Surface Wave Solutions in 1D and 2D for the Broer–Kaup–Boussinesq–Kupershmidt System

ABSTRACT The Broer–Kaup–Boussinesq–Kupershmidt (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 . These terms also make the BKBK system completely integrable as a Hamiltonian system. Remarkably, when , the BKBK system may be transformed into the focusing nonlinear Schrödinger equation. Thus, the BKBK system with its real parameter 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 behavior in certain computational simulations of regularized versions of the 1D and 2D BKBK systems.