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The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is investigated in this study. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is an extension of Calogero–Bogoyavlenskii–Schiff equation that describes the movement of Riemann waves along y-axis while long waves moves along the x-axis. The dynamics of Riemann waves is one of the most significant applications including tsunami in rivers, internal waves in oceans and magento-sound waves in plasmas. Finding new precise solutions with the assistance of a relatively new extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique is the primary objective of this effort. The suggested techniques are important tools in the fields of mathematical physics. Successful extraction of hyperbolic, rational, and trigonometric function solutions are achieved by using the proposed analytical methods. The extended (2 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff equation is studied for the first time using extended G ′ G 2 -expansion approach and exp (- φ (ζ)) -expansion technique in this work and novel solutions are observed. 3D plots, contour plots and 2D plots are used to depict the dynamics of the extracted solutions. [ABSTRACT FROM AUTHOR]