Application of the Cole-Hopf Transformation for Finding Exact Solutions to Several Forms of the Seventh-Order KdV Equation

Abstract

We use a generalized Cole-Hopf transformation to obtain a condition that allows us to find exact solutions for several forms of the general seventh-order KdV equation (KdV7). A remarkable fact is that this condition is satisfied by three well-known particular cases of the KdV7. We also show some solutions in these cases. In the particular case of the seventh-order Kaup-Kupershmidt KdV equation we obtain other solutions by some ansatzes different from the Cole-Hopf transformation.

1. Introduction

During the last years scientists have seen a great interest in the investigation of nonlinear processes. The reason for this is that they appear in various branches of natural sciences and particularly in almost all branches of physics: fluid dynamics, plasma physics, field theory, nonlinear optics, and condensed matter physics. In this sense, the study of nonlinear partial differential equations NLPDEs and their solutions has great relevance today. Some analytical methods such as Hirota method [1] and scattering inverse method [2] have been used to solve some NLPDEs. However, the use of these analytical methods is not an easy task. Therefore, several computational methods have been implemented to obtain exact solutions for these models. It is clear that the knowledge of closed-form solutions of (NLPDEs) facilitates the testing of numerical solvers, helps physicists to better understand the mechanism that governs the physic models, provides knowledge of the physic problem, provides possible applications, and aids mathematicians in the stability analysis of solutions. The following are some of the most important computational methods used to obtain exact solutions to NLPDEs: the tanh method [3], the generalized tanh method [4, 5], the extended tanh method [6, 7], the improved tanh-coth method [8, 9], and the Exp-function method [10, 11]. Practically, there is no unified method that could be used to handle all types of nonlinear problems. All previous computational methods are based on the reduction of the original equation to equations in fewer dependent or independent variables. The main idea is to find such variables and, by passing to them, to obtain simpler equations. In particular, finding exact solutions of some partial differential equations in two independent variables may be reduced to the problem of finding solutions of an appropriate ordinary differential equation (or a system of ordinary differential equations). Naturally, the ordinary differential equation thus obtained does not give all solutions of the original partial differential equation, but provides a class of solutions with some specific properties.

The simplest classes of exact solutions to a given partial differential equation are those obtained from a traveling-wave transformation.

The general seventh-order Korteweg de Vries equation (KdV7) [12] 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) which has been introduced by Pomeau et al. [13] for discussing the structural stability of standard Korteweg de Vries equation (KdV) under a singular perturbation. Some well-known particular cases of (1.1) are the following:

(i)seventh-order Sawada-Kotera-Ito equation [12, 14–16] (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, (1.2)(ii)seventh-order Lax equation [12, 17] (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, (1.3)(iii)seventh-order Kaup-Kupershmidt equation [12, 18] (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)

Exact solutions for several forms of (1.1) have been obtained by other authors using the Hirota method [19], the tanh-coth method [19], and He's variational iteration method [20]. However, the principal objective of this work consists on presenting a condition over the coefficients of (1.1) to obtain new exact traveling-wave solutions different from those in [19, 20] by using a Cole-Hopf transformation. We show that some of the previous models (1.2)–(1.4) satisfy this condition, and therefore in these cases we obtain new exact solutions for them.

This paper is organized as follow: In Section 2, we make use of a Cole-Hopf transformation to derive a condition over the coefficients of (1.1) for the existence of traveling-wave solutions. In Section 3 we present new exact solutions for several forms of (1.1) which satisfy the condition given in Section 2. In Section 4, new exact solutions to (1.4) are obtained using other anzatzes. Finally, some conclusions are given.

2. Using the Cole-Hopf Transformation

The main purpose of this section consists on establishing a polynomial equation involving the coefficients 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, and 𝑔 of (1.1) that allows us to find traveling wave solutions (soliton and periodic solutions) using a special Cole-Hopf transformation. To this end, we seek solutions to (1.1) in the form 𝑢=𝑢(𝑥,𝑡)=𝑢(𝜉), where

𝜉=𝑘𝑥+𝜔𝑡+𝛿,𝑘,𝜔,with𝑘≠0,𝜔≠0,𝛿=const. (2.1)

From (1.1) and (2.1) we obtain the following seventh-order nonlinear ordinary differential equation:

𝑏𝑘³(𝑢^′(𝜉))³+𝑑𝑘³𝑢²(𝜉)𝑢^′′′(𝜉)+𝑒𝑘⁵𝑢^′′(𝜉)𝑢^′′′(𝜉)+𝑢^′(𝜉)(𝜔+𝑎𝑘𝑢³(𝜉)+𝑐𝑘³𝑢(𝜉)𝑢^′′(𝜉)+𝑓𝑘⁵𝑢⁽⁴⁾(𝜉))+𝑔𝑘⁵𝑢(𝜉)𝑢⁽⁵⁾(𝜉)+𝑘⁷𝑢⁽⁷⁾(𝜉)=0. (2.2) With the aim to find exact solutions to (2.2) we use the following Cole-Hopf transformation [21, 22]:

𝑑𝑢(𝜉)=𝐴²𝑑𝜉²ln(1+exp(𝜉))+𝐵, (2.3) where 𝐴≠0 and 𝐵 are arbitrary real constants.

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, the following algebraic system is obtained:

𝑎𝐵³𝑘+𝐵²𝑑𝑘³+𝐵𝑔𝑘⁵+𝑘⁷+𝜔=0,3𝑎𝐴𝐵²𝑘+6𝑎𝐵³𝑘+𝐴𝑐𝐵𝑘³+2𝐴𝐵𝑑𝑘³+𝐴𝑒𝑘⁵+𝐴𝑓𝑘⁵+𝐴𝑔𝑘⁵−6𝐵²𝑑𝑘³−54𝐵𝑔𝑘⁵−246𝑘⁷+6𝜔=0,3𝑎𝐴²𝐵𝑘+12𝑎𝐴𝐵²𝑘+15𝑎𝐵³𝑘+𝐴²𝑏𝑘³+𝐴²𝑐𝑘³+𝐴²𝑐𝑘³−2𝐴𝑐𝐵𝑘³−16𝐴𝐵𝑑𝑘³−26𝐴𝑓𝑘⁵−56𝐴𝑔𝑘⁵−14𝐴𝑒𝑘⁵−33𝐵²𝑑𝑘³+135𝐵𝑔𝑘⁵+4047𝑘⁷+15𝜔=0,𝑎𝐴³𝑘+6𝑎𝐴²𝐵𝑘+18𝑎𝐴𝐵²𝑘+20𝑎𝐵³𝑘−2𝐴²𝑏𝑘³−4𝐴²𝑐𝑘³−10𝐴²𝑑𝑘³−6𝐴𝑐𝐵𝑘³−36𝐴𝐵𝑑𝑘³+66𝐴𝑓𝑘⁵+246𝐴𝑔𝑘⁵+42𝐴𝑒𝑘⁵−52𝐵²𝑑𝑘³+380𝐵𝑔𝑘⁵−11572𝑘⁷+20𝜔=0. (2.4)(2.5)(2.6)(2.7)

From equations (2.4) and (2.5), we obtain

𝜔=−(𝑎𝐵³𝑘+𝐵²𝑑𝑘³+𝐵𝑔𝑘⁵+𝑘⁷),𝐴=−6𝜔+246𝑘⁷−6𝐵³𝑎𝑘+54𝐵𝑔𝑘⁵+6𝐵²𝑑𝑘³𝑘⁵𝑒+𝑓𝑘⁵+𝑔𝑘⁵+3𝐵²𝑎𝑘+𝐵𝑐𝑘³+2𝐵𝑑𝑘³. (2.8)(2.9)

Now, we substitute expression for 𝜔 in (2.8) into (2.6) and (2.7) and solve them for 𝑏 and 𝑒 to obtain

1𝑏=−𝐴²𝑘²(𝑎𝐴³+15𝑎𝐴²𝐵+54𝑎𝐴𝐵²−𝐴²𝑐𝑘²−7𝐴²𝑑𝑘²−12𝐴𝐵𝑐𝑘²−84𝐴𝐵𝑑𝑘²−12𝐴𝑓𝑘⁴+78𝐴𝑔𝑘⁴−216𝐵²𝑑𝑘²+720𝐵𝑔𝑘⁴+504𝑘⁶1),𝑒=−14𝐴𝑘⁴(𝑎𝐴³+12𝑎𝐴²𝐵+42𝑎𝐴𝐵²−2𝐴²𝑐𝑘²−8𝐴²𝑑𝑘²−10𝐴𝐵𝑐𝑘²−68𝐴𝐵𝑑𝑘²+14𝐴𝑓𝑘⁴+134𝐴𝑔𝑘⁴−168𝐵²𝑑𝑘²+600𝐵𝑔𝑘⁴−3528𝑘⁶). (2.10)(2.11) Substitution of (2.10) and (2.11) into (2.9) gives

