Computing Exact Solutions to a Generalized Lax-Sawada-Kotera-Ito Seventh-Order KdV Equation

Abstract

The Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7).

1. Introduction

Many direct and computational methods have been used to handle nonlinear partial differential equations (NLPDE’s). Some methods used in a satisfactory way to obtain exact solutions to NLPDE’s are inverse scattering method [1], Hirota bilinear method [2, 3], Backlund transformations [4], Painlevé analysis [5], Lie groups [6], the tanh method [7], the generalized tanh method [8, 9], the extended tanh method [10–12], the improved tanh-coth method [13, 14], the Exp-function method [15–17], the projective Riccati equation method [18], the generalized projective Riccati equations method [19–24], the extended hyperbolic function method [25], variational iteration method [26, 27], He’s polynomials [28], homotopy perturbation method [29], and many other methods [30]. However, there is not a unified method that could be used to handle all NLPDE’s; in this sense, the implementation of new methods or variants of the some well-known methods is relevant. The principal objective of this paper consists in obtaining exact traveling wave solutions which include periodic and soliton solutions to a particular case of the general seventh-order KdV(KdV7), which is a generalization of the seventh-order Sawada-Kotera-Ito (SKI-KdV7) equation, by using a variant of the exp-function method. The general seventh-order KdV (KdV7) equation [31] reads 𝑢_𝑡+𝑎𝑢³𝑢_𝑥+𝑏𝑢³_𝑥+𝑐𝑢𝑢_𝑥𝑢_𝑥𝑥+𝑑𝑢²𝑢_𝑥𝑥𝑥+𝑒𝑢_2𝑥𝑢_3𝑥+𝑓𝑢_𝑥𝑢_4𝑥+𝑔𝑢𝑢_5𝑥+𝑢_7𝑥=0. (1.1) The (KdV7) was introduced initially by Pomeau et al. [32] for discussing the structural stability of KdV equation under a singular perturbation. Some particular cases of (1.1) are (i)seventh-order Lax equation [1, 6] (𝑎=140,𝑏=70,𝑐=280,𝑑=70,𝑒=70,𝑓=42,𝑔=14): 𝑢_𝑡+140𝑢³𝑢_𝑥+70𝑢³_𝑥+280𝑢𝑢_𝑥𝑢_𝑥𝑥+70𝑢²𝑢_𝑥𝑥𝑥+70𝑢_2𝑥𝑢_3𝑥+42𝑢_𝑥𝑢_4𝑥+14𝑢𝑢_5𝑥+𝑢_7𝑥=0; (1.2)(ii)seventh-order Sawada-Kotera-Ito equation [1, 8–10] (𝑎=252,𝑏=63,𝑐=378,𝑑=126,𝑒=63,𝑓=42,𝑔=21): 𝑢_𝑡+252𝑢³𝑢_𝑥+63𝑢³_𝑥+378𝑢𝑢_𝑥𝑢_𝑥𝑥+126𝑢²𝑢_𝑥𝑥𝑥+63𝑢_2𝑥𝑢_3𝑥+42𝑢_𝑥𝑢_4𝑥+21𝑢𝑢_5𝑥+𝑢_7𝑥=0; (1.3)(iii)seventh-order Kaup-Kupershmidt equation [1, 7] (𝑎=2016,𝑏=630,𝑐=2268,𝑑=504,𝑒=252,𝑓=147,𝑔=42): 𝑢_𝑡+2016𝑢³𝑢_𝑥+630𝑢³_𝑥+2268𝑢𝑢_𝑥𝑢_𝑥𝑥+504𝑢²𝑢_𝑥𝑥𝑥+252𝑢_2𝑥𝑢_3𝑥+147𝑢_𝑥𝑢_4𝑥+42𝑢𝑢_5𝑥+𝑢_7𝑥=0. (1.4)

