Integrability and lump solutions to an extended (2+1)-dimensional KdV equation

The purpose of this paper is to investigate an extended KdV equation in (2+1)-dimensions which cannot be directly bilinearized. The equation contains many important integrable models as its special cases. On the basis of the exchange identities for Hirota’s bilinear operators and the existing research results, a bilinear Bäcklund transformation is presented for the extended equation. And then, associated with the obtained bilinear Bäcklund transformations, we derive a Lax pair and a modified equation in detail, which implies that the introduced equation is also integrable. Finally, two kinds of nonsingular rational solutions are generated from the nonlinear superposition formula and arbitrary travelling wave solutions. The first class of rational solutions shows us that the presented equation possesses a general class of lump solutions with negative coefficients of two second-order linear dispersion terms. The second class of nonsingular rational solutions is essentially travelling wave solutions due to special solution structures of the presented equation.