Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

In this paper, we study a Generalized (2+1)-dimensional Boussinesq-type equation. Using the Hirota bilinear method, we present the $N$-order bright soliton solutions and dark soliton solutions. For the one-soliton solution, the bright soliton solution and the dark soliton solution share the same limit line but have different extreme values. Building on the soliton solutions, we derive higher-order bright and dark breather solutions as well as mixed solutions. The dynamic behavior is characterized using visual representations. Furthermore, through the long-wave limit method, we obtain the bright and dark lump solutions. Notably, they share the same extreme points but have different extreme values. Additionally, we derive two semi-rational solutions as lump-soliton and lump-breather. It is found that the lump moves along the peak amplitude of the soliton wave.