2. Generalization of the Lax KdV7 and the Sawada-Kotera-Ito KdV7

Observe that (1.2) and (1.3) satisfy the relation𝑑𝑎=63(𝑒+𝑓+𝑔). (2.1) For this reason we will study equation𝑢_𝑡+𝑑63(𝑒+𝑓+𝑔)𝑢³𝑢_𝑥+𝑏𝑢³_𝑥+𝑐𝑢𝑢_𝑥𝑢_𝑥𝑥+𝑑𝑢²𝑢_𝑥𝑥𝑥+𝑒𝑢_2𝑥𝑢_3𝑥+𝑓𝑢_𝑥𝑢_4𝑥+𝑔𝑢𝑢_5𝑥+𝑢_7𝑥=0. (2.2) We seek solutions to (2.2) in the Cole-Hopf form 𝑢(𝑡,𝑥)=𝐴𝜕_𝑥tanh(𝜉), (2.3) where 𝐴 is some constant to be determined later and 𝜉=𝜉(𝑡,𝑥)=𝜇(𝑥+𝜆𝑡+𝛿),𝜇,𝛿,𝜆=const. (2.4) Substituting (2.3) into (2.2), we obtain a polynomial equation in the variable 𝜁=exp(𝜉). Equating the coefficients of the different powers of 𝜁 to zero, we obtain following algebraic system:𝜆+64𝜇⁶=0,64𝜇⁵(𝐴(𝑒+𝑓+𝑔)−247𝜇)+5𝜆=0,64𝜇⁴(𝐴²(𝑏+𝑐+𝑑)−3𝐴𝜇(5𝑒+9𝑓+19𝑔)+4293𝜇²)+9𝜆=0,64𝜇³(𝐴³𝑑(𝑒+𝑓+𝑔)−63𝐴²𝜇(3𝑏+5𝑐+11𝑑)+126𝐴𝜇²(28𝑒+46𝑓+151𝑔)−983997𝜇³)+315𝜆=0. (2.5) Eliminating 𝐴,𝜆, and 𝜇 from system (2.5) gives1𝑏=𝑑+5126(𝑒+𝑓+𝑔)(𝑒−5𝑓+10𝑔),𝑐=21𝑔(𝑒+𝑓+𝑔)−2𝑑. (2.6) It is easy to verify that (1.2) and (1.3) are particular cases of general KdV7 equation (1.1) subject to (2.1) and (2.6). This motivates us to define the generalized Lax-Sawada-Kotera-Ito seventh-order equation (LSKI KdV7) as follows:𝑢_𝑡+163𝑑(𝑒+𝑓+𝑔)𝑢³𝑢_𝑥1+(𝑑+126(𝑒+𝑓+𝑔)(𝑒−5𝑓+10𝑔))𝑢³_𝑥5+(21𝑔(𝑒+𝑓+𝑔)−2𝑑)𝑢𝑢_𝑥𝑢_𝑥𝑥+𝑑𝑢²𝑢_𝑥𝑥𝑥+𝑒𝑢_2𝑥𝑢_3𝑥+𝑓𝑢_𝑥𝑢_4𝑥+𝑔𝑢𝑢_5𝑥+𝑢_7𝑥=0. (2.7)

