Open AccessArticle

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

Feng Ye

¹,

Jian Tian

^1,*,

Xiaoting Zhang

¹,

Chunling Jiang

¹,

Tong Ouyang

¹ and

Yongyi Gu

^2,*

Department of Basic Courses Teaching, Software Engineering Institute of Guangzhou, Guangzhou 510990, China

School of statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510006, China

Authors to whom correspondence should be addressed.

Axioms 2022, 11(7), 330; https://doi.org/10.3390/axioms11070330

Submission received: 1 June 2022 / Revised: 28 June 2022 / Accepted: 29 June 2022 / Published: 7 July 2022

(This article belongs to the Special Issue Differential Equations and Genetic Algorithms)

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Abstract

In this article, we prove that the

⟨ p, q ⟩

condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions

u_{r} (x + ν t)

and simply periodic solutions

u_{s, 1} (x + ν t)

of Kawahara equation are solitary wave solutions, while simply periodic solutions

u_{s, 2} (x + ν t)

are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods.

Keywords:

Kawahara equation; complex method; exact solution; elliptic function

MSC:

30D35; 34A05

1. Introduction and Main Results

In this article, we study all traveling wave exact solutions of the Kawahara equation [1]:

u_{t} + a u u_{x} + c u_{x x x} - d u_{x x x x x} = 0,

(1)

where

a, c, d

are constants, x and t represent spatial and time variables,

u u_{x}

is the nonlinear disturbance term, and

u_{x x x}

and

u_{x x x x x}

are the dispersion terms of order three and five, respectively. The solutions of Equation (1) could be used to analyze and interpret a lot of nonlinear dispersive phenomena that can arise in optical fiber, ocean, plasma physics, etc. [2,3]. Given the importance of this equation, it remains the subject of study for many researchers. Many different techniques were devoted to studying various solutions of Equation (1) and its family [4,5,6,7,8,9,10,11,12,13]. For example, Kudryashov [14], obtained exact meromorphic solutions of the Kawahara equation using the Laurent series. Wazwaz [15], found some different solutions (compacton and solitons) to Equation (1) in terms of trigonometric functions. Khan [16], analyzed the Kawahara equation using the variational approach and derived new conditions for obtaining solitary wave solutions. Using the traveling wave ansatz, Baiswas [17], studied the generalized Kawahara equation and derived a solitary wave solution for the family of the Kawahara equation. A lot of effective methods are applied to study the exact solutions, which makes the research more abundant [18,19,20,21,22,23]. Wang [24] used ansatz method to derive the exact solitary wave solution for the generalized Korteweg–de Vries–Kawahara (GKdV-K) equation. Aiman Zara [25] studied numerical approximation of the modified Kawahara equation using the Kernel smoothing method. El-Tantawy [26] derived a set of novel exact and approximate analytic solutions to the family of the forced damped Kawahara equation (KE) using the ansatz method.

In recent years, Yuan et al. [27,28] introduced the

〈 p, q 〉

condition to study a class of constant-coefficient complex algebraic differential equations with dominant terms, and obtained all possible meromorphic solutions which belong to the class W. That is to say, the general representation of rational function solutions, finitely growing simply periodic solutions and elliptic solutions, and both rational function solutions and simply periodic solutions can be obtained by degenerating elliptic general solutions. Futhermore, a complex method for obtaining exact solutions in the mathematical physics is presented. Using the complex method [27,28], we prove that all meromorphic solutions of complex Kawahara equation belong to the class W, and then get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Equation (1). This method is simpler than other methods in the process of finding solutions, and can also be applied to the solution of other nonlinear differential equations.

Substitute the traveling wave transform

u = u (x, t) = w (z), z = x + ν t

(2)

into Equation (1); integrate it to get

C_{0} + ν w + \frac{1}{2} a w^{2} + c w^{″} - d w^{(4)} = 0,

(3)

where

C_{0}

is an integral constant.

Multiply Equation (3) by

w^{'}

, and integrate it; thus, we get the complex ordinary differential Kawahara equation

w^{'} w^{‴} - \frac{1}{2} {(w^{″})}^{2} - A w^{3} + B {(w^{'})}^{2} + C w^{2} + D w + E = 0,

(4)

where

A = \frac{a}{6 d}, B = - \frac{c}{2 d}, C = - \frac{ν}{2 d}, D = - \frac{C_{0}}{d}, E = - \frac{C_{1}}{d}

C_{0}

and

C_{1}

are integral constants.

Now, we give the main results in our paper.

Theorem 1.

Let

A \neq 0

, then all meromorphic solutions

w (z)

of Equation (4) belong to the class W. Here, class W consists of elliptic functions and their degenerations with the form

R (z)

R (e^{α z})

α \in C

, where R is a rational function.

