Darboux Transformation And Explicit Solutions Of Variant Boussinisq Equation

A hierachy of a variant Bousinessq equation is studied by mean of mapping method. We obtained a new (2+1)dimensional variant Bousinessq equation with its Lax pair. The three kinds Darboux transformations are constructed by means of the Lax pairs. Some new kinds exact solutions are obtained.There are four parts in this paper, the first part is an introduction; In the second part ,we considered a spectral problem associat with the variant Bousinessq equation:The 3×3 Lenard operators K, J and Lwnard sequence {gn} were reduced by means of a map: σ : R~3 → sl(2). The hierarchy of variant, Boussinesq equation was generated as:The conecponding Lax pairs is :when N = 1. , we have variant Boussinesq equation :Its Lax pair is :Let. N = 2, we have:If the soliton equations ( 2.5 ) and ( 2.7 ) are compatible, then ,their solution should satisfy the following (2 + 1)dimensional soliton equation(y ≡ t1,t ≡t2)In the third part, we obtained three kinds Darboux transformations T1, T2, T for (2+1) dimensional variant Boussinesq equation. We also proved that T1 · T2 = T.In the forth part, some interesting solutions of the (2+1) dimensional variant Boussi-nesq equation were constructed by the Darboux transformation.