Breather wave and lump soliton solutions under the influence of unstable background waves in the (2+1)-dimensional KdV equation

Abstract Breather wave and lump soliton solutions under the influence of unstable background waves are presented for the (2+1)-dimensional KdV equation. The results indicate that the existence conditions and dynamic behaviors of the breather wave solution depend strongly on the unstable background waves. The breather wave solution v can be manifested as a single peak bright breather wave or double peak bright breather wave. However, the breather wave solution u always features a single peak dark breather wave structure. Furthermore, we investigate the degenerate behaviors of breather wave solutions, and derive the lump soliton solution under the influence of unstable background waves. The character of lump soliton solution depend entirely on the unstable background waves. With the variation of unstable background waves, the bright lump soliton solution v can evolve from a single peak wave structure into a double peak wave structure. However, the lump soliton solution u always features a single hole dark lump soliton structure. Combined with the linear instability analysis result, we demonstrate the phase diagram for different types of the breather wave and lump soliton solutions. It is shown that when u 0 changes from negative to positive at u 0 = 0, the slightest change in u 0 leads to an essential change in the dynamic behavior of the obtained solution. The system undergoes a pitchfork bifurcation. The unstable region is split into two parts through the bifurcation diagram. One of them is exactly the existence region of localized wave solutions. These results further enrich the structures and dynamical behaviors of bright and dark localized waves.