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

Abstract

A generalized Sawada-Kotera equation and its Lax pairs are proposed. With the help of the gauge transformation between spectral problems, a Darboux transformation for the generalized SK equation is constructed. As an application of the Darboux transformation, we give some explicit solutions of the generalized SK equation such as the rational solutions, soliton solutions, and periodic solutions.

1. Introduction

The Sawada-Kotera (SK) equation 𝑢_𝑡=−𝑢_{𝑥𝑥𝑥𝑥𝑥}+15𝑢𝑢_𝑥𝑥−𝑢³_𝑥 (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 [4–8]. The aim of the present paper is using the Darboux transformation [9–12] to study a generalized SK equation: 𝑢_𝑡=−𝑢_{𝑥𝑥𝑥𝑥𝑥}+15𝑢𝑢_𝑥𝑥−𝑢³_𝑥−15𝑣𝑢_𝑥𝑥−10𝑣𝑣_𝑥,𝑣_𝑡=−𝑣_{𝑥𝑥𝑥𝑥𝑥}+15𝑢𝑣_𝑥𝑥+𝑣𝑣_𝑥−3𝑣𝑢²_𝑥+30𝑣𝑢_𝑥𝑥𝑥. (2)

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.

2. 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: ℒ𝜓=𝜆𝜓,𝜓_𝑡=ℬ𝜓, (3) where operators ℒ and ℬ are defined as follows: ℒ=𝜕³−3𝑢𝜕+𝑣,ℬ=9𝜕⁵−45𝑢𝜕³+15𝑣−3𝑢_𝑥𝜕²+153𝑢²−2𝑢_𝑥𝑥+𝑣_𝑥𝜕+10𝑣_𝑥𝑥−3𝑢𝑣. (4) Then the compatibility condition between the two equations of (3) yields the Lax equation, ℒ_𝑡=[ℬ,ℒ], which is equivalent to the generalized SK equation: 𝑢_𝑡=−𝑢_{𝑥𝑥𝑥𝑥𝑥}+15𝑢𝑢_𝑥𝑥−𝑢³_𝑥−15𝑣𝑢_𝑥𝑥−10𝑣𝑣_𝑥,𝑣_𝑡=−𝑣_{𝑥𝑥𝑥𝑥𝑥}+15𝑢𝑣_𝑥𝑥+𝑣𝑣_𝑥−3𝑢²𝑣_𝑥+30𝑢_𝑥𝑣_𝑥𝑥. (5) If we choose 𝑣=0 and 𝑣=−(3/2)𝑢_𝑥, (5) can be, respectively, reduced to the SK equation: 𝑢_𝑡=−𝑢_{𝑥𝑥𝑥𝑥𝑥}−45𝑢²𝑢_𝑥+15𝑢𝑢_𝑥𝑥𝑥+15𝑢_𝑥𝑢_𝑥𝑥 (6) and the Kaup-Kupershmidt equation: 𝑢_𝑡=−𝑢_{𝑥𝑥𝑥𝑥𝑥}−45𝑢²𝑢_𝑥+15𝑢𝑢_𝑥𝑥𝑥+752𝑢_𝑥𝑢_𝑥𝑥. (7)

Theorem 1. Let 𝑓 satisfy (3) with 𝜆=𝜆₀ and 𝐴=−(ln𝑓)_𝑥. Then the following Darboux transformation gives the relation about the original solutions 𝑢, 𝑣 of (5) and its new ones 𝑢, 𝑣: 𝑢=𝑢+𝐴_𝑥,𝑣=𝑣−3𝑢_𝑥+3𝐴𝐴_𝑥−3𝐴_𝑥𝑥. (8)

