Surface Wave Solutions in 1D and 2D for the Broer-Kaup-Boussinesq-Kupershmidt (BKBK) System

The BKBK system is a singular perturbation of the classical shallow water equations which modifies their transport velocity to depend on wave elevation slope. This modification introduces backward diffusion terms proportional to a real parameter $\kappa$. These terms also make BKBK completely integrable as a Hamiltonian system. Remarkably, when $\kappa =i\mathrm{/}2$ the BKBK system may be transformed into the focusing nonlinear Schrödinger (NLS). Thus, the BKBK system with its real parameter $\kappa$ is complementary to the traditional modulational approach for water waves. We investigate the Lie algebraic and variational properties of the BKBK system in this paper and we study its solution behaviour in certain computational simulations of regularised versions of the 1D and 2D BKBK systems.