𝐴(𝐴+12𝐵)(𝑎𝐴²−2𝐴𝑐𝑘²−8𝐴𝑑𝑘²+120𝑔𝑘⁴)𝑎𝐴³+12𝑎𝐴²𝐵−2𝐴²𝑐𝑘²−8𝐴²𝑑𝑘²−24𝐴𝐵𝑐𝑘²−96𝐴𝐵𝑑𝑘²+120𝐴𝑔𝑘⁴−168𝐵²𝑑𝑘²+600𝐵𝑔𝑘⁴−3528𝑘⁶=0. (2.12) From this last equation we obtain, in particular,

𝐴=−12𝐵, (2.13) or

𝑘𝐴=²𝑎(𝑐+4𝑑±(𝑐+4𝑑)²−120𝑎𝑔). (2.14) These last two expressions are valid for

𝐵²𝑑+5𝐵𝑔𝑘²+21𝑘⁴≠0. (2.15) Suppose that 𝐴=−12𝐵. Substituting this expression into (2.10) and (2.11) gives

𝑏=−−216𝑎𝐵³−216𝐵²𝑑𝑘²+144𝐵𝑓𝑘⁴−216𝐵𝑔𝑘⁴+504𝑘⁶144𝐵²𝑘²,𝑒=−504𝑎𝐵³−168𝐵²𝑐𝑘²−504𝐵²𝑑𝑘²−168𝐵𝑓𝑘⁴−1008𝐵𝑔𝑘⁴−3528𝑘⁶168𝐵𝑘⁴. (2.16)(2.17) Eliminating 𝐵, 𝑘, and 𝜔 from (2.8), (2.16), and (2.17), we obtain the following polynomial equation in the variables 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓 and 𝑔:

28224𝑎²+𝑃(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)𝑎+𝑄(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)=0, (2.18) where

𝑃(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)=3(56𝑏(𝑒+19𝑓−21𝑔)−28𝑐(3𝑒−7𝑓+33𝑔)+(𝑒+3(𝑓+𝑔))((𝑒−5𝑓+15𝑔)(𝑒+3(𝑓+𝑔))−336𝑑)),𝑄(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)=(2𝑏+𝑐)(252𝑏²−2𝑏(42𝑐+504𝑑−(𝑒+𝑓+6𝑔)(𝑒−5𝑓+15𝑔))+7𝑐²+𝑐(168𝑑−(2𝑓−3𝑔)(𝑒−5𝑓+15𝑔))+3𝑑(336𝑑−(𝑒−5𝑓+15𝑔)(𝑒+3(𝑓+𝑔)))). (2.19) We will call (2.18) the first discriminant equation. It is remarkable the fact that (1.2), (1.3), and (1.4) satisfy this equation. All seventh order equations that satisfy the first discriminant equation (2.18) may be solved exactly by means of the Cole-Hopf transformation (2.3).

Now, suppose that 𝐴≠−12𝐵. Since 𝐴≠0, then (2.14) holds. Reasoning in a similar way, from (2.14) we obtain the following second polynomial equation in terms of the coefficients of (1.1), which we shall call the second discriminant equation for (1.1) associated to Cole-Hopf transformation (2.3):

17640𝑎²+𝑃(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)𝑎+𝑄(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)=0, (2.20) where

𝑃(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)=−3(280𝑏𝑔+14𝑐(3𝑒+5(𝑓+6𝑔))+56𝑑(3𝑒+5(𝑓+3𝑔))−𝑔(3𝑒+5(𝑓+2𝑔))²),𝑄(𝑏,𝑐,𝑑,𝑒,𝑓,𝑔)=(𝑏+𝑐+𝑑)(10𝑔²(𝑏−3𝑑)−𝑔(𝑐+4𝑑)(3𝑒+5𝑓)+14(𝑐+4𝑑)²). (2.21)

Direct calculations show that (1.2) and (1.3) satisfy not only the first discriminant equation (2.18), but also the second one. However, (1.4) does not satisfy the second discriminant equation (2.20). All seventh-order equations that satisfy (2.20) can be solved exactly by using the Cole-Hopf transformation (2.3).

