The Periodic Solutions to Kawahara Equation by Means of the Auxiliary Equation with a Sixth-Degree Nonlinear Term

Abstract

It is well known that different types of exact solutions of an auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term are presented to obtain novel exact solutions of the Kawahara equation. By the aid of the solutions of the original auxiliary equation, some other physically important nonlinear equations can be solved to construct novel exact solutions.

1. Introduction

Over the past decades a number of approximate methods for finding travelling wave solutions to nonlinear evolution equations have been proposed or developed and furthermore modified. The solutions to various evolution equations have been found by one or other of these methods. References [1–24] and references within cite some of this accomplishment. Among these methods, one of the current methods is so called auxiliary equation method [20–24]. The technique of this method consist of the solutions of the nonlinear evolution equations such that the target solutions of the nonlinear evolution equations can be expressed as a polynomial in a linearly independent elementary function which satisfies a particular ordinary differential equation which is named as auxiliary equation in general. Recently, to determine the solutions of nonlinear evolution equations, many exact solutions of various auxiliary equations have been utilized [20–32].

A concise observation shows that novel exact solutions of the auxiliary equations may fabricate original exact travelling wave solutions of nonlinear equations. Therefore, it is very important to find novel exact solutions of the auxiliary equations.

In this paper, we will examine the consequences of the choice of the auxiliary equation for determining the solutions of the nonlinear evolution equation in consideration and to seek more types of new exact solutions of nonlinear differential equations which satisfy a first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term.

2. Auxiliary Equation Technique and Some Remarks

We assume that the given nonlinear partial differential equation for 𝑢(𝑥,𝑡) to be in the form 𝑃𝑢,𝑢_𝑥,𝑢_𝑡,𝑢_𝑥𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,…=0, (1) where 𝑃 is a polynomial in its arguments, which include nonlinear terms and the highest-order derivatives.

Then using the transformation 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝜇(𝑥−𝑐𝑡) one can reduce (1) to the ordinary differential equation 𝑄𝑢,𝑢_𝜉,𝑢_𝜉𝜉,𝑢_𝜉𝜉𝜉,…=0. (2) Now, we seek the exact travelling wave solutions of (2) by means of auxiliary equation method.

As it is well known, most exact travelling wave solutions [4–24] of nonlinear evolution equations were obtained on the assumption that the exact solution can be expressed as a finite expansion of linearly independent elementary functions. One common way of obtaining linearly independent elementary functions is to assume them as a solution of appropriate algebraic or differential equations which are exactly solvable.

Thus, it is natural to ask the following question: What type of finite expansion or what kind of function can be used to construct new travelling wave solutions?

Hence, for the solution of (2), let us assume that the exact solution can be expressed as simple expansion in the form of 𝑢(𝜉)=^𝑁_𝑖=0𝑎_𝑖𝑧^𝑖(𝜉), (3) where 𝑎_𝑖, (𝑖=0,1,…,𝑁) are constants which will be determined later and function 𝑧(𝜉) is an appropriate function that yields new travelling wave solutions and is defined by the solution of the auxiliary equation considered.

Now, let us remember the process for finding the unknown coefficients 𝑎_𝑖, (𝑖=0,1,…,𝑁), where 𝑢(𝜉)=∑^𝑁_𝑖=0𝑎_𝑖𝑧^𝑖(𝜉). Substituting the auxiliary equation into the given nonlinear equation and equating each coefficient of a power of 𝑧(𝜉) to zero yield an algebraic system. Hence, all coefficients 𝑎_𝑖 can be determined by solving the algebraic system and the parameter 𝑁 is a positive integer and can be determined by balancing the highest order derivative terms and the highest power nonlinear terms in (3) in general.

In most auxiliary equation methods, the function 𝑧(𝜉) is defined as the solution of an auxiliary ordinary differential equation. Even if it is possible to construct more complex types of exact solutions for many auxiliary ordinary differential equations, one also encounters some difficulties. Hence, the following vital question arise: How do we find a variety of exact solutions of the auxiliary equations? Unfortunately, the answer of this question is not unique and various authors utilize different auxiliary ordinary differential equations which are exactly solvable.