In addition, the solutions of Equation (4) have the forms:

(1) All elliptic function solutions

w_{d} (z) = \frac{35}{2 A} {[\frac{℘^{'} (z) + M}{℘ (z) - N}]}^{4} - [\frac{140 (℘ (z) + N)}{A} - \frac{70 B}{39 A}] {[\frac{℘^{'} (z) + M}{℘ (z) - N}]}^{2} + \frac{280}{A} {[℘ (z) + N]}^{2} - \frac{280 B}{39 A} [℘ (z) + N] + \frac{C}{3 A} - \frac{62 B^{2}}{1521 A} - \frac{28}{A} g_{2},

(5)

where

g_{3} = \frac{31 B^{3}}{593190} - \frac{7 B}{390} g_{2}, M^{2} = 4 N^{3} - g_{2} N - g_{3},

g_{2}^{2} - \frac{2 B^{2}}{507} g_{2} + \frac{1457}{41384889} B^{4} - \frac{A D}{1932} - \frac{C^{2}}{5796} = 0,

E = \frac{1}{A^{2}} [(\frac{1148}{69} A D + \frac{1148}{207} C^{2} - \frac{661248}{656903} B^{4}) g_{2} + \frac{23808}{4826809} B^{6}

+ (\frac{20}{1521} A D + \frac{20}{4563} C^{2}) B^{2} - \frac{1}{3} A C D - \frac{2}{27} C^{3}],

D and N are constants.

(2) All simply periodic solutions

\begin{matrix} w_{s, 1} (z) = \frac{2 B^{2}}{169 A} [35 {coth}^{4} \frac{\sqrt{26 B} i}{26} (z - z_{0}) - 70 {coth}^{2} \frac{\sqrt{26 B} i}{26} (z - z_{0}) + 23] + \frac{C}{3 A}, \end{matrix}

(6)

and

w_{s, 2} (z) = \frac{B^{2}}{507 A γ^{2}} [210 {coth}^{4} \sqrt{\frac{B}{26 γ}} (z - z_{0}) - (140 γ + 560) {coth}^{2} \sqrt{\frac{B}{26 γ}} (z - z_{0}) + 166 γ + 352] + \frac{C}{3 A},

(7)

where

z_{0} \in C; D = \frac{1}{85683} \frac{5184 B^{4} - 28561 C^{2}}{A},

E = \frac{1}{130323843} \frac{746496 B^{6} - 2628288 B^{4} C + 4826809 C^{3}}{A^{2}}

w_{s, 1} (z)

, and

D = \frac{1}{23905557} \frac{(217620 γ - 59724) B^{4}}{γ^{4} A} - \frac{C^{2}}{3 A},

E = \frac{1}{A^{2}} [\frac{16 B^{6}}{448893237} \frac{3001 - 7007 γ}{γ^{6}} - \frac{B^{4} C}{71716671} \frac{217620 γ - 59724}{γ^{4}} + \frac{C^{3}}{27}],

γ is a root of

31 z^{2} - 31 z + 10 = 0

w_{s, 2} (z)

(3) All rational function solutions

w_{r} (z) = \frac{280}{A} \frac{1}{{(z - z_{0})}^{4}} + \frac{C}{3 A},

(8)

where

B = 0, D = - \frac{C^{2}}{3 A}, E = \frac{C^{3}}{27 A^{2}}

and

z_{0} \in C

All traveling wave exact solutions of Equation (1) are obtained by substituting (2) into meromorphic solutions w of Equation (4). So, we get the following Theorem.

Theorem 2.

Let

a d \neq 0

, all traveling wave exact solutions

u (x, t)

of Equation (1) have the following forms:

(1) All elliptic function solutions

\begin{matrix} u_{d} (x + ν t) = \frac{105 d}{a} {[\frac{℘^{'} (x + ν t) + M}{℘ (x + ν t) - N}]}^{4} - (\frac{840 d (℘ (x + ν t) + N)}{a} + \frac{70 c}{13 a}) {[\frac{℘^{'} (x + ν t) + M}{℘ (x + ν t) - N}]}^{2} + \\ \frac{1680 d}{a} {[℘ (x + ν t) + N]}^{2} + \frac{280 c}{13 a} [℘ (x + ν t] + N) - \frac{168 d}{a} g_{2} - \frac{31 c^{2}}{507 a d} - \frac{ν}{a}, \end{matrix}

(9)

where

M^{3} = 4 N^{3} - g_{2} N - g_{3}, g_{3} = - \frac{31 c^{3}}{4745520 d^{3}} + \frac{7 c}{780 d} g_{2},

g_{2}^{2} - \frac{c^{2}}{1014 d^{2}} g_{2} + \frac{1457 c^{4}}{662158224 d^{4}} + \frac{a C_{0}}{11592 d^{2}} - \frac{ν^{2}}{23184 d^{2}} = 0,

and

C_{0}, N

are constants.

