The new extended direct algebraic technique is used to attain the new exact solitary wave solutions of the nonlinear fractional couple Drinfeld–Sokolov–Wilson system (DSWS). This DSWS has a lot of applications in water waves, fluid mechanics, quantum mechanics, chemistry, and biology. This system is developed in the perspective of fractional theory of calculus, for this purpose Riemann–Liouville fractional derivative (RLFD), β -fractional differential operator and Atangana–Baleanu fractional derivative (ABFD) are utilized. The attained solutions contain some new families such as trigonometric functions, hyperbolic functions, rational functions, logarithmic functions, and exponential functions. The graphically analysis and comparison of solutions are plotted by taking appropriate values of the constant parameters to enhance the understanding of physical real-life phenomena. The approach is said to be compatible and effective as a scientific approach for the examination of the various spacetime fractional models in engineering and physics concerning real-life issues. [ABSTRACT FROM AUTHOR]