3. Solutions to the KdV7

Suppose that 𝐴=−12𝐵. Then (1.1) satisfies (2.18). In this case, we obtain the following solution to (1.1):

𝑢₁𝐴(𝑥,𝑡)=−12(1−3sech²(12𝑘𝜉)),𝜉=𝜉(𝑥,𝑡)=𝑘𝑥+𝜔𝑡+𝛿,𝜔=−1728(−𝑎𝐴³+12𝐴²𝑑𝑘²−144𝐴𝑔𝑘⁴+1728𝑘⁶1),𝑏=8𝐴²𝑘²(−𝑎𝐴³+12𝐴²𝑑𝑘²+96𝐴𝑓𝑘⁴−144𝐴𝑔𝑘⁴−4032𝑘⁶1),𝑒=−48𝐴𝑘⁴(𝑎𝐴³−4𝐴²𝑐𝑘²−12𝐴²𝑑𝑘²+48𝐴𝑓𝑘⁴+288𝐴𝑔𝑘⁴−12096𝑘⁶). (3.1) So that, from 𝑢₁ we can obtain explicit solutions to (1.2), (1.3), and (1.4). They are

(i)Sawada-Kotera-Ito equation (1.2): 𝐴=2𝑘²1,𝜔=6𝑘⁷,1𝑢(𝑥,𝑡)=−6𝑘²+12𝑘²sech²(121𝑘𝑥+𝑘12⁷𝑡+𝛿), (3.2)(3.3)(ii)Lax equation (1.3): 𝐴=2𝑘²1,𝜔=𝑘27⁷,1𝑢(𝑥,𝑡)=−6𝑘²+12𝑘²sech²(121𝑘𝑥+𝑘54⁷𝑡+𝛿), (3.4)(iii)Kaup-Kupershmidt equation (1.4): 1𝐴=2𝑘²1,𝜔=𝑘48⁷,1𝑢(𝑥,𝑡)=−𝑘24²+18𝑘²sech²(121𝑘𝑥+𝑘96⁷𝑡+𝛿). (3.5)(3.6)

Figure 1 shows the graph of (3.6) for 𝑘=1.9, 𝛿=−1, −3≤𝑥≤3 and 0≤𝑡≤3.

Figure 1 

Graph of solution given by (3.6).

It may be verified that condition (2.15) holds when 𝐴 is given by either (3.2) or (3.5) since 𝑘≠0.

Now, let us assume that 𝐴≠−12𝐵. In this case, 𝐴 is given by (2.14). Then (1.1) satisfies (2.20). We obtain the following solution to (1.1):

𝑢₂𝑘(𝑥,𝑡)=²4𝑎(𝑐+4𝑑±(𝑐+4𝑑)²−120𝑎𝑔)sech²(12𝜉)+𝐵,𝜉=𝜉(𝑥,𝑡)=𝑘𝑥+𝜔𝑡+𝛿,𝜔=−(𝑎𝐵³𝑘+𝐵²𝑑𝑘³+𝐵𝑔𝑘⁵+𝑘⁷1),𝑏=2𝐴²𝑘²(6𝑘²(𝑑(𝐴²+12𝐴𝐵+72𝐵²)+2𝐴𝑘²(2𝑓−3𝑔)−168𝑘⁴)−𝑎𝐴(𝐴²+18𝐴𝐵+108𝐵²1)),𝑒=−2𝐴𝑘⁴(−𝑎𝐴𝐵(𝐴+6𝐵)+4𝐵𝑑𝑘²(𝐴+6𝐵)−2𝐴𝑘⁴(𝑓+𝑔)+504𝑘⁶). (3.7) So that, from 𝑢₂ we can obtain explicit solutions to (1.2) and (1.3). The first is