(2) All simply periodic solutions

\begin{matrix} u_{s, 1} (x + ν t) = \frac{3 c^{2}}{169 a d} [35 {coth}^{4} (\frac{1}{2} \sqrt{\frac{c}{13 d}} (x - x_{0} + ν (t - t_{0}))) \\ - 70 {coth}^{2} (\frac{1}{2} \sqrt{\frac{c}{13 d}} (x - x_{0} + ν (t - t_{0}))) + 23] - \frac{ν}{a}, \end{matrix}

(10)

and

\begin{matrix} u_{s, 2} (x + ν t) = \frac{c^{2}}{338 a d γ^{2}} [210 {coth}^{4} (\frac{1}{2} \sqrt{- \frac{c}{13 d γ}} (x - x_{0} + ν (t - t_{0}))) \\ - (140 γ + 560) {coth}^{2} (\frac{1}{2} \sqrt{- \frac{c}{13 d γ}} (x - x_{0} + ν (t - t_{0}))) \\ + 166 γ + 352] - \frac{ν}{a}, \end{matrix}

(11)

where γ is a root of

31 z^{2} - 31 z + 10 = 0

(3) All rational function solutions

u_{r} (x + ν t) = \frac{1680 d}{a} \frac{1}{{(x - x_{0} + ν (t - t_{0}))}^{4}} - \frac{ν}{a},

(12)

where

c = 0, x_{0}

and

t_{0}

are arbitrary real constants.

The rest of the article is organized in the following order: In Section 2, we will present the relevant lemmas and methodology. Section 3 gives the detailed proof process of Theorem 1 and concise proof method of Theorem 2. Section 4 illustrates our main results using computer simulations. In the last section, some conclusions are given.

2. Preliminary Lemmas and the Complex Method

We need some definitions and lemmas in order to prove Theorem 1 and present the complex method.

Set

k, n \in N : = {1, 2, 3, . . .}, r_{j} \in N_{0} = N \cup {0}

r = (r_{0}, r_{1}, . . ., r_{k})

j = 0, 1, . . ., k

M_{r} [w] (z) : = {[w (z)]}^{r_{0}} {[w^{'} (z)]}^{r_{1}} {[w^{″} (z)]}^{r_{2}} \dots {[w^{(k)} (z)]}^{r_{k}} .

(13)

The degree of

M_{r} [w]

defined by

d (r) : = r_{0} + r_{1} + \dots + r_{k}

Definition 1 ([28]).

A differential polynomial is defined by

P [w] : = \sum_{r \in Λ} b_{r} M_{r} [w],

(14)

where Λ is a finite index set, and

b_{r}

are constants.

deg P [w] : = {max}_{r \in Λ} {d (r)}

is called the degree of

P [w]

Consider the differential equation

E (z, w) : = P [w] - a w^{n} = 0,

(15)

where

a \neq 0

is a constant.

The dominant part of

E (z, w)

composes all dominant terms which can determine the multiplicity q of w in

E (z, w)

, and is denoted by

\hat{E} (z, w)

D (q)

and

D_{r} (q)

represent the multiplicity of pole of each term in

\hat{E} (z, w)

and the multiplicity of pole of each monomial in

E (z, w) - \hat{E} (z, w)

, respectively.

Obviously

D_{r} (q) = q d (r) + r_{1} + 2 r_{2} + \dots + k r_{k} < D (q) .

(16)

Definition 2 ([28]).

The derivative of

\hat{E} (z, w)

with respect to w can be calculated by the following formula, for any

χ,

{\hat{E}}^{'} (z, w) χ = lim_{λ \to 0} \frac{\hat{E} (z, w + λ χ) - \hat{E} (z, w)}{λ} .

(17)

Definition 3 ([28]).

Substituting Laurent series

w (z) = \sum_{l = - q}^{\infty} c_{l} z^{l},

(18)

into Equation (15), where

c_{- q} \neq 0, q > 0

. Then, we can get p different principle

\sum_{l = - q}^{- 1} c_{l} z^{l}

with pole of multiplicity q at

z = 0

, Equation (15) is said to satisfy weak

〈 p, q 〉

condition. If Equation (15) has p different meromorphic solutions with pole of multiplicity q at

z = 0

, Equation (15) satisfies

〈 p, q 〉

condition.

Definition 4 ([28]).

Let

T_{1}, T_{2}

be two given complex numbers, such that

I m \frac{T_{1}}{T_{2}} > 0, L = L [2 T_{1}, 2 T_{2}]

is discrete subset

L [2 T_{1}, 2 T_{2}] = {T | T = 2 m T_{1} + 2 n T_{2}, m, n \in Z}

, which is isomorphic to

Z \times Z

. The discriminant

Δ = Δ (c_{1}, c_{2}) : = c_{1}^{3} - 27 c_{2}^{2}

and

s_{n} = s_{n} (L) : = \sum_{T \in L \ {0}} \frac{1}{T^{n}} .

(19)

Weierstrass elliptic function

℘ (z) : = ℘ (z, g_{2}, g_{3})

is a meromorphic function with double periods

2 T_{1}, 2 T_{2}

, which satisfies the following formula

{(℘^{'} (z))}^{2} = 4 ℘ {(z)}^{3} - g_{2} ℘ (z) - g_{3},

(20)

where

g_{2} = 60 s_{4}, g_{3} = 140 s_{6}

and

Δ (g_{2}, g_{3}) \neq 0 .

