Hamiltonian structure, optimal classification, optimal solutions and conservation laws of the classical Boussinesq–Burgers system

Abstract

Shallow water waves phenomena have been vastly studied in the literature for many decades. Understanding their intricacies is not only important in predicting the timing and magnitude of their destructive nature (such as tsunami) but also in knowing how to prevent their occurrences. Among the equations used for gaining accurate and deep understanding of shallow water waves for two-layered fluid flow is the classical Boussinesq–Burgers system. Its solutions have been studied in the literature using various methods; however, no study has considered its connection with (some of) the Painlevé transcendents which lead to infinite rational and special function solutions. The present study explore this avenue through the Lie group method — a powerful and versatile method that has been effectively and extensively used for various analyses of nonlinear differential equations. Invariant, rational and special function solutions of the classical Boussinesq–Burgers equations are derived and non-trivial conserved quantities are also constructed.