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

Abstract

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.