(i)Sawada-Kotera-Ito equation (1.2): 𝐴=2𝑘²,𝜔=−252𝐵³𝑘−126𝐵²𝑘³−21𝐵𝑘⁵−𝑘⁷,1𝑢(𝑥,𝑡)=2𝑘²sech²(12(𝑘𝑥+𝜔𝑡+𝛿))+𝐵, (3.8) where 𝑘𝐵≠−²3𝑘,𝐵≠−²2. (3.9) Restriction (1.2) guarantees condition (2.15). However, when 𝐵=−𝑘²/2, we obtain following solution to (1.2): 𝐴=2𝑘²,𝜔=192𝑘⁷,1𝑢(𝑥,𝑡)=−2𝑘²tanh²(12(𝑘𝑥+192𝑘⁷𝑡+𝛿)), (3.10) and when 𝐵=−𝑘²/3, another solution is given by 𝐴=2𝑘²4,𝜔=3𝑘⁷,𝑘𝑢(𝑥,𝑡)=−²3+12𝑘²sech²(124(𝑘𝑥+3𝑘⁷𝑡+𝛿)), (3.11)(ii)The second solution is Lax equation (1.3) 𝐴=2𝑘²,𝜔=−140𝐵³𝑘−70𝐵²𝑘³−14𝐵𝑘⁵−𝑘⁷,1𝑢(𝑥,𝑡)=2𝑘²sech²(12(𝑘𝑥+𝜔𝑡+𝛿))+𝐵, (3.12) for any real number 𝐵.

4. Other Exact Solutions to the Seventh-Order Kaup-Kupershmidt Equation

As we remarked in the previous section, the seventh-order Kaup-Kupershmidt equation (1.4) does not satisfy the second discriminant equation (2.20). However, we can use other methods to obtain exact solutions different from those we already obtained, for instance,

(i)an exp rational ansatz: 𝑏𝑢(𝜉)=𝑎+1+𝑝exp(𝜉)+𝑞exp(−𝜉),𝜉=𝑘𝑥+𝜔𝑡+𝛿, (4.1)(ii)the tanh-coth ansatz [23]: 𝑢(𝜉)=𝑝+𝑎tanh(𝜉)+𝑏coth(𝜉)+𝑐tanh²(𝜉)+𝑑coth²(𝜉),𝜉=𝑘𝑥+𝜔𝑡+𝛿, (4.2) where 𝑎, 𝑏, 𝑐, 𝑑, 𝑘, 𝑝, 𝑞, 𝑘, 𝜔, and 𝛿 are constants.

4.1. Solutions by the exp Ansatz

From (1.4) and (4.1) we obtain a polynomial equation in the variable 𝜁=exp(𝜉). Solving it, we get the solution

1𝑢(𝑥,𝑡)=−𝑘24²+𝑞𝑘²exp(𝑘𝑥+(1/48)𝑘⁷𝑡+𝛿)+4𝑞²exp(−(𝑘𝑥+(1/48)𝑘⁷.𝑡+𝛿))+4𝑞 (4.3) From (4.3) the following solutions are obtained:

1𝑢(𝑥,𝑡)=−𝑘24²+18𝑘²sech²(121(𝑘𝑥+𝑘48⁷1𝑡+𝛿)),𝑞=2. (4.4) Observe that solutions given by (3.6) and (4.4) coincide. Consider

1𝑢(𝑥,𝑡)=−𝑘24²−18𝑘²csch²(121(𝑘𝑥+𝑘48⁷1𝑡+𝛿)),𝑞=−2,1𝑢(𝑥,𝑡)=𝑘24²−14𝑘²11−sin(𝑘𝑥−(1/48)𝑘⁷√𝑡+𝛿),𝑞=−12,𝑘→𝑘√−1,𝛿→𝛿√1−1,𝑢(𝑥,𝑡)=𝑘24²−14𝑘²11+sin(𝑘𝑥−(1/48)𝑘⁷√𝑡+𝛿),𝑞=−−12,𝑘→𝑘√−1,𝛿→𝛿√−1. (4.5)

4.2. Solutions by the tanh-coth Ansatz

We change the tanh and coth functions to their exponential form and then we substitute (4.2) into (1.4). We obtain a polynomial equation in the variable 𝜁=exp(𝜉). Equating the coefficients of the different powers of 𝜁 to zero results in an algebraic system in the variables 𝑎, 𝑏, 𝑐, 𝑑, 𝑝, 𝑘, and 𝜔. Solving it with the aid of a computer, we obtain the following solutions of (1.4).

