A $\bar{\partial}$-method for the $(2+1)$-dimensional coupled Boussinesq equation and its integrable extension

The content of this paper is divided into two parts. Starting from the Lax pair with a spectral function $\psi(x,y,t,k)$, the $\bar{\partial}$-dressing method is used to investigate the $(2+1)$-dimensional coupled Boussinesq equation, thereby constructing the scattering equation in the form of a linear $\bar{\partial}$ problem, and ultimately deriving the reconstruction formula for the solutions. By complexifying each independent variable of the $(2+1)$-dimensional coupled Boussinesq equation, we construct its generalizations to $(4+2)$ dimensions. The spectral analysis of the $t$-independent part of the Lax pair with a spectral function $\chi(x,y,t,k)$ together with the nonlocal $\bar{\partial}$ formalism yield the representation for the solution of the $\bar{\partial}$ problem. Additionally, the nonlinear Fourier transform pair comprising both direct and inverse transforms is successfully worked out.