Dynamical Behavior of the Solutions of Coupled Boussinesq–Burgers Equations Occurring at the Seaside Beaches

Since water lies at the heart of sustainable development and climate change adaptation, therefore, the objective of the present research work is to derive a new variety of analytical solutions for a system of partial differential equations that depicts the propagation of shallow water waves at seaside beaches or in lakes. The (1+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+1)$$\end{document}-Boussinesq–Burgers system is solved by using the classical Lie-symmetry analysis and optimal subalgebra using a direct algorithm. Using the one-parameter optimal system, one-dimensional and two-dimensional optimal subalgebras are generated for the system to get a greater variety of solutions. Analytic solutions in this study are different from the nature of research reported earlier. Constructed solutions are represented graphically and show parabolic, multisoliton, periodic, dark, and bright solitons and progressive behaviors. The profiles of solitons could have some implications for port and coastal architecture. Additionally, conserved vectors demonstrate that the system is integrable. As far as the authors are aware, the conserved vectors are calculated and the optimal subalgebra technique is employed first time for the system. Coastal and civil engineers can use the solutions of the system to frame the architecture of the coasts.