Nonlinear evolution equations in one, two and three dimensions are the corner stone to understand and analyze numerous physical phenomena. The engineering applications comprise ocean waves, shallow water waves, plasma waves, fluid dynamics and many other applications. One of the most known models describing evolution equations is Hirota-Satsuma coupled system of equation in (1+1)-dimensions which describes the interaction of two long waves in shallow water. As the system is coupled and nonlinear, the group transformation technique was employed to transform the system of partial differential equations (PDEs) into a simpler system of ordinary differential equations (ODEs). The resultant system is numerically solved using Runge-Kutta and shooting methods. Two cases of similarity variable were attained leading to the interacted two waves. The results were illustrated graphically. The first solution led to two-solitons solution with large span as the wave conserves its energy and dissipates slowly. The second solution led to multiple solitons solution with higher lumps which enforces the wave to dissipate quicker than the first case.