Symbolic-computation study of integrable properties for the (2 + 1)-dimensional Gardner equation with the two-singular manifold method

The singular manifold method from the Painlevé analysiscan be used to investigate many important integrable propertiesfor the non-linear partial differential equations. In this paper,the two-singular manifold method is applied to the (2 + 1)-dimensionalGardner equation with two Painlevé expansion branchesto determine the Hirota bilinear form, Bäcklund transformation,Lax pairs and Darboux transformation. Based on the obtainedLax pairs, the binary Darboux transformation is constructedand the N x N Grammian solution is also derived by performingthe iterative algorithm N times with symbolic computation.