Symbolic computation: Analytical solutions and dynamics of a shallow water wave equation in coastal engineering

Abstract This study investigates a shallow water wave-like scalar equation relevant to coastal engineering, employing an innovative computational framework and Mathematica software to derive the analytical solutions. Rational solutions are constructed by using polynomial expansions about <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> </m:math> x and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>t</m:mi> </m:math> t . Furthermore, double-periodic soliton solutions are obtained through direct functional Ansätz techniques, while hyperbolic-type and trigonometric-type localized wave solutions are derived by using an improved <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo accent="false">′</m:mo> <m:mo>∕</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(G^{\prime} /G) -expansion method. In addition to these solutions, we also provided the corresponding Mathematica software calculation code. The dynamic behaviors of these solutions including amplitude modulation, phase interactions, and waveform stability are analyzed and visualized via three-dimensional map and contour map. This study connects theoretical progress in integrable systems with real-world uses in modeling coastal water dynamics.