3. Solutions to Generalized LSKI KdV7

In order to look for solutions to (2.7), we will use the exp ansatz 𝑞𝑢(𝜉)=𝑝+,1+𝑟exp(−𝜉)+𝑠exp(𝜉) (3.1) where 𝑝, 𝑞, 𝑟, and 𝑠 are some constants. Substituting (3.1) into (2.7) gives an algebraic system. Solving it, we obtain 1𝜆=−63𝑑(𝑒+𝑓+𝑔)𝑝³−𝜇²(𝑑𝑝²+𝑔𝑝𝜇²+𝜇⁴),𝑞=126𝜇²1𝑒+𝑓+𝑔,𝑠=4𝑟,𝑟=𝑟,𝜇=𝜇. (3.2) From (2.4), (3.1), and (3.2), we obtain following solution to (2.7) subject:𝑢(𝑥,𝑡)=𝑝+126𝜇²,1(𝑒+𝑓+𝑔)(1+𝑟exp(𝜉)+(1/4𝑟)exp(−𝜉))𝜉=𝜇(𝑥+𝜆𝑡+𝛿),𝜆=−63𝑑(𝑒+𝑓+𝑔)𝑝³−𝜇²(𝑑𝑝²+𝑔𝑝𝜇²+𝜇⁴). (3.3) In particular, if 𝑟=1/2, equation (3.3) gives𝑢(𝑥,𝑡)=𝑝+63𝜇²𝑒+𝑓+𝑔sech²(𝜇21(𝑥+𝜆𝑡+𝛿)),𝜆=−63𝑑(𝑒+𝑓+𝑔)𝑝³−(𝑑𝑝²+𝑔𝑝𝜇²+𝜇⁴)𝜇². (3.4) Replacing 𝜇 with 𝜇√−1 gives the following periodic solutions: 𝑢(𝑥,𝑡)=𝑝−63𝜇²𝑒+𝑓+𝑔sec²(𝜇21(𝑥+𝜆𝑡+𝛿)),𝜆=−63𝑑(𝑒+𝑓+𝑔)𝑝³+(𝑑𝑝²−𝑔𝑝𝜇²+𝜇⁴)𝜇². (3.5) On the other hand, if 𝑟=−1/2, equation (3.3) gives𝑢(𝑥,𝑡)=𝑝−63𝜇²𝑒+𝑓+𝑔csch²(𝜇21(𝑥+𝜆𝑡+𝛿)),𝜆=−63𝑑(𝑒+𝑓+𝑔)𝑝³−(𝑑𝑝²+𝑔𝑝𝜇²+𝜇⁴)𝜇². (3.6) Replacing 𝜇 with 𝜇√−1 gives the following periodic solutions:𝑢(𝑥,𝑡)=𝑝−63𝜇²𝑒+𝑓+𝑔csc²(𝜇21(𝑥+𝜆𝑡+𝛿)),𝜆=−63𝑑(𝑒+𝑓+𝑔)𝑝³+(𝑑𝑝²−𝑔𝑝𝜇²+𝜇⁴)𝜇². (3.7)

4. Solutions to Sawada-Kotera-Ito KdV7 Equation

From (3.3)–(3.7) with 𝑑=126,𝑒=63,𝑓=42, and 𝑔=21, we obtain the following analytic solutions to equation (1.3): 𝑢(𝑥,𝑡)=𝑝+4𝑟𝜇²exp(𝜇(𝑥+𝜆𝑡+𝛿))(1+2𝑟exp(𝜇(𝑥+𝜆𝑡+𝛿)))²,𝜆=−252𝑝³−126𝑝²𝜇²−21𝑝𝜇⁴−𝜇⁶,1𝑢(𝑥,𝑡)=𝑝+2𝜇²sech²(12𝜇(𝑥+𝜆𝑡+𝛿)),𝜆=−252𝑝³−126𝑝²𝜇²−21𝑝𝜇⁴−𝜇⁶,1𝑢(𝑥,𝑡)=𝑝−2𝜇²sec²(12𝜇(𝑥+𝜆𝑡+𝛿)),𝜆=−252𝑝³+126𝑝²𝜇²−21𝑝𝜇⁴+𝜇⁶,1𝑢(𝑥,𝑡)=𝑝−2𝜇²csch²(12𝜇(𝑥+𝜆𝑡+𝛿)),𝜆=−252𝑝³−126𝑝²𝜇²−21𝑝𝜇⁴−𝜇⁶. (4.1)

5. Conclusions

We exhibited an equation that generalizes both seventh-order Lax equation and seventh-order Sawada-Kotera-Ito equation. At the same time, we obtained exact solutions to these equations with the aid of a Cole-Hopf ansatz. These same ideas are suitable for the seventh-order Kaup-Kupershmidt equation. We think that some of the solutions in this work are new in the open literature. We may apply other methods to find exact solutions to a variety of nonlinear PDE’s. See [3, 12–52].