Multi-lump, lump-kink and interaction of breather with other nonlinear waves of a couple Boussinesq system

TL;DR

This work studies nonlinear wave interactions in the two-component CB system $u_t + v_x + 2 u u_x = 0$, $v_t + 2 (u v)_x + u_{xxx}=0$ using the Hirota bilinear method. It systematically builds lump solitons, lump-kink couplings, and breather interactions (including first- and second-order breathers and breather-breather collisions) by transforming to a bilinear form $(D_x D_t + D_x^3)(f g)=0$ and appropriate rational-function ansatz for $f$ and $g$; Breathers are generated via second-order perturbations with complex-conjugate wave numbers and exhibit interactions with periodic, kink, and bright solitons. The main contributions are explicit lump and double-lump solutions, lump-kink and lump-with-two-kink configurations, and a comprehensive breather-interaction framework, all illustrated through extensive 2D, 3D, and contour plots that reveal diffusion-like and fission-type dynamics. These results enhance understanding of multimodal nonlinear dynamics in coupled Boussinesq systems and have potential implications for shallow-water, optical, and plasma contexts.

Abstract

We investigate the interaction characteristics of nonlinear coherent structures in the couple Boussinesq (CB) system using the Hirota bilinear approach. First, we derive the lump solutions using a positive quadratic polynomial within the Hirota perturbation technique. Next, We study the explicit interactions between lumps and one or two-kink waves, and observe double lump patterns. Furthermore, we discover the interactions of breathers, revealing a diffusion-like behavior. We notice that breather waves can interact with periodic, kink, and bright solitons for specific parameter sets in the CB system. Interactions between two chains of double breathers are also found. We analyze our results using a combination of symbolic computations and graphical representations, providing a deeper understanding of their behavior. This study reveals previously unreported nonlinear dynamics in the CB system.

Figures (8)

• Figure 1: Lump solutions for \ref{['lump2']} when the parameters are $c_0=-3, c_2=4, c_3=5.$ (a), (d) are the 3D plots; (b), (e) are the wave propagation for different times; and (c), (f) are the corresponding contour plots for $u$ and $v$, respectively.
• Figure 2: Double lump solutions for \ref{['lump3']}. The 3D plots are represented by (a)-(d), the 2D plots by (e)-(h), and the contour plots (i)-(l) for the parameter set $c_0=-3, c_2=4.$
• Figure 3: Lump with one kink for solution \ref{['lump1kinks1']}, considering the first case of parameter set 1: $c_0=-3, c_2=2, c_3=5, d_0=7.$ (a), (d) are the 3D plots, (b), (e) are the 2D plots, and (c), (f) are the corresponding contour plots for $u$ when $c_5=0.2,$ and $v$ when $c_5=-4$, respectively.
• Figure 4: Lump with one-kink for solution \ref{['lump1kinks2']}, considering the first case of set 2. (a), (d) are the 3D plots, (b), (e) are the 2D plots, and (c), (f) are the corresponding contour plots of $u$ when $c_0=-10, c_2=8, c_3=-1, c_4=-3, d_0=15$ and $v$ for $c_0=-1, c_2=2, c_3=5, c_4=-3, d_0=0.05$, respectively.
• Figure 5: Lump with two-kink for solution \ref{['lump2kink-sol']}, when $m_1=0.9, b_2=0.05, b_3=0.5, b_4=2, k_1=1.57, k_3=5.3, k_4=3, k_5=4.2, k_6=2.$ (a), (d) are the 3D plots, (b), (e) are the 2D plots, and (c), (f) are the corresponding contour plots of $u$ and $v$, respectively. For $b_1=10, k_7=0.5, k_8=6$, we have (a), (b), (c), and for $b_1=3.5, k_7=2.5, k_8=1.5$, we have (d), (e), (f).