Proof. Assume that 𝜓 satisfies (3) and 𝐴=−(ln𝑓)_𝑥. Let 𝜓=𝜓_𝑥+𝐴𝜓. (9) Using the first expression of (3), a direct calculation gives the following equations: 𝜓_𝑥=𝜓_𝑥𝑥+𝐴𝜓_𝑥+𝐴_𝑥𝜓,𝜓_𝑥𝑥=𝐴𝜓_𝑥𝑥+3𝑢+2𝐴_𝑥𝜓_𝑥+𝐴_𝑥𝑥−𝑣+𝜆𝜓,𝜓_𝑥𝑥𝑥=3𝑢+𝐴_𝑥𝜓_𝑥𝑥+3𝑢_𝑥−𝑣+3𝐴_𝑥𝑥+3𝑢𝐴+𝜆𝜓_𝑥+𝐴_𝑥𝑥𝑥−𝑣_𝑥−𝑣𝐴+𝜆𝐴𝜓. (10) Substituting (9) and (10) into the following equation: ℒ𝜓=𝜆𝜓, (11) where ℒ=𝜕³−3𝑢𝜕+𝑣 (12) and comparing the coefficients of 𝜓,  𝜓_𝑥,  and 𝜓_𝑥𝑥, we obtain the following: 𝑢−𝑢−𝐴_𝑥3=0,𝑢𝐴−𝑣−3𝑢_𝑥+𝑣−3𝐴_𝑥𝑥3−3𝑢𝐴=0,𝑢𝐴_𝑥−𝑣𝐴−𝐴_𝑥𝑥𝑥+𝑣_𝑥+𝑣𝐴=0. (13)(14) Equation (13) implies the following: 𝑢=𝑢+𝐴_𝑥,𝑣=𝑣−3𝑢_𝑥+3𝐴𝐴_𝑥−3𝐴_𝑥𝑥. (15) Substituting (15) into (14) and integrating it once, we have the following: 3𝑢𝐴+3𝐴𝐴_𝑥−𝐴³−𝐴_𝑥𝑥+𝑣=𝜆₀, (16) where 𝜆₀ is a constant of integration. Through direct calculations, we arrive at the following: −𝑓_𝑥𝑥𝑓=𝐴_𝑥−𝐴²,−𝑓_𝑥𝑥𝑥𝑓=𝐴_𝑥𝑥−3𝐴𝐴_𝑥+𝐴³,−𝑓_4𝑥𝑓=𝐴_𝑥𝑥𝑥−4𝐴𝐴_𝑥𝑥−3𝐴²_𝑥+6𝐴²𝐴_𝑥−𝐴⁴,−𝑓_5𝑥𝑓=𝐴_{𝑥𝑥𝑥𝑥}−5𝐴𝐴_𝑥𝑥𝑥−10𝐴_𝑥𝐴_𝑥𝑥+15𝐴𝐴²_𝑥+10𝐴²𝐴_𝑥𝑥−10𝐴³𝐴_𝑥+𝐴⁵. (17) Using (17) and 𝐴=−(ln𝑓)_𝑥, a simple reduction shows that (16) gives rise to the following: ℒ𝑓=𝜆₀𝑓. (18)
Similarly, we consider the following equation: 𝜓_𝑡=ℬ𝜓, (19) where ℬ=9𝜕⁵−45𝑢𝜕³+15𝑣−3𝑢_𝑥𝜕²+153𝑢²−2𝑢_𝑥𝑥+𝑣_𝑥𝜕+10𝑣_𝑥𝑥−3𝑢𝑣. (20) Seeing (3), (8), and (9), a direct calculation shows that (19) gives the following: 𝐴_𝑡=9𝐴_{𝑥𝑥𝑥𝑥𝑥}−10𝑣_𝑥𝑥𝑥−45𝑢𝐴_𝑥𝑥𝑥+30𝑢𝑣_𝑥+15𝑣𝐴_𝑥𝑥−90𝐴²_𝑥𝑥+30𝑢_𝑥𝑣−90𝑢_𝑥𝐴_𝑥𝑥+45𝑢²𝐴_𝑥+90𝑢𝐴²_𝑥+90𝐴³_𝑥−75𝑢_𝑥𝑥𝐴_𝑥−120𝐴_𝑥𝐴_𝑥𝑥𝑥+15𝑣_𝑥𝐴_𝑥+270𝐴𝐴_𝑥𝐴_𝑥𝑥+30𝐴²𝐴_𝑥𝑥𝑥−30𝐴𝐴_{𝑥𝑥𝑥𝑥}−30𝑢_𝑥𝑥𝑥𝐴−90𝑢𝐴²𝐴_𝑥+90𝑢𝐴𝐴_𝑥𝑥+90𝑢𝑢_𝑥𝐴−30𝑣𝐴𝐴_𝑥−90𝐴²𝐴²_𝑥+90𝑢_𝑥𝐴𝐴_𝑥, (21) which together with (17) implies the following: 𝑓_𝑡=ℬ𝑓. (22)
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 𝑢, 𝑣 of the generalized SK equation (5) are mapped into its new ones 𝑢, 𝑣 by the Darboux transformation (8).

3. 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).

