In this article, the nonlinear (3+1) dimensional Wazwaz Benjamin Bona Mahony (WBBM) equation is considered for analysis which is related to some specific undular bore evolution through a long wave in shallow water. One-dimensional optimal system of Lie infinitesimal generators, associated vector field, commutation relations, and adjoint representation for the WBBM equation are illustrated. Moreover, the symmetry reductions are made, and closed-form solutions of the WBBM equation are obtained based on the optimal system. In addition, we make use of the transnational symmetries to reduce the governing equation to a nonlinear ordinary differential equation which is solved using the new extended direct algebraic method (NEDAM) to obtain the traveling wave profile of the WBBM equation.
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