Discussion on integrable properties for higher-dimensional variable-coefficient nonlinear partial differential equations

Abstract

In this paper we introduce two new higher-dimensional variable-coefficient partial differential equations. One is a (2+1)-dimensional equation which can be reduced to the well-known KP equation which first occurs to the paper B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary waves in weakly dispersive media,” Sov. Phys. Dokl. 15, 539 (1970), whose bilinear representation, Lax pairs, Bäcklund transformations, and infinite conservation laws are obtained respectively by using the Bell polynomials. Another one is a (3+1)-dimensional equation whose integrability is also investigated by us and whose Lax pairs, Bäcklund transformations, and infinite conservation laws are obtained, respectively.