Exact Solutions to the Generalized Hirota - Satsuma KdV Equations Using the Extended Trial Equation Method

In this paper, we study the traveling wave solutions of the generalized Hirota - Satsuma KdV equations by using the modified extended trial equation method. We construct the exact solutions for the nonlinear partial differential equations when the balance number is a positive integer via the generalized Hirota-Satsuma KdV equations using different types of functions such as: hyperbolic functions, trigonometric functions, Jacobi elliptic functions, and rational functional. The performance of this method is reliable, effective, and powerful for solving more complicated nonlinear partial differential equations in mathematical physics. The balance amount in this method is not constant and changes whenever the derivative definition of the trial equation changes. This method allowed us to construct many new types of exact solutions. We show by using the Maple software package that all obtained solutions satisfy the original partial differential equations.