Darboux Transformation and Explicit Solutions for a Generalized Sawada-Kotera Equation

was first proposed by Sawada and Kotera when they gave a method for finding N-soliton solutions of the KdV equation and the KdV-like equation [1]. In [2], Caudrey et al. showed that (1) was a member of a new hierarchy of KdV equations. The SK equation’s physical importance was illustrated by Aiyer et al. in [3]. Then, the equation has been investigated by many authors [4–8]. The aim of the present paper is using the Darboux transformation [9–12] to study a generalized SK equation:


Introduction
The Sawada-Kotera (SK) equation   = −  + 15(  −  3 )  (1) was first proposed by Sawada and Kotera when they gave a method for finding N-soliton solutions of the KdV equation and the KdV-like equation [1].In [2], Caudrey et al. showed that (1) was a member of a new hierarchy of KdV equations.
The SK equation's physical importance was illustrated by Aiyer et al. in [3].Then, the equation has been investigated by many authors [4][5][6][7][8].The aim of the present paper is using the Darboux transformation [9][10][11][12] to study a generalized SK equation: The present paper is organized as follows.In Section 2, with the aid of the Lax pairs of the SK equation [13,14] and extending them by adding one potential function, we propose a generalized SK equation and its Lax pairs.Based on the gauge transformation between spectral problems, we derive a Darboux transformation of the generalized SK equation.In Section 3, the Darboux transformation is applied to the generalized SK equation, by which explicit solutions (we have verified the correctness of the solutions by using the Mathematic 5.0.) of the generalized SK equation are derived, including rational solutions, soliton solutions, and periodic solutions.

Darboux Transformation of the Generalized Sawada-Kotera Equation
In this section, we will derive a generalized SK equation and its Darboux transformation.To this end, we first introduce the Lax pairs: where operators L and B are defined as follows: Then the compatibility condition between the two equations of (3) yields the Lax equation, L  = [B, L], which is equivalent to the generalized SK equation: ( If we choose V = 0 and V = −(3/2)  , (5) can be, respectively, reduced to the SK equation: and the Kaup-Kupershmidt equation: Theorem 1.Let  satisfy (3) with  =  0 and  = −(ln )  .
Then the following Darboux transformation gives the relation about the original solutions , V of (5) and its new ones , V: Proof.Assume that  satisfies (3) and  = −(ln )  .Let Using the first expression of (3), a direct calculation gives the following equations: Substituting ( 9) and ( 10) into the following equation: where and comparing the coefficients of ,   , and   , we obtain the following: Equation ( 13) implies the following: Substituting (15) into ( 14) and integrating it once, we have the following: where  0 is a constant of integration.Through direct calculations, we arrive at the following: Using ( 17) and  = −(ln )  , a simple reduction shows that (16) gives rise to the following: Similarly, we consider the following equation: where Seeing ( 3), (8), and ( 9), a direct calculation shows that (19) gives the following: which together with (17) implies the following: This means that both of the Lax pairs (3) and ( 11) and ( 19) have the same form; that is, they lead to the same equation (5).Therefore, original solutions , V of the generalized SK equation ( 5) are mapped into its new ones , V by the Darboux transformation (8).

Explicit Solutions of the Generalized Sawada-Kotera Equation
In this section, we will construct explicit solutions of the generalized SK equation ( 5) by using the Darboux transformation (8).