Definition 5 ([29]).

The Fuchs index of Equation (15) can be defined as the root of the equation

P (i) = lim_{z \to 0} z^{- i + D (q)} {\hat{E}}^{'} (z, c_{- q} z^{- q}) z^{i - q} = 0

(21)

Lemma 1 ([28,30,31,32]).

Set

p, q, m, n \in N, deg P [w] < n

. If Equation (15) satisfies

〈 p, q 〉

condition, all meromorphic solutions

w (z)

of Equation (15) belong to class W.

Any elliptic function solution with pole at

z = 0

is given in the form

w (z) = \sum_{i = 1}^{m - 1} \sum_{j = 2}^{q} \frac{{(- 1)}^{j} c_{- i j}}{(j - 1)!} \frac{d^{j - 2}}{d z^{j - 2}} (\frac{1}{4} {[\frac{℘^{'} (z) + M_{i}}{℘ (z) - N_{i}}]}^{2} - ℘ (z)) + \sum_{i = 1}^{m - 1} \frac{c_{- i 1}}{2} \frac{℘^{'} (z) + M_{i}}{℘ (z) - N_{i}} + \sum_{j = 2}^{q} \frac{{(- 1)}^{j} c_{- m j}}{(j - 1)!} \frac{d^{j - 2}}{d z^{j - 2}} ℘ (z) + k_{0},

(22)

where

c_{- i j}

can be determined by (18),

c_{- i j}

and

k_{0}

are constants.

M_{i}^{2} = 4 N_{i}^{3} - g_{2} N_{i} - g_{3}

and

\sum_{i = 1}^{m} c_{- i 1} = 0 .

Any rational function solution

w : = R (z)

is expressed as

R (z) = \sum_{i = 1}^{m} \sum_{j = 1}^{q} \frac{c_{i j}}{{(z - z_{i})}^{j}} + k_{0},

(23)

which has

m (\leq p)

different poles of multiplicity q.

Any simply periodic solution

w : = R (ξ)

is a rational function of

ξ = e^{α z} (α \in C)

and can be given in the form

R (ξ) = \sum_{i = 1}^{m} \sum_{j = 1}^{q} \frac{c_{i j}}{{(ξ - ξ_{i})}^{j}} + k_{0},

(24)

which has

m (\leq p)

different poles of multiplicity q.

Lemma 2 ([33,34]).

Weierstrass elliptic functions

℘ (z)

have the addition formula and two successive degeneracies, as shown below:

(I) Addition formula

℘ (z - z_{0}) = - ℘ (z) - ℘ (z_{0}) + \frac{1}{4} {[\frac{℘^{'} (z) + ℘^{'} (z_{0})}{℘ (z) - ℘ (z_{0})}]}^{2} .

(25)

(II) If

Δ (g_{2}, g_{3}) = 0

, Weierstrass elliptic functions degenerate to simply periodic functions, which can be expressed as

℘ (z, 3 δ^{2}, - δ^{3}) = 2 δ - \frac{3 δ}{2} {coth}^{2} \sqrt{\frac{3 δ}{2}} z .

(26)

(III) If

g_{2} = g_{3} = 0

, Weierstrass elliptic functions degenerate to rational functions of z, which can be expressed as

℘ (z, 0, 0) = \frac{1}{z^{2}} .

(27)

Next, we give the complex method.

Substituting the transform $T : u (x, t) \to w (z), (x, t) \to z$ into a given PDE yields a nonlinear ODE: Equation (4) here.
Insert (18) into Equation (4) here to determine that weak $〈 p, q 〉$ condition holds.
By (22)–(24), we obtain all meromorphic solutions $w (z)$ of Equation (4) here with pole at $z = 0$ .
Get all meromorphic solutions $w (z - z_{0})$ by Lemmas 1 and 2.
Inserting the inverse transform $T^{- 1}$ into $w (z - z_{0})$ , we obtain all exact solutions $u (x, t)$ of the given partial differential equation.

3. Proof of Theorem 1

From balance the order of the poles in Equation (4), yields

(q + 1) + (q + 3) = 2 (q + 2) = 3 q,

and we get

q = 4

Substitute (18) into Equation (4),and set the cofficients to zero, we have

c_{- 4} = \frac{280}{A}, c_{- 3} = 0, c_{- 2} = \frac{280 B}{39 A}, c_{- 1} = 0,

c_{0} = - \frac{62 B^{2} - 507 C}{1521 A}, c_{1} = 0, c_{2} = \frac{62 B^{3}}{59319 A}, c_{3} = 0,

c_{4} = \frac{1}{425673144} \frac{- 11780 B^{4} + 257049 A D + 85683 C^{2}}{A}, \dots .

(28)

and then determine

p = 1 .

Therefore, Equation (4) is said to satisfy weak

〈 1, 4 〉

condition. We also get that all meromorphic solutions of Equation (4) belong to W if Equation (4) satisfies

〈 1, 4 〉

condition by Lemma 1.

