Extended Gram-type determinant solutions to the Kadomtsev-Petviashvili equation

In this paper, we would give a broad set of sufficient conditions of systems of linear partial differential equations which guarantee that the Grammian determinant solves the KP (Kadomtsev-Petviashvili) equation in the bilinear form. A systematic analysis of linear partial differential equations is made for solving the resultant linear systems. A special method is constructed to solve the representative systems.