For example, Nickel [25] has utilized the nonlinear differential equation 𝑑𝑧𝑑𝜉²=𝛼𝑧⁴+4𝛽𝑧³+6𝛾𝑧²+4𝛿𝑧+𝜀≡𝑅(𝑧) (4) as an auxiliary ordinary differential equation which transforms (2) into polynomial equation 𝑃(𝑧)=0, 𝑧^′𝑄(𝑧)=0, and 𝑃(𝑧)+𝑧^′𝑄(𝑧)=0, respectively. Vanishing coefficients in the polynomials 𝑃(𝑧)=0, 𝑧^′𝑄(𝑧)=0 and 𝑃(𝑧)+𝑧^′𝑄(𝑧)=0, respectively, lead to equations which partially determine the coefficients 𝛼, 𝛽, 𝛾, 𝛿, and 𝜀 in (4). We think that Nickel preferred (4) due to it is analytically solvability. Because, Whittaker and Watson [26] showed that the solution 𝑧(𝜉) of (4) can be expressed in terms of Weierstrass’ elliptic function.

At the same time, Fan [27] developed a new direct algebraic method to seek additional types of new exact solutions of evolutionary nonlinear differential equations that can be expressed as polynomials in several elementary or special functions which satisfy a first-order nonlinear ordinary differential equation with a fourth-degree nonlinear term 𝑑𝑧𝑑𝜉²=𝑎₀+𝑎₁𝑧(𝜉)+𝑎₂𝑧²(𝜉)+𝑎₃𝑧³(𝜉)+𝑎₄𝑧⁴(𝜉), (5) where 𝑎_𝑖, (𝑖=0,1,3,4) are constants. The method has been applied to find many exact traveling wave solutions of various types of nonlinear equations [27–29]. Sirendaorejii [21] and Abdou [30] utilized (5) for the case 𝑎_𝑖, (𝑖=2,3,4). Perhaps, it is significant to point out that the choice of the case, that is, 𝑎_𝑖(𝑖=2,3,4), permits analytical solution of the first-order nonlinear ordinary differential equation with a fourth-degree nonlinear term as Nickel’s. This property of the nonlinear ordinary differential equation with a fourth-degree nonlinear term used in large in [20, 23], for example, for the case 𝑎_𝑖(𝑖=0,2,4) and in [21, 22] for the case 𝑎_𝑖(𝑖=2,3,4) for (5) to find many exact traveling wave solutions of various types of nonlinear equations.

In this paper, we seek for the solution of (2) in terms of ansatz (3) with 𝑧(𝜉) satisfying the following new auxiliary equation with a sixth-degree nonlinear term, that is, 𝑑𝑧𝑑𝜉²=𝑎₀+𝑎₂𝑧²(𝜉)+𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉), (6) where 𝑎_𝑖(𝑖=0,2,4,6) are real constants. Cases 1–8 report the new solutions of the auxiliary equation (6) with a sixth-degree nonlinear term under 8 distinct cases which include solutions including Bessel function and Lambert function and one can use these new solutions to seek exact travelling wave solutions for nonlinear equations.

Case 1. If 𝑑𝑧𝑑𝜉²=𝑎₀+𝑎₆𝑧⁶(𝜉) (7) then 𝑧(𝜉)1=Bessel𝐽6,2𝑧⁶5𝑧𝜋+3𝜉Γ6. (8)

Case 2. If 𝑑𝑧𝑑𝜉²=𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉) (9) then 𝑧(𝜉)=−𝑎₆+𝑎²₄𝜉²−2𝑎²₄𝜉𝑐₁+𝑎²₄𝑐²₁𝑎₄−𝑎₆+𝑎²₄𝜉²−2𝑎²₄𝜉𝑐₁+𝑎²₄𝑐²₁. (10)

Case 3. If 𝑑𝑧𝑑𝜉²=𝑎₀+𝑎₂𝑧²(𝜉)+𝑎₆𝑧⁶(𝜉) (11) then𝑧(𝜉)=−(3/8)𝜉−10𝑎₆𝑎₂𝑎₂/𝑎₆^3/2+(3/8)−10𝑎₆𝑎₂𝑎₂/𝑎₆^3/2𝑐₁+𝑎₀𝑎₀−8+9𝑎₂𝜉²−18𝑎₂𝜉𝑐₁+9𝑎₂𝑐²₁/𝑎₂/8𝑎₆𝑎²₂^1/3𝑎₂+−10𝑎₆𝑎₂𝑎₂2𝑎₆−(3/8)𝜉−10𝑎₆𝑎₂𝑎₂/𝑎₆^3/2+(3/8)−10𝑎₆𝑎₂𝑎₂/𝑎₆^3/2𝑐₁+𝑎₀𝑎₀−8+9𝑎₂𝜉²−18𝑎₂𝜉𝑐₁+9𝑎₂𝑐²₁/𝑎₂/8𝑎₆𝑎²₂^1/3. (12)