In fact, since (17) and

\hat{E} (z, w) = w^{'} w^{‴} - \frac{1}{2} {(w^{″})}^{2} - A w^{3}

of Equation (4), we have

D (4) = 12

D_{r} (4) < 12, A c_{- 4} = 280,

(29)

and

{\hat{E}}^{'} (x, c_{- 4} x^{- 4}) x^{i - 4} = ({(c_{- 4} x^{- 4})}^{'} \frac{d^{3}}{d x^{3}} + {(c_{- 4} x^{- 4})}^{‴} \frac{d}{d x} - {(c_{- 4} x^{- 4})}^{″} \frac{d^{2}}{d x^{2}} - 3 A {(c_{- 4} x^{- 4})}^{2}) x^{i - 4} = - (4 (i - 1) (i - 4) (i - 5) + 120 i + 360) c_{- 4} x^{i - 12} .

(30)

Thus, from (29), (30) and (21), we infer that the Fuchs index of Equation (4) are zeros of the function

P (i) = - (4 (i - 1) (i - 4) (i - 5) + 120 i + 360) c_{- 4} .

(31)

It is easy to prove that

f (x) : = 4 (x - 1) (x - 4) (x - 5) + 120 x + 360 > 0

x > - 1 .

Hence, if

i \in N

, then

P (i) \neq 0

. That is to say, the Fuchs index of Equation (4) cannot be a positive integer. This implies that (ref. [29], p. 90): The principle part of the Laurent series of

w (z)

determines the whole Laurent series of

w (z)

. Therefore, weak

〈 1, 4 〉

condition implies that

〈 1, 4 〉

condition holds.

From, (23), we get the indeterminant form of rational function solutions of Equation (4) at pole

z = 0

R_{1} (z) = \frac{c_{- 4}}{z^{4}} + \frac{c_{- 2}}{z^{2}} + k_{0} = \frac{280}{A} \frac{1}{z^{4}} + \frac{280 B}{39 A} \frac{1}{z^{2}} + k_{0} .

(32)

Inserting (32) into Equation (4), and then setting the coefficients to zero, we get the system of Equations (1) which are given in Appendix A. Computing the system of Equations (1), we get

k_{0} = \frac{C}{3 A}, B = 0, D = - \frac{C^{2}}{3 A}, E = \frac{C^{3}}{27 A^{2}} .

So we find that the rational function solutions of Equation (4) are

w_{r 0} (z) = \frac{280}{A} \frac{1}{z^{4}} + \frac{C}{3 A},

(33)

at pole

z = 0

, here

B = 0, D = - \frac{C^{2}}{3 A}, E = \frac{C^{3}}{27 A^{2}}

Hence, all rational function solutions of Equation (4) are

w_{r} (z) = \frac{280}{A} \frac{1}{{(z - z_{0})}^{4}} + \frac{C}{3 A},

(34)

where

B = 0, D = - \frac{C^{2}}{3 A}, E = \frac{C^{3}}{27 A^{2}}

and

z_{0} \in C

By (22) and (26), we get the indeterminant form of simply periodic solutions of Equation (4) at pole

z = 0

w_{s 0} (z) = \frac{1}{6} c_{- 4} ℘^{″} (z, 3 δ^{2}, - δ^{3}) + c_{- 2} ℘ (z, 3 δ^{2}, - δ^{3}) + k_{0} = \frac{280}{6 A} ℘^{″} (z, 3 δ^{2}, - δ^{3}) + \frac{280 B}{39 A} ℘ (z, 3 δ^{2}, - δ^{3}) + k_{0},

(35)

where

℘ (z, 3 δ^{2}, - δ^{3}) = 2 δ - \frac{3}{2} δ {coth}^{2} \sqrt{\frac{3 δ}{2}} z .

Expanding

℘ (z, 3 δ^{2}, - δ^{3})

z_{0} = 0

, we have

℘ (z, 3 δ^{2}, - δ^{3}) = \frac{1}{z^{2}} + \frac{3 δ^{2}}{20} z^{2} - \frac{δ^{3}}{28} z^{4} + o (z^{4})

(36)

Substituting (36) into (35), we get

\begin{matrix} w (z) = \frac{280}{A} \frac{1}{z^{4}} + \frac{280 B}{39 A} \frac{1}{z^{2}} + \frac{14 δ^{2}}{A} + k_{0} + (\frac{14 B δ^{2}}{13 A} - \frac{20 δ^{3}}{A}) z^{2} \\ + (- \frac{10 B δ^{3}}{39 A} + \frac{21 δ^{4}}{2 A}) z^{4} + o (z^{4}) \end{matrix}

(37)

Comparing coefficients of (37) and (28), we obtain the system of Equations (2) which are showed in Appendix A. Solving the system of Equations (2), we have

δ = - \frac{B}{39}, k_{0} = - \frac{76 B^{2}}{1521 A} + \frac{C}{3 A}, D = \frac{1}{85683} \frac{5184 B^{4} - 28561 C^{2}}{A},

(38)

and

δ = \frac{B}{39 γ}, k_{0} = \frac{B^{2}}{1521 A} \frac{6 - 62 γ}{γ^{2}} + \frac{C}{3 A},

D = \frac{1}{23905557} \frac{(217620 γ - 59724) B^{4}}{γ^{4} A} - \frac{C^{2}}{3 A} . .

