<p>This research investigates the propagation of solitary waves i.e., so called (Equal Width (EW) and Modified Equal Width (MEW) equations) using a trigonometric quintic B-spline collocation method integrated with a Finite Difference (FD) scheme. The numerical approach employs the Crank-Nicolson method to discretize the spatial derivative terms, while the Finite Difference method is applied for the time derivatives. The Rubin-Graves linearization technique is employed for linearization of the non-liner terms. The stability of the trigonometric quintic B-spline collocation (TQBC) method is analyzed using the Von Neumann approach. This method is accurate to convergence order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O\left({h}^{4}+\Delta {\text{t}}^{2}\right)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <msup> <mrow> <mi>h</mi> </mrow> <mn>4</mn> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mrow> <mtext>t</mtext> </mrow> <mn>2</mn> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. Six test cases are analyzed to validate the present study, including a single solitary wave, two- and three-wave interactions, the Maxwellian initial condition, and the undular bore phenomena associated with the EW and MEW equations. The method’s efficiency is validated through error norms <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({L}_{2},{L}_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>L</mi> <mi>∞</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and root mean square error (RMS). It’s computational effectiveness is further emphasized by the order of convergence and computational time cost, demonstrating minimal memory usage.</p>