Nonlinear excitations and “peakons” of a (2+1)-dimensional generalized Broer-Kaup system

Abstract

Shallow water waves and a host of long wave phenomena are commonly investigated by various models of nonlinear evolution equations. Examples include the Korteweg-de Vries, the Camassa-Holm, and the Whitham-Broer-Kaup (WBK) equations. Here a generalized WBK system is studied via the multi-linear variable separation approach. A special class of wave profiles with discontinuous derivatives ("peakons") is developed. Peakons of various features, e.g. periodic, pulsating or fractal, are investigated and interactions of such entities are studied.

Objective Reduction Solutions to Higher-Order Boussinesq System in (2+1)-Dimensions

“…As these special reduction cases based on the derived variable separation solutions, it is shown that some novel soliton structures like single linearity soliton structure, breath soliton structure, single linearity y-periodic solitary wave structure, libration dromion structure and kink-like multisoliton structure with actual physical meaning exist in the (2+1)-dimensional higher-order Boussinesq system by prescribing different values of the arbitrary parameters similar to Refs. [12], [13], and [16]. Fially, we must point out that although the special conditional similarity reduction equations obtained by this method are not as many as that by other similarity reduction approach, but through this approach, one can obtain many explicit and exact solutions of the given partial differential equations.…”

“…which is an extension version of the Whitham-Broer-Kaup system using Painleve analysis. [16] Whitham-Broer-Kaup system is valuable model for long waves by incorporating or mimicking convective, dispersive and viscous effects. The HOB system is also named higher-order Broer-Kaup system obtained from the inner parameter dependent symmetry constraints of the KP equation.…”

Solitary waves for a nonlinear dispersive long wave equation

“…For example, when α = 0, β = 0, system (1) becomes a classical long wave equation describing shallow water with dispersive, while when α = 1, β = 0, the system becomes variant Boussinesq equation. Furthermore, many important models are extensions of WBK equation, such as the generalized Broer-Kaup equation [20]. Up to now, many researchers have devoted considerable efforts to the WBK equation and obtained a lot of wave solutions accordingly [23,24].…”

“…On the basis of a more accurate mode, Dai and Liu [19] revealed how the singular points affect the wave solutions in a compressible mooney-rivlin rod. Recently, various features of peakons were investigated to explore the interactions between the waves [20], and a new theory was developed for computing solitary waves with high precision [21].…”