(39)

Substituting (35), (38) and (39) into Equation (4), from the correlation of coefficients we can obtain, respectively,

E = \frac{1}{130323843} \frac{746496 B^{6} - 2628288 B^{4} C + 4826809 C^{3}}{A^{2}}

and

E = \frac{1}{A^{2}} [\frac{16 B^{6}}{448893237} \frac{3001 - 7007 γ}{γ^{6}} - \frac{B^{4} C}{71716671} \frac{217620 γ - 59724}{γ^{4}} + \frac{C^{3}}{27}],

where

γ

is a root of

31 z^{2} - 31 z + 10 = 0 .

Hence, we obtain the simply periodic solutions of Equation (4) at pole

z = 0

are

\begin{matrix} w_{s, 1} (z) = \frac{2 B^{2}}{169 A} [35 {coth}^{4} \frac{\sqrt{26 B} i}{26} z - 70 {coth}^{2} \frac{\sqrt{26 B} i}{26} z + 23] + \frac{C}{3 A}, \end{matrix}

(40)

and

\begin{matrix} w_{s, 2} (z) = \frac{B^{2}}{507 A γ^{2}} [210 {coth}^{4} \sqrt{\frac{B}{26 γ}} z - (140 γ + 560) {coth}^{2} \sqrt{\frac{B}{26 γ}} z \\ + 166 γ + 352] + \frac{C}{3 A}, \end{matrix}

(41)

where

δ = - \frac{B}{39}, k_{0} = - \frac{76 B^{2}}{1521 A} + \frac{C}{3 A},

D = \frac{1}{85683} \frac{5184 B^{4} - 28561 C^{2}}{A},

E = \frac{1}{130323843} \frac{746496 B^{6} - 2628288 B^{4} C + 4826809 C^{3}}{A^{2}}

w_{s, 1} (z)

, and

D = \frac{1}{23905557} \frac{(217620 γ - 59724) B^{4}}{γ^{4} A} - \frac{C^{2}}{3 A},

E = \frac{1}{A^{2}} [\frac{16 B^{6}}{448893237} \frac{3001 - 7007 γ}{γ^{6}} - \frac{B^{4} C}{71716671} \frac{217620 γ - 59724}{γ^{4}} + \frac{C^{3}}{27}],

γ

is a root of

31 z^{2} - 31 z + 10 = 0

w_{s, 2} (z)

Furthermore, all simply periodic solutions of Equation (4) are given by

\begin{matrix} w_{s, 1} (z) = \frac{2 B^{2}}{169 A} [35 {coth}^{4} \frac{\sqrt{26 B} i}{26} (z - z_{0}) - 70 {coth}^{2} \frac{\sqrt{26 B} i}{26} (z - z_{0}) + 23] + \frac{C}{3 A}, \end{matrix}

(42)

and

\begin{matrix} w_{s, 2} (z) = \frac{B^{2}}{507 A γ^{2}} [210 {coth}^{4} \sqrt{\frac{B}{26 γ}} (z - z_{0}) - (140 γ + 560) {coth}^{2} \sqrt{\frac{B}{26 γ}} (z - z_{0}) \\ + 166 γ + 352] + \frac{C}{3 A}, \end{matrix}

(43)

where

z_{0} \in C,

D = \frac{1}{85683} \frac{5184 B^{4} - 28561 C^{2}}{A},

E = \frac{1}{130323843} \frac{746496 B^{6} - 2628288 B^{4} C + 4826809 C^{3}}{A^{2}}

w_{s, 1} (z)

, and

D = \frac{1}{23905557} \frac{(217620 γ - 59724) B^{4}}{γ^{4} A} - \frac{C^{2}}{3 A},

E = \frac{1}{A^{2}} [\frac{16 B^{6}}{448893237} \frac{3001 - 7007 γ}{γ^{6}} - \frac{B^{4} C}{71716671} \frac{217620 γ - 59724}{γ^{4}} + \frac{C^{3}}{27}],

γ

is a root of

31 z^{2} - 31 z + 10 = 0

w_{s, 2} (z)

From (22), we infer the indeterminant forms of elliptic function solutions of Equation (4) are

w_{d 0} (z) = \frac{c_{- 4}}{6} ℘^{″} (z) + c_{- 2} ℘ (z) + k_{0} = \frac{280}{6 A} ℘^{″} + \frac{280 B}{39 A} ℘ + k_{0} .

(44)

with pole at

z = 0

Expanding

℘ (z, g_{2}, g_{3})

z = 0

, we have

℘ (z, g_{2}, g_{3}) = \frac{1}{z^{2}} + \frac{g_{2}}{20} z^{2} + \frac{g_{3}}{28} z^{4} + o (z^{4}) .