(i)𝑎=0,𝑏=0,𝑐=0,𝑑=−(1/2)𝑘²,𝑝=(1/3)𝑘²,𝜔=(4/3)𝑘⁷: 1𝑢(𝑥,𝑡)=3𝑘²−12𝑘²coth²4(𝑘(𝑥+3𝑘⁶1𝑡)).𝑢(𝑥,𝑡)=−3𝑘²−12𝑘²cot²4(𝑘(𝑥−3𝑘⁶𝑡)). (4.6)(ii)𝑎=0,𝑏=0,𝑐=−(1/2)𝑘²,𝑑=0,𝑝=(1/3)𝑘²,𝜔=(4/3)𝑘⁷: 1𝑢(𝑥,𝑡)=3𝑘²−12𝑘²tanh²4(𝑘(𝑥+3𝑘⁶𝑡)). (4.7) Figure 2 shows the graph of𝑢₃(𝑥,𝑡) for𝑘=−0.68,−7≤𝑥≤10, and 0≤𝑡≤16.1𝑢(𝑥,𝑡)=−3𝑘²−12𝑘²tan²4(𝑘(𝑥−3𝑘⁶𝑡)). (4.8) Figure 3 shows the graph of𝑢₄(𝑥,𝑡) for𝑘=1.18,−3≤𝑥≤2, and0≤𝑡≤2(iii)𝑎=0,𝑏=0,𝑐=−(1/2)𝑘²,𝑑=−(1/2)𝑘²,𝑝=(1/3)𝑘²,𝜔=(256/3)𝑘⁷: 𝑘𝑢(𝑥,𝑡)=²3−𝑘²2coth²(𝑘(𝑥+2563𝑘⁶𝑘𝑡))−²2tanh²(𝑘(𝑥+2563𝑘⁶𝑘𝑡)),𝑢(𝑥,𝑡)=−²3−𝑘²2cot²(𝑘(𝑥−2563𝑘⁶𝑘𝑡))−²2tan²(𝑘(𝑥−2563𝑘⁶𝑡)). (4.9)

Figure 2 

Graph of solution (4.7).

Figure 3 

Graph of solution (4.8).

4.3. Other Exact Solutions

We may employ other methods to find exact solutions to the seventh-order Kaup-Kupershmidt equation. Thus, a solution to (1.4) of the form

𝑑𝑢(𝜉)=𝐴²𝑑𝜉²log(𝑠+exp(𝜉)+exp(2𝜉))+𝐵,𝜉=𝑘𝑥+𝜔𝑡+𝛿 (4.10) is

𝑢(𝜉)=−𝑘²𝑠²(16𝑠−1)+2𝑘²𝑠(8𝑠−5)𝑒^𝜉+𝑘²(160𝑠²−62𝑠+1)𝑒^2𝜉+2𝑘²(8𝑠−5)𝑒^3𝜉−𝑘²(16𝑠−1)𝑒^4𝜉24𝑠²(4𝑠−1)+48𝑠(4𝑠−1)𝑒^𝜉+24(2𝑠+1)(4𝑠−1)𝑒^2𝜉+48(4𝑠−1)𝑒^3𝜉+24(4𝑠−1)𝑒^4𝜉,𝜉=𝜉(𝑥,𝑡)=𝑘𝑥+𝜔𝑡+𝛿,𝜔=4096𝑠³+960𝑠²+264𝑠−148(4𝑠−1)³𝑘⁷1,𝐴=2𝑘²𝑘,𝐵=−²(16𝑠−1).24(4𝑠−1) (4.11) In particular, taking 𝑠=±1, we get

𝑘𝑢(𝑥,𝑡)=²(4cosh(𝑘𝑥+(197/48)𝑘⁷𝑡+𝛿)−10cosh(2(𝑘𝑥+(197/48)𝑘⁷𝑡+𝛿))+33)24(2cosh(𝑘𝑥+(197/48)𝑘⁷𝑡+𝛿)+1)²,𝑘𝑢(𝑥,𝑡)=²(52sinh(𝑘𝑥+(3401/6000)𝑘⁷𝑡+𝛿)−34cosh(2(𝑘𝑥+(3401/6000)𝑘⁷𝑡+𝛿))−223)120(2sinh(𝑘𝑥+(3401/6000)𝑘⁷𝑡+𝛿)+1)². (4.12)(4.13) Following solution results from (4.12) by replacing 𝑘 with 𝑘√−1 and 𝛿 with 𝛿√−1∶

