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In this paper, the two-mode version of the generalized KdV (gTMKdV) equation is presented. The new model arises in weakly dispersive and nonlinear wave medium and describes the motion of either symmetric or asymmetric binary-waves regarding each wave’s height, which mainly depends on the nonlinearity and dispersion parameters. The overlapping of such binary-waves is affected by the phase-velocity parameter. Two recent effective schemes are proposed to extract novel explicit solutions for arbitrary values of the nonlinearity and dispersive factors. In addition, comprehensive graphical analysis is conducted to explain the nonlinear dynamics of the obtained bidirectional solitary solutions of gTMKdV. Finally, all the recovery solutions reported in this work are verified by direct substitution in the governing model.