(45)

Put (45) into (44) and apply (20), we get

\begin{matrix} w (z) = \frac{280}{A} \frac{1}{z^{4}} + \frac{280 B}{39 A} \frac{1}{z^{2}} + \frac{14 g_{2}}{3 A} + k_{0} + (\frac{14 B g_{2}}{39 A} + \frac{20 g_{3}}{A}) z^{2} \\ + (\frac{10 B g_{3}}{39 A} + \frac{7 g_{2}^{2}}{6 A}) z^{4} + o (z^{4}) . \end{matrix}

(46)

Comparing coefficients of

w (z)

and (28), we obtain the system of Equations (3). which are shown in the Appendix. Computing the system of Equations (3), we derive

k_{0} = \frac{C}{3 A} - \frac{14}{3 A} g_{2} - \frac{62 B^{2}}{1521 A}, g_{3} = \frac{31 B^{3}}{593190} - \frac{7 B}{390} g_{2} .

(47)

and

g_{2}

satisfies

g_{2}^{2} - \frac{2 B^{2}}{507} g_{2} + \frac{1457 B^{4}}{41384889} - \frac{A D}{1932} - \frac{C^{2}}{5796} = 0 .

Substituting (44) and (47) into Equation (4), from the correlation of coefficients, we can get

\begin{matrix} E = \frac{1}{A^{2}} {(\frac{1148}{69} A D + \frac{1148}{207} C^{2} - \frac{661248}{656903} B^{4}) g_{2} + \frac{23808}{4826809} B^{6} \\ + (\frac{20}{1521} A D + \frac{20}{4563} C^{2}) B^{2} - \frac{1}{3} A C D - \frac{2}{27} C^{3}}, \end{matrix}

then we have

w_{d 0} (z) = \frac{280}{A} ℘^{2} (z) + \frac{280 B}{39 A} ℘ (z) + \frac{C}{3 A} - \frac{62 B^{2}}{1521 A} - \frac{28}{A} g_{2} .

(48)

Thus, all elliptic function solutions of Equation (4) are

w_{d} (z) = \frac{280}{A} ℘^{2} (z - z_{0}) + \frac{280 B}{39 A} ℘ (z - z_{0}) + \frac{C}{3 A} - \frac{62 B^{2}}{1521 A} - \frac{28}{A} g_{2},

(49)

where

z_{0} \in C, g_{3} = \frac{31 B^{3}}{593190} - \frac{7 B}{390} g_{2},

\begin{matrix} E = \frac{1}{A^{2}} {(\frac{1148}{69} A D + \frac{1148}{207} C^{2} - \frac{661248}{656903} B^{4}) g_{2} + \frac{23808}{4826809} B^{6} \\ + (\frac{20}{1521} A D + \frac{20}{4563} C^{2}) B^{2} - \frac{1}{3} A C D - \frac{2}{27} C^{3}}, \end{matrix}

and

g_{2}

satisfies

g_{2}^{2} - \frac{2 B^{2}}{507} g_{2} + \frac{1457 B^{4}}{41384889} - \frac{A D}{1932} - \frac{C^{2}}{5796} = 0 .

By using the addition formula, we can get another representation of it as

\begin{matrix} w_{d} (z) = \frac{35}{2 A} {[\frac{℘^{'} (z) + M}{℘ (z) - N}]}^{4} - [\frac{140 (℘ (z) + N)}{A} - \frac{70 B}{39 A}] {[\frac{℘^{'} (z) + M}{℘ (z) - N}]}^{2} \\ + \frac{280}{A} {[℘ (z) + N]}^{2} - \frac{280 B}{39 A} [℘ (z) + N] + \frac{C}{3 A} - \frac{62 B^{2}}{1521 A} - \frac{28}{A} g_{2}, \end{matrix}

(50)

where

M^{2} = 4 N^{3} - g_{2} N - g_{3}, g_{3} = \frac{31 B^{3}}{593190} - \frac{7 B}{390} g_{2},

g_{2}^{2} - \frac{2 B^{2}}{507} g_{2} + \frac{1457}{41384889} B^{4} - \frac{A D}{1932} - \frac{C^{2}}{5796} = 0,

E = \frac{1}{A^{2}} [(\frac{1148}{69} A D + \frac{1148}{207} C^{2} - \frac{661248}{656903} B^{4}) g_{2} + \frac{23808}{4826809} B^{6}

+ (\frac{20}{1521} A D + \frac{20}{4563} C^{2}) B^{2} - \frac{1}{3} A C D - \frac{2}{27} C^{3}],

D and N are constants.

So far, we have completed the proof of Theorem 1.

Substituting (2) into all meromorphic solutions

w (z)

of Equation (4), we obtained Theorem 2. According to the theorem in ref. [35], we can get the following corollary.

Corollary. All rational solutions

u_{r} (x + ν t)

and simply periodic solutions

u_{s, 1} (x + ν t)

of Equation (1) are real valued, while simply periodic solutions

u_{s, 2} (x + ν t)

are not real valued.

