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.