(I) We choose a trivial solution 𝑢=0, 𝑣=0 of (5). Then (3) with 𝜆=𝜆₀ is reduced to the following: 𝜓_𝑥𝑥𝑥=𝜆₀𝜓,𝜓_𝑡=9𝜓_{𝑥𝑥𝑥𝑥𝑥}. (23) Let 𝜆₀=−𝑘³(𝑘≠0). We can see that (23) has a general solution: 𝑓=𝑐₁expΔ₁+𝑐₂−1exp2Δ₁cosΔ₂+𝑐₃−1exp2Δ₁sinΔ₂, (24) where 𝑐_𝑗, (𝑗=1,2,3) are constants and Δ₁=−𝑘𝑥−9𝑘⁵𝑡,Δ₂=√32𝑘𝑥−9𝑘⁵𝑡. (25) Using the Darboux transformation (8), we get an explicit solution of (5)3𝑐𝑢=²²+𝑐²₃𝑘²exp−3Δ₁−6𝑐₁𝑘²exp−(3/2)Δ₁𝑐₂+√3𝑐₃cosΔ₂−√3𝑐₂−𝑐₃sinΔ₂4𝑐₁+𝑐₂exp−(3/2)Δ₁cosΔ₂+𝑐₃exp−(3/2)Δ₁sinΔ₂²,−−3𝑣=9exp2Δ₁𝑘³4𝑐²₁𝑐₂−√3𝑐₃cosΔ₂+√3𝑐₂+𝑐₃sinΔ₂+exp−3Δ₁𝑐₂−√3𝑐₃cosΔ₂+√3𝑐₂+𝑐₃sinΔ₂𝑐²₂+𝑐²₃−3+2exp2Δ₁𝑐₁6+cos2Δ₂−√3sin2Δ₂𝑐²₂+2sin2Δ₂+√3cos2Δ₂𝑐₂𝑐₃+6−cos2Δ₂+√3sin2Δ₂𝑐²₃×8𝑐₁+𝑐₂−3exp2Δ₁cosΔ₂+𝑐₃−3exp2Δ₁sinΔ₂³⁻¹. (26)

Particularly, when we choose 𝑐₁=0, 𝑐₂=𝑐₃=1, we can get a periodic solution of (5): 𝑢=3𝑘²2cosΔ₂+sinΔ₂²,𝑣=−3𝑘³+3𝑘³12+√32cosΔ₂−sinΔ₂cosΔ₂+sinΔ₂³+3𝑘³2sin2Δ₂−1cosΔ₂+sinΔ₂². (27) Plots of the solutions are given in Figures 1 and 2.

Figure 1 

𝑢.

Figure 2 

𝑣.

(II) We consider the trivial solution 𝑢=0, 𝑣=1 of (5). Then (3) with 𝜆=𝜆₀ is reduced to the following: 𝜓_𝑥𝑥𝑥=𝜆₀−1𝜓,𝜓_𝑡=9𝜓_{𝑥𝑥𝑥𝑥𝑥}+15𝜓_𝑥𝑥. (28)

Case 1. When 𝜆₀=1, it is easy to see that (28) has a general solution: 𝑓=𝛼𝑥²+𝛽𝑥+𝛾+30𝛼𝑡, (29) where 𝛼, 𝛽, and 𝛾 are constants. Using the Darboux transformation (8), we get a rational solution of the generalized SK equation (5): 𝑢=2𝛼²𝑥²+2𝛼𝛽𝑥+𝛽²−2𝛼𝛾−60𝛼²𝑡𝛼𝑥²+𝛽𝑥+𝛾+30𝛼𝑡²,3𝛽𝑣=1+²−4𝛼𝛾+30𝛼𝑡𝛽+2𝛼𝑥𝛼𝑥²+𝛽𝑥+𝛾+30𝛼𝑡³. (30)