Case 4. If 𝑑𝑧𝑑𝜉²=𝑎₂𝑧²(𝜉)+𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉) (13) then 𝑧(𝜉)=𝑒^{(√𝑎2(−2𝜉+𝑐1))}√2−𝑎×2⁶𝑒^{√(2𝑎2(−2𝜉+𝑐1))}𝑎₄Lambert𝑊−2𝑎₆𝑒^{√(2𝑎2(−2𝜉+𝑐1))}/3𝑎₄^−1/2. (14)

Case 5. If 𝑑𝑧𝑑𝜉²=𝑎₀+𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉) (15) then 𝑧(𝜉)=𝜉−^𝑧(𝜉)𝑑𝑎𝑎₀+𝑎₄𝑎⁴+𝑎₆𝑎⁶−𝑐₁=0. (16)

Case 6. If 𝑑𝑧𝑑𝜉²=𝑎₂𝑧²(𝜉)+𝑎₆𝑧⁶(𝜉) (17) then 𝑧(𝜉)=𝑒^{(√𝑎2(−𝜉+𝑐1))}2^1/4×𝑎₆𝑒^{√(4𝑎2(−𝜉+𝑐1))}𝑎₂Lambert𝑊−𝑎₆𝑒^{√(4𝑎2(−𝜉+𝑐1))}/2𝑎₂^−1/4. (18)

Case 7. If 𝑑𝑧𝑑𝜉²=𝑎₆𝑧⁶(𝜉) (19) then 𝑧(𝜉)=1−2𝜉√𝑎₆+2𝑐₁√𝑎₆. (20)

Case 8. If 𝑑𝑧𝑑𝜉²=𝑎₀+𝑎₂𝑧²(𝜉)+𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉) (21) then 𝑧(𝜉)=𝑎₆81^2/3−38𝜉𝑎₀^3/2+38𝑎₀^3/2𝑐₁+3√8𝑎₀8×𝑎₀−8𝑎₆+9𝑎²₄𝜉²−18𝑎²₄𝜉𝑐₁+9𝑎²₄𝑐²₁𝑎²₄×𝑎⁴₄𝑎²₆^−11/318𝑎²₄⁻¹+𝑎₀81^1/32−38𝜉𝑎₀^3/2+38𝑎₀^3/2𝑐₁+3√8𝑎₀8×𝑎₀−8𝑎₆+9𝑎²₄𝜉²−18𝑎²₄𝜉𝑐₁+9𝑎²₄𝑐²₁𝑎²₄×𝑎⁴₄𝑎²₆^{−1−1/3−1}. (22)
As it is noticeable, by choosing the particular values of 𝑎_𝑖(𝑖=0,2,4,6), (6) provides numerous types of special solutions. Due to limited space, we omit those results; instead we affirm the following remarks.
Hence, in ansatz (3), the parameter 𝑁 is a positive integer and can be determined by balancing the higher-order derivative term and the highest power nonlinear terms in (2). The highest degree of 𝜕^𝑝𝑢/𝜕𝜉^𝑝 can be calculated by 𝒪𝜕^𝑝𝑢𝜕𝜉^𝑝𝒪𝑢=𝑁+𝑝,𝑝=0,1,2,…,_𝑞𝜕^𝑝𝑢𝜕𝜉^𝑝=𝑞𝑁+𝑝,𝑞,𝑝=0,1,2,…. (23) Consequently substituting (3) and (6) into (2) and equating the coefficients of all powers of 𝑧(𝜉) and 𝑧^𝑗(𝜉)𝑎₀+𝑎₂𝑧²(𝜉)+𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉)(𝑗=0,1,2,…) to zero in the resulting equation, several algebraic equations will be obtained. Then solving these algebraic equations by the symbolic computation system Maple, and combining (3) and the auxiliary equation (6), we can get the exact solutions for (1).

3. Exact Travelling Wave Solutions

In this section, we are going to combine the auxiliary equation method with the new solutions of original auxiliary equation (6) with a six-degree nonlinear term to consider the new solutions of the generalized nonlinear Klein-Gordon equation which is used as a test problem by a number of researchers [20, 21, 24].

The Kawahara equation is 𝑢_𝑡+𝑢𝑢_𝑥+𝛼𝑢_𝑥𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥𝑥}=0, (24) where 𝛼 is nonzero arbitrary constant. Equation (24), proposed first by Kawahara in 1972, occurs in the theory of shallow water waves and plays an important role in the modeling of many physical phenomena such as plasma waves and magnetoacoustic waves. Making the transformation 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥−𝑐𝑡 and integrating once with respect to 𝜉, (24) becomes 1−𝑐𝑢+2𝑢²+𝛼𝑢^′′−𝑢⁽⁴⁾=0. (25) From (23), we have 𝑁=4. Therefore, the ansatz yields 𝑢(𝜉)=𝑔₀+𝑔₁𝑧(𝜉)+𝑔₂𝑧²(𝜉)+𝑔₃𝑧³(𝜉)+𝑔₄𝑧⁴(𝜉) (26) and 𝑧(𝜉) may be determined from one of the Cases 1–8.

