On the Kaup-Broer-Kupershmidt systems

Abstract

The aim of this paper is to survey and complete, mostly by numerical simulations, results on a remarkable Boussinesq system describing weakly nonlinear, long surface water waves. It is the only member of the so-called (abcd) family of Boussinesq systems known to be completely integrable.

Figures (19)

• Figure 1: Difference between numerical and exact solution for soliton initial data (\ref{['soliton']}) with $C=0.8$ for $t=1$, on the left $\eta$, on the right $v$.
• Figure 2: The solution to the KBK system for perturbed soliton initial data of the form (\ref{['pert']}) with $\lambda=1.01$ and $\mu=1$ at the final time $t=5$ in blue and a fitted soliton in green, on the left $v$, on the right $\eta$.
• Figure 3: The solution to the KBK system for perturbed soliton initial data of the form (\ref{['pert']}) with $\lambda=0.99$ and $\mu=1$ at the final time $t=5$ in blue and a fitted soliton in green, on the left $v$, on the right $\eta$.
• Figure 4: The solution to the KBK system for perturbed soliton initial data of the form (\ref{['pert']}) with $\lambda=1$ and $\mu=1.01$ in the upper row and $\mu=0.99$ in the lower row fitted soliton in green, on the left $v$, on the right $\eta$.
• Figure 5: The solution to the KBK system for the perturbed stationary solution of the form (\ref{['pert']}) with $\lambda=1$ and $\mu=1.01$ in the upper row and $\mu=0.99$ in the lower row at the final time $t=5$ in blue and a fitted stationary solution in green, on the left $v$, on the right $\eta$.

Theorems & Definitions (15)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 4.1