The singular manifold method from the Painleve analysis can be used to investigate many important integrable properties for the nonlinear partial differential equations.In this paper, the two-singular-manifold method is applied to the (2+1)-dimensional Gardner equation with two Painleve expansion branches to determine the Hirota bilinear form, Backlund transformation, Lax pairs and Darboux transformation. Based on the obtained Lax pairs, the binary Darboux transformation is constructed and the N N Grammian solution is also derived by performing the iterative algorithm Ntimes with symbolic computation.
Arising from the Painlevé analysis proposed by Weiss, Tabor and Carnevale [1], the singular manifold method (SMM) has been successfully used to investigate the typical integrable properties for many integrable nonlinear partial differential equations (NPDEs), such as the Lax pair [2,3], auto-Bäcklund transformation [2,3,4], nonclassical Lie symmetry [5] and Hirota bilinear formulation [6]. In Refs. [4,7], it is shown the SMM has turned out to be capable of obtaining some special classes of solutions for non-integrable NPDEs. However, due to the existence of several Painlevé expansion branches for some given NPDEs like the modified Korteweg-de Vries (mKdV) equation [8], Sine-Gordon (SG) equation [8] and modified Kadomtsev-Petviashvili (KP) equation [9], in this situation the SMM is not feasible to exploit the integrable properties of these equations. Therefore, Refs. [8,10,11] have generalized the SMM and developed the two-singular-manifold method to uncover information about integrable character.
Different from the usual expansion, the two-singular-manifold method involves two truncated Painlevé expansions at the constant level term, which contains two different singular manifolds at a time. This approach has been applied to the mKdV equation [8,12], SG equation [8], classical Boussinesq system [10,11], Mikhailov-Shabat system [10], generalized dispersive long wave equation [9], modified KP equation [9], and so on. With this method, not only the auto-Bäcklund transformation and Lax pair can be obtained, but also the Darboux transformation can be constructed in terms of the truncated Painlevé expansions in both the NPDE and its Lax pair [9,13,14]. In addition, the relationship relating the singular manifolds and Hirota τ -function can be precisely established [9,10,11,12].
Permeation of symbolic computation among various fields of science and engineering remarkably helps the investigations on the nonlinear partial differential equations (NPDEs) [15,16,17,18]. Symbolic computation has increased the ability of a computer to deal with a large amount of complicated and tedious algebraic calculations.
In this paper, by virtue of the symbolic computation, we will investigate the integrable properties for the (2+1)-dimensional Gardner equation [19,20]
where α and β are two arbitrary constants. When u y = 0, Eqn. (1.1) reduces to the wellknown (1+1)-dimensional Gardner equation. For α = 0, Eqn. (1.1) is the KP equation, while it is the modified KP equation with β = 0. Therefore, the (2+1)-dimensional Gardner equation could be regarded as a combined KP and modified KP equation. Eqn. (1.1) is completely integrable in the sense that it has been solved by the inverse spectral transform method [20]. Refs. [20,21,22,23,24] have presented its wide classes of analytical solutions including the rational solution, quasi-periodic solution , soliton solution and non-decaying real solutions.
In the following sections, with the help of symbolic computation, we will apply the two-singular-manifold method to the (2+1)-dimensional Gardner equation to determine the Hirota bilinear form, Bäcklund transformation and Lax pairs. Based on the obtained Lax pairs, we will construct the binary Darboux transformation and perform symbolic computation on the iterative algorithm to generate the Grammian solutions.
To begin with, we rewrite Eqn. (1.1) as the following system
)
Then, we expand the solutions of System (2.1) in a generalized Laurent series
where χ = χ(x, y, t), and u j = u j (x, y, t), v j = v j (x, y, t) are analytical functions in the neighborhood of a non-characteristic movable singularity manifold χ(x, y, t) = 0, while a and b are two integers to be determined. By the analysis of the leading terms, we obtain
where ǫ = ±1. It is easy to see that u 0 and v 0 can take two values so that System (2.1) has two different Painlevé expansion branches. By using two different singular manifolds φ and ϕ [9, 13, 14], we take the truncated Painlevé expansion at the constant level term
where the singular manifold φ corresponds to ǫ = 1 and ϕ to ǫ = -1, which can also be regarded as an auto-Bäcklund transformation between two different solutions (u ′ , v ′ ) and (u, v) for System (2.1), when singular manifolds φ and ϕ satisfy the truncation conditions. Motivated by Expressions (2.4), we introduce the dependent variable transformations
to transform System (2.1) into the Hirota bilinear form. Substituting Expressions (2.5) back into System (2.1), we obtain
where D is the well-known Hirota bilinear operator [25]
which are the Hirota bilinear form of Eqn. (1.1). By the perturbation technique, one can assume the functions f and g in powers of a small parameter ε [25] to obtain the multi-soliton solutions of Eqn. (1.1) from Eqns. (2.8) and (2.9). It is noted that the key step for the Hirota method is to seek for the suitable dependent variable transformation for a given NPDE to be transformed into the Hirota bilinear form. If we do not
