The <inline-formula><tex-math id="M1">$ \bar{\partial} $</tex-math></inline-formula>-dressing method for the (2+1)-dimensional potential Korteweg-de Vries type equation

In this work, we extend the $ \bar{\partial} $-dressing method to investigate the (2+1)-dimensional potential Korteweg-de Vries type equation. By means of the eigenfunctions and Green's functions associated with its Lax pair representation, we introduce a new $ \bar{\partial} $-problem formulation to construct the solution. This procedure is accomplished through the application of the Cauchy-Green formula and the selection of an appropriate spectral transformation. Additionally, we derive a formal solution to the (2+1)-dimensional potential Korteweg-de Vries type equation via the inverse spectral problem, and effectively overcome the challenge of spectral data evolution by determining the time evolution of the spectral data.