM and RSGEM. The physical significance of the parametric dependencies of these solutions is also examined. Additionally, several simulations, including three-diemensional (3D) visualizations and revolutionary wave behaviors, are presented, based on different parameter selections. Revolutionary surfaces, defined by height and radius as independent variables, are extracted to further illustrate the wave dynamics.