We apply the Simple Equations Method (SEsM) for obtaining exact traveling wave solutions of the Korteweg—deVries (KdV)-type equation. We construct the general solution of this equation as a composite function of two simpler functions which are presented as finite series of the solutions of two simple equations. We express these solutions by special functions V, which are solutions of appropriate ordinary differential equations. For this study we select ordinary differential equations (ODEs) of second order as simple equations. In this way, we obtain various particular solutions of the studied equation presented by elementary functions, which are solutions in the form of the function \(1/\cosh ^{m}(\kappa > x+\omega t)\) and the elliptic functions of Jaccobi. One of these solutions is illustrated numerically to show the possibility of emergence of different multi–solitons depending on the free coefficients in the presented analytical solutions.