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

TL;DR

The paper analyzes the Broer-Kaup-Boussinesq-Kupershmidt (BKBK) system as a dispersive, integrable perturbation of shallow water dynamics, focusing on its geometric structure (Lie--Poisson and Euler--Poincaré formalisms) and the role of the transport-velocity shift $\mathbf{v}=\mathbf{u}-\kappa\nabla\ln\eta$. It shows how the 1D system connects to the focusing nonlinear Schrödinger equation via a Madelung-type transformation when $\kappa=i/2$, and derives 2D generalisations including potential vorticity conservation and energy-Casimir stability analysis. Owing to backward diffusion from real $\kappa$, the 1D model is ill-posed at high wavenumbers, which motivates regularisation: a fourth-order dissipation in 1D and a gradient-penalty Hamiltonian in 2D that preserves key geometric properties. Numerical simulations demonstrate the necessity and effectiveness of these regularisations, revealing traveling waves, ring and vortex-like structures, and PV-influenced dynamics, thereby linking integrability, instability mechanisms, and regularisation in a unified geometric framework.

Abstract

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 $κ$. These terms also make BKBK completely integrable as a Hamiltonian system. Remarkably, when $κ=i/2$ the BKBK system may be transformed into the focusing nonlinear Schrödinger (NLS). Thus, the BKBK system with its real parameter $κ$ 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.

Figures (8)

• Figure 1: Space–time evolution of the velocity $u$ (black) and depth $\eta$ (orange) for the 1D BKBK system regularized with symmetric fourth-order dissipation, cf. \ref{['BK-System-BAK3-reg2']}. The figures show that fourth-order dissipation with $0<\nu\ll1$ prevents the ill-posed growth observed in the unregularized system.
• Figure 2: Waterfall plot of the regularized 1D BKBK system with symmetric fourth-order diffusion, cf. \ref{['BK-System-BAK3-reg2']}. For the Gaussian initial condition $u=0$, $\eta=1+\exp(-(x-24)^2/8),\nu=0.01$, the solution rapidly splits into two dominant crests which propagate in opposite directions. While this bidirectional splitting occurs for both $\kappa= -0.5$ and $\kappa=-0.1$, the case $\kappa=-0.1$ in the right panel exhibits a small leading depression ahead of the main crest, with $\eta<\eta_0$, which is absent for $\kappa=-0.5$ in the left panel.
• Figure 3: Simulation of the 2D BKBK equation initialised with two Gaussian ridges, for negative $\kappa=-0.05$. Top: velocity magnitude $|\vec{u}|$. Bottom: surface height $\eta$ as a 3D surface. At $t=0$, the free surface consists of two initial Gaussian ridges. By $t=0.25$, the two peaks merge into a single crest, generating outward-propagating ring waves. At $t=0.50$, the surface splits along the $y$-direction, producing two secondary peaks while the ring pattern continues to expand. At $t=0.75$, a depression forms at the center, accompanied by strong outward-propagating velocity oscillations; the free-surface height also exhibits significant fluctuations.
• Figure 4: Comparison of simulations with different values of $\kappa=-0.5$. The outer rings are smaller at $t=1$, indicating slower outward wave propagation compared to $\kappa= -0.05$. At $t=0.25$, the surface profile also collapses more slowly, and the resulting ring structures remain more compact.
• Figure 5: Simulation of the 2D BKBK equation with a localized tanh-segment initialization, for negative $\kappa=-0.5$. Top: velocity magnitude $|\vec{u}|$. Bottom: surface height $\eta$ as a 3D surface. At $t= 0$, the flow is initialized with a narrow rectangular segment in $u_x$. By $t=0.1$, the disturbance begins to tilt and induces small surface deflections. At $t=0.5$, the velocity field splits into two lobes with a vortex dipole while the surface height develops two crests and moves in the opposite direction. At $t= 1$, outward-propagating oscillations appear and the central structure weakens. By $t= 2$, the disturbance has radiated into compact concentric wavefronts.

Theorems & Definitions (5)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Proposition 4.1
• proof