Grammian-type determinant solutions to generalized KP and BKP equations

Grammian-type determinant identities of the bilinear KP hierarchy and a useful property on the derivatives of Grammians are proposed. Based on these Grammian-type determinant identities, three Grammian structures are furnished to a large class of generalized nonlinear KP and BKP type equations. All the generating functions for matrix entries satisfy a system of combined linear partial differential equations. In the first case of the class of generalized KP and BKP equations, general rational solutions are given, making use of its Grammian formulation.