Case 2. When 𝜆₀=1−𝑘³(𝑘≠0), (28) has a general solution: 𝑓=𝑐₁expΔ₁+𝑐₂−1exp2Δ₁cosΔ₂+𝑐₃−1exp2Δ₁sinΔ₂, (31) where 𝑐_𝑗, (𝑗=1,2,3) are constants and Δ₁=−𝑘𝑥−9𝑘⁵−15𝑘²Δ𝑡,₂=√32𝑘𝑥−9𝑘⁵𝑡+15𝑘²𝑡. (32) Using the Darboux transformation (8), we get an explicit solution of (5):3𝑐𝑢=²²+𝑐²₃𝑘²exp−3Δ₁−6𝑐₁𝑘²exp−(3/2)Δ₁𝑐₂+√3𝑐₃cosΔ₂−√3𝑐₂−𝑐₃sinΔ₂4𝑐₁+𝑐₂exp−(3/2)Δ₁cosΔ₂+𝑐₃exp−(3/2)Δ₁sinΔ₂²,−3𝑣=1−9exp2Δ₁𝑘³4𝑐²₁𝑐₂−√3𝑐₃cosΔ₂+√3𝑐₂+𝑐₃sinΔ₂+exp−3Δ₁𝑐₂−√3𝑐₃cosΔ₂+√3𝑐₂+𝑐₃sinΔ₂𝑐²₂+𝑐²₃−3+2exp2Δ₁𝑐₁6+cos2Δ₂−√3sin2Δ₂𝑐²₂+2sin2Δ₂+√3cos2Δ₂𝑐₂𝑐₃+6−cos2Δ₂+√3sin2Δ₂𝑐²₃×8𝑐₁+𝑐₂−3exp2Δ₁cosΔ₂𝑐₃−3exp2Δ₁sinΔ₂³⁻¹. (33)
(III) We choose another trivial solution 𝑢=1, 𝑣=0 of (5). Then (3) with 𝜆=𝜆₀ is reduced to the following: 𝜓_𝑥𝑥𝑥=3𝜓_𝑥+𝜆₀𝜓,𝜓_𝑡=9𝜓_{𝑥𝑥𝑥𝑥𝑥}−45𝜓_𝑥𝑥𝑥+45𝜓_𝑥. (34)

Case 1. For 𝜆₀=0, a direct calculation gives a general solution of (34): 𝑓=𝑐₁exp(Δ)+𝑐₂exp(−Δ), (35) where 𝑐₁, 𝑐₂ are constants and Δ=√3𝑥−9√3𝑡. Using the Darboux transformation (8), we get a soliton solution of the generalized SK equation (5) (𝑐₁=𝑐₂=1): 𝑢=3[tanh(Δ)]²−2,𝑣=9√3[tanh(Δ)]³−9√3tanh(Δ). (36) Plots of the solutions are given in Figures 3 and 4.

Figure 3 

𝑢.

Figure 4 

𝑣.

Case 2. For 𝜆₀=𝑘³−3𝑘(𝑘≠0), (34) has a general solution: 𝑓=𝑐₁expΔ₁+𝑐₂expΔ₂+𝑐₃expΔ₃, (37) where 𝑐_𝑗, (𝑗=1,2,3) are constants and Δ₁=𝑘𝑥+9𝑘⁵−45𝑘³Δ+45𝑘𝑡,₂=−𝑘+12−3𝑘²2𝑥−925𝑘−5𝑘³+𝑘⁵+12−3𝑘²−3𝑘²12−3𝑘²+𝑘⁴12−3𝑘²Δ𝑡,₃=−𝑘−12−3𝑘²2𝑥+92−5𝑘+5𝑘³−𝑘⁵+12−3𝑘²−3𝑘²12−3𝑘²+𝑘⁴12−3𝑘²𝑡. (38) Using the Darboux transformation (8), we get an explicit solution of (5): 3𝑢=1+22𝑐₂𝑐₃𝑘²−4expΔ₂+Δ₃−𝑐₁𝑐₂2+𝑘²−𝑘12−3𝑘²expΔ₁+Δ₂−𝑐₁𝑐₃2+𝑘²+𝑘12−3𝑘²expΔ₁+Δ₃𝑐₁expΔ₁+𝑐₂expΔ₂+𝑐₃expΔ₃²,𝑣=3𝑘³Δ−9𝑘+3𝑐₁expΔ₁+𝑐₂expΔ₂+𝑐₃expΔ₃³, (39) where Δ=𝑘3−𝑘²𝑐³₁exp3Δ₁+𝑐³₂exp3Δ₂+𝑐³₃exp3Δ₃−12𝑐₁𝑐₂𝑐₃expΔ₁+Δ₂+Δ₃+32𝑐²₂𝑐₃exp2Δ₂+Δ₃×2𝑘−𝑘³−412−3𝑘²+𝑘²12−3𝑘²+32𝑐₂𝑐²₃expΔ₂+2Δ₃×2𝑘−𝑘³+412−3𝑘²−𝑘²12−3𝑘²−32𝑐²₁𝑐₂exp2Δ₁+Δ₂1+𝑘²𝑘−12−3𝑘²−32𝑐²₁𝑐₃exp2Δ₁+Δ₃1+𝑘²𝑘+12−3𝑘²+32𝑐₁𝑐²₃expΔ₁+2Δ₃𝑘8−𝑘²+𝑘12−3𝑘²+32𝑐₁𝑐²₂expΔ₁+2Δ₂𝑘8−𝑘²−𝑘12−3𝑘². (40)

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11171312) and Henan Natural Science Foundation of Basic Research (no. 102300410214).