<p>In this paper, the sixth-order Benney-Luke equation as a nonlinear partial differential equation that determines how nonlinear waves travel through analyzing wave tension in physical systems and studying the stress of water surface is studied. The method employs Hirota’s bilinear form to construct diverse solution models, including the multi waves, breather waves, Ma-breather, Kuznetsov-Ma-breather, periodic cross-kink solutions. These solutions offer critical insights into the impact of behaviours on nonlinear wave dynamics, particularly in the stress of water surface. Also, in this study, some standard, compatible, and useful wave solutions with a high score of accuracy and applicability, designated in terms of hyperbolic, trigonometric, and rational functions of the stated models using the generalized <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tan (\phi /2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>tan</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-expansion approach are established. The model is adapted to univariate wave equations through wave transformation erstwhile to investigation. To enhance visualization, we use Maple software to plot three-dimensional, two-dimensional, and density graphs, illustrating the intricate effects of solitons and other kinds of solutions on the obtained solutions. This work contributes to the understanding of soliton dynamics in higher-dimensional nonlinear systems and their relevance in practical applications. The obtained solutions are compared with the exact or existing numerical results in the literature to verify the applicability, efficiency and accuracy of the method. The acquired results reveal the efficiency and simplicity of the suggested techniques which are more conscientious to solve higher dimensional nonlinear ordinary as well as partial differential equations.</p>