Now, for convenience, we give the calculation of typical two cases only for the practical purposes and the rest can be determined in a similar manner.

Example 1. We first consider the auxiliary equation with six-degree nonlinearity, that is, 𝑑𝑧𝑑𝜉²=𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉). (27) in Cases 1–8. The solution of (27) 𝑧(𝜉)=−𝑎₆+𝑎²₄𝜉²−2𝑎²₄𝜉𝑐₁+𝑎²₄𝑐²₁𝑎₄−𝑎₆+𝑎²₄𝜉²−2𝑎²₄𝜉𝑐₁+𝑎²₄𝑐²₁ (28) is, we belive, a new solution in the literature.
Hence, substituting (26) and (27) into (25) and letting each coefficient of 𝑧^𝑖(𝜉)𝑎₂𝑧²(𝜉)+𝑎₆𝑧⁶(𝜉), (0≤𝑖≤8) to be zero, we obtain −384𝑔₂𝑎₆−2688𝑔₄𝑎₄=0,−105𝑔₁𝑎₆−1260𝑔₃𝑎₄=0,24𝑎₆𝛼𝑔₄+12𝑔³₄−840𝑔₄𝑎²₄−480𝑎₆𝑎₄𝑔₂=0,15𝑎₆𝛼𝑔₃+𝑔₃𝑔₄−360𝑔₃𝑎²₄−120𝑎₆𝑎₄𝑔₁=0,20𝑎₄𝛼𝑔₄+12𝑔²₃−120𝑔₂𝑎²₄+8𝑎₆𝛼𝑔₂+𝑎𝑔₄𝑔₂=0,3𝑎₆𝛼𝑔₁+𝑔₁𝑔₄+𝑔₂𝑔₃−24𝑔₁𝑎²₄+12𝛼𝑎₄𝑔₃𝑔=0,₀𝑔₄+𝑔₁𝑔₃−𝑐𝑔₄+12𝑔²₂+6𝛼𝑎₄𝑔₂𝑔=0,₀𝑔₃+𝑔₁𝑔₂−𝑐𝑔₃+2𝛼𝑎₄𝑔₁𝑔=0,₀𝑔₂−𝑐𝑔₂+12𝑔²₁𝑔=0,₀−𝑐=0. (29)
Solving the system (29) by the aid of Maple 13, we can determine the coefficients: 𝑔₀=4𝑐3,𝑔₂=𝑔₁=𝑔₃𝑎=0,𝜇=𝜇,₄=𝑎₄,𝑎₆=𝑎₆,𝑐=8064𝑎₄−72𝛼𝑎₆−32𝑔₄+2520𝑎²₄−60𝛼𝑎₄. (30) Substituting the above coefficients into ansatz (26) with the solution (28) of auxiliary equation, we obtain one of the new solutions of Kawahara equation: 𝑢(𝑥,𝑡)∶=10752𝑎₄−96𝛼𝑎₆−2𝑔₄+3360𝑎²₄−80𝑎₄𝛼+𝑔₄𝑎²₄×−𝑎₆+𝑎²₄𝑥+8064𝑎₄−72𝛼𝑎₆−32𝑔₄+2520𝑎²₄−60𝑎₄𝛼𝑡²−2_Cl𝑎²₄𝑥+8064𝑎₄−72𝛼𝑎₆−32𝑔₄+2520𝑎²₄−60𝑎₄𝛼𝑡+𝑎²₄_Cl²⁻². (31) The graph of (31) is illustrated for selected values of coefficients given in Figure 1.

Figure 1 

Graph of (31) for 𝑔4=1, 𝛼=𝜇=1, 𝑎4=tan𝑥, 𝑎6=−1.