𝑘𝑢(𝑥,𝑡)=²(10cos(2𝑘𝑥−(197/24)𝑘⁷𝑡+2𝛿)−4cos(𝑘𝑥−(197/48)𝑘⁷𝑡+𝛿)−33)24(2cos(𝑘𝑥−197𝑘⁷𝑡/48+𝛿)+1)². (4.14) The choice 𝑠=1/16 gives

𝑢(𝑥,𝑡)=8𝑘²(15sinh(𝑘𝑥−𝑘⁷𝑡+𝛿)+17cosh(𝑘𝑥−𝑘⁷𝑡+𝛿)+4)(15sinh(𝑘𝑥−𝑘⁷𝑡+𝛿)+17cosh(𝑘𝑥−𝑘⁷𝑡+𝛿)+16)². (4.15) Finally, by using the ansatze

𝑢(𝜉)=𝑝+𝑞cosh(2𝜉)(𝑚+sinh(𝜉))²,𝑢(𝜉)=𝑟+𝑠cosh(2𝜉)(𝑚+cosh(𝜉))², (4.16) we obtain

𝑢(𝑥,𝑡)=−5𝑘²(11+5cos(2𝑘𝑥−19𝑘⁷𝑡+2𝛿))8(√10±5sin(𝑘𝑥−(19/2)𝑘⁷𝑡+𝛿))²,𝑢(𝑥,𝑡)=5𝑘²(11−5cosh(2𝑘𝑥+19𝑘⁷𝑡+2𝛿))8(√10±5cosh(𝑘𝑥+(19/2)𝑘⁷𝑡+𝛿))². (4.17)(4.18) The correctness of the solutions given in this work may be checked with the aid of either Mathematica 7.0 or Maple 12. For example, to see that function defined by (4.12) is a solution to the seventh-order Kaup-Kupershmidt (1.4), we may use Mathematica 7.0 as follows:

In [1]:= kk7 = ∂𝚝 # + 2016 #³∂𝚡 # + 630 (∂𝚡 #)³ + 2268 # ∂𝚡 # ∂𝚡,𝚡 # + 504 #²∂𝚡,𝚡,𝚡 # + 252 ∂𝚡,𝚡 # ∂𝚡,𝚡,𝚡 # + 147 ∂𝚡 # ∂𝚡,𝚡,𝚡,𝚡 # + 42 # ∂𝚡,𝚡,𝚡,𝚡,𝚡 # + ∂𝚡,𝚡,𝚡,𝚡,𝚡,𝚡,𝚡 # &;

(*This defines the differential operator associated with the seventh-order Kaup-Kupershmidt equation*)𝚔𝚜𝚘𝚕=^𝟸(𝟹𝟹+𝟺𝚌𝚘𝚜𝚑[𝚔𝚡+𝟷𝟿𝟽𝚔^𝟸𝚝𝟺𝟾+𝛿]−𝟷𝟶𝚌𝚘𝚜𝚑[𝟸(𝚔𝚡+𝟷𝟿𝟽𝚔^𝟽𝚝𝟺𝟾+𝛿)])𝟸𝟺(𝟷+𝟸𝚌𝚘𝚜𝚑[𝚔𝚡+𝟷𝟿𝟽𝚔^𝟸𝚝𝟺𝟾+𝛿])^𝟸 (4.19)(*This defines the solution*)

In [3]:=Simplify [kk7 [TrigToExp [sol]]]

(*Here we apply the differential operator to the solution and then we make use of the Simplify command*)

Out [3]:= 0

5. Conclusions

In this work we have obtained two conditions associated to Cole-Hopf transformation (2.3) for the existence of exact solutions to the general KdV7. Two cases have been analyzed. The first one is given by (2.13) and the other by (2.14). The first case leads to first discriminant equation (2.18) which is satisfied by Sawada-Kotera-Ito equation (1.2), Lax equation (1.3), and Kaup-Kupershmidt equation (1.4). The second case gives the second discriminant equation (2.20), which is fulfilled only by (1.2) and (1.3). However, when both conditions (2.13) and (2.14) hold, we may obtain solutions different from those that result from the two cases we mentioned. We did not consider this last case. Other works that related the problem of finding exact solutions of nonlinear PDE's may be found in [24, 25].

Acknowledgments

The authors are grateful to their anonymous referees for their valuable help, time, objective analysis, and patience in the correction of this work. Their comments and observations helped them to simplify results of calculations, to clarify the key ideas, and to improve their manuscript. They also thank the editor for giving them the opportunity to publish this paper.