Example 2. We, next, consider the auiliary equation with six-degree nonlinearity, that is, 𝑑𝑧𝑑𝜉²=𝑎₂𝑧²(𝜉)+𝑎₆𝑧⁶(𝜉) (32) in Cases 1–8. The solution of (32) 𝑧(𝜉)=𝑒^{(√𝑎2(−𝜉+𝑐1))}2^1/4×𝑎₆𝑒^{√(4𝑎2(−𝜉+𝑐1))}𝑎₂−𝑎Lambert𝑊⁶𝑒^{√(4𝑎2(−𝜉+𝑐1))}2𝑎₂^−1−1/4 (33) is, we belive, a new solution in the literature.
Substituting (26) and (32) into (25) and letting each coefficient of 𝑧^𝑖(𝜉)𝑎₂𝑧²(𝜉)+𝑎₆𝑧⁶(𝜉), (0≤𝑖≤8) to be zero, we obtain 50𝛼−3188𝑎₂=0,30𝛼𝑎₂−354𝑎²₂−4𝑐=0. (34)
Solving the system (34) by the aid of Maple 13, we can determine the coefficients: 𝑐=487125𝛼²5081672,𝑔₀=𝑔₀,𝑔₁=𝑔₁,𝑔₂=𝑔₂,𝑔₃=𝑔₃,𝑔₄=𝑔₄𝑎,𝜇=𝜇,₂=25𝛼1594,𝑎₆=𝑎₆. (35)
Substituting the above coefficients into ansatz (26) with the solution (33) of auxiliary equation, we obtain another new solution of Kawahara equation:

𝑢(𝑥,𝑡)∶=𝑔₀+𝑔₁𝐞⁻14^{lambert𝑊−}797𝑎25⁶𝐞2√25√1594√𝛼−𝑥−487125𝛼²𝑡+5081672_Cl797𝛼⁻√25√1594√𝛼𝑥+487125𝛼²𝑡50816721594⁺√25√1594√𝛼_Cl1594+𝑔₂𝐞⁻14^{lambert𝑊−}797𝑎25⁶𝐞2√25√1594√𝛼−𝑥−487125𝛼²𝑡+5081672_Cl797𝛼⁻√25√1594√𝛼𝑥+487125𝛼²𝑡50816721594⁺√25√1594√𝛼_Cl1594²+𝑔₃𝐞⁻14^{lambert𝑊−}797𝑎25⁶𝐞2√25√1594√𝛼−𝑥−487125𝛼²𝑡+5081672_Cl797𝛼⁻√25√1594√𝛼𝑥+487125𝛼²𝑡50816721594⁺√25√1594√𝛼_Cl1594³+𝑔₄𝐞⁻14^{lambert𝑊−}797𝑎25⁶𝐞2√25√1594√𝛼−𝑥−487125𝛼²𝑡+5081672_Cl797𝛼⁻√25√1594√𝛼𝑥+487125𝛼²𝑡50816721594⁺√25√1594√𝛼_Cl1594⁴. (36)

The graph of (36) is illustrated for selected values of coefficients given in Figure 2.

Figure 2 

Graph of (36) for 𝛼=1/2, 𝑎6=𝑔0=𝑔2=1, 𝑔1=𝑔3=−1, 𝑔4=sin𝑥.

4. Conclusion

As is seen, the key idea of obtaining new travelling wave solutions for the nonlinear equations is constructing different types of solutions of the given auxiliary equation. In this paper, the exact solutions of the auxiliary equation with six-degree nonlinearity (6) is 𝑑𝑧𝑑𝜉²=𝑎₀+𝑎₂𝑧²(𝜉)+𝑎₄𝑧⁴(𝜉)+𝑎₆𝑧⁶(𝜉), (37) where 𝑎_𝑖(𝑖=0,2,4,6) are real constants, used to construct the solutions of Klein-Gordon equation [32]. Using these solutions, we have successfully obtained some new exact periodic solutions of the Kawahara equation. For comparative purposes, it is noticeable that Wu et al. in [20] and Lv et al. in [23] have utilized (5) for 𝑎_𝑖(𝑖=0,2,4), and Sirendaoreji in [21] and Jang in [22] have applied the ansatz (𝑑𝑧/𝑑𝜉)²=𝑎₂𝑧²(𝜉)+𝑎₃𝑧³(𝜉)+𝑎₄𝑧⁴(𝜉) which only covers, at most, fourth-order nonlinear term in auxiliary ordinary differential equation, and therefore their solutions are limited in the quantity to the order considered.

In this letter, we have obtained exact solutions of the auxiliary equation with sixth-degree nonlinearity (6) for distinct cases (see Cases 1–8). Therefore, our approach gives additional new solutions beside the solutions obtained via fourth order auxiliary equation, and theoretically, some of our solutions may coincide with the solutions of Wu et al. in [20], Sirendaoreji in [21], Jang in [22], Lv et al. in [23], and some cases of Yomba [24] for certain choice of the parameters which we have left as an exercise.

However, it is well known that different types of auxiliary equations produce new travelling wave solutions for many nonlinear problems. Hence this also is our future work. The presented method could lead to finding new exact travelling wave solutions for other nonlinear problems.