4. Computer Simulations

This subsection will show our results through computer simulations of

u_{r} (x + ν t)

and

u_{s, 1} (x + ν t)

, as demonstrated in the following figures.

By applying the complex method, we are able to achieve the rational solution $u_{r} (x + ν t)$ of Equation (4). Figure 1 describes the 3D graphs of solution $u_{r} (x + ν t)$ for $a = 1$ , $ν = 1$ , and $d = \frac{1}{1680}$ within the interval $- 5 \leq x, t \leq 5$ . Figure 2 shows the 2D graphs of solution $u_{r} (x + ν t)$ for $a = 1$ , $ν = 1$ , and $d = \frac{1}{1680}$ within the interval $- 10 \leq x \leq 10$ when $t = 0$ . It could be observed that they have one generation pole, which is shown by Figure 1 and Figure 2.
By employing the complex method, we are able to obtain the simply periodic solutions $u_{s, 1} (x + ν t)$ and $u_{s, 2} (x + ν t)$ of Equation (4). Figure 3 shows the 3D graphs of solution $u_{s, 1} (x + ν t)$ for $a = 3$ , $c = 13$ , $d = 1$ , and $ν = 3$ within the interval $- 2 π \leq x, t \leq 2 π$ . Figure 4 describes the 2D graphs of solution $u_{s, 1} (x + ν t)$ for $a = 3$ , $c = 13$ , $d = 1$ , and $ν = 3$ within the interval $- 2 π \leq x \leq 2 π$ when $t = 0$ .

5. Conclusions

In this paper, we are the first to utilize the complex method to prove that all meromorphic solutions of complex Kawahara equation belong to the class W, and then we get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Equation (1). In addition, we find that all rational solutions

u_{r} (x + ν t)

and simply periodic solutions

u_{s, 1} (x + ν t)

of Equation (1) are solitary wave solutions, which could be used to analyze and interpret a lot of nonlinear dispersive phenomena. This research enriches the methods of solving differential equations. Our results also reveal that the complex method of looking for traveling wave exact solutions is general and feasible, and can be applied to other nonlinear partial differential equations.

Author Contributions

Conceptualization, J.T. and Y.G.; methodology, J.T. and Y.G.; software, C.J.; writing—original draft, F.Y. and X.Z.; writing—review, editing, T.O. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by the NSF of China (11901111), Young Innovative Talents Project of Guangdong Universities (2021KQNCX130), Science Research Group Project of SEIG (ST202101).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors would like to express their hearty thanks to the editor and referees for their very helpful comments and useful suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

The system of Equations (1):

$- A k_{0}^{3} + C k_{0}^{2} + D k_{0} + E = 0,$

$- \frac{280}{13} B k_{0}^{2} + \frac{560}{39} \frac{B C}{A} k_{0} + \frac{280 B D}{39 A} = 0,$

$- 840 k_{0}^{2} + (\frac{560 C}{A} - \frac{78400 B^{2}}{507 A}) k_{0} + \frac{78400 B^{2} C}{1521 A^{2}} + \frac{280 D}{A} = 0,$

$- \frac{156800 B}{13 A} k_{0} - \frac{9721600 B^{3}}{59319 A^{2}} + \frac{156800 B C}{39 A^{2}} = 0,$

$- \frac{235200}{A} k_{0} - \frac{4860800 B^{2}}{507 A^{2}} + \frac{78400 C}{A^{2}} = 0 .$
The system of Equations (2):

$\frac{14 δ^{2}}{A} + k_{0} = - \frac{62 B^{2} - 507 C}{1521 A},$

$\frac{14 B δ^{2}}{13 A} - \frac{20 δ^{3}}{A} = \frac{62 B^{3}}{59319 A},$

$- \frac{10 B δ^{3}}{39 A} + \frac{21 δ^{4}}{2 A} = \frac{- 11780 B^{4} + 257049 A D + 85683 C^{2}}{425673144} .$
The system of Equations (3):

$\frac{14 g_{2}}{3 A} + k_{0} = - \frac{62 B^{2} - 507 C}{1521 A},$

$\frac{14 B g_{2}}{39 A} + \frac{20 g_{3}}{A} = \frac{62 B^{3}}{59319 A},$

$\frac{10 B g_{3}}{39 A} + \frac{7 g_{2}^{2}}{6 A} = \frac{- 11780 B^{4} + 257049 A D + 85683 C^{2}}{425673144} .$

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Figure 1. The 3D profiles of solution

u_{r} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 5, t_{0} = 3

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 5, t_{0} = - 3

Figure 1. The 3D profiles of solution

u_{r} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 5, t_{0} = 3

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 5, t_{0} = - 3

Figure 2. The 2D profiles of solution

u_{r} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 5, t_{0} = 0

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 5, t_{0} = 0

Figure 2. The 2D profiles of solution

u_{r} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 5, t_{0} = 0

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 5, t_{0} = 0

Figure 3. The 3D profiles of solution

u_{s, 1} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 1, t_{0} = 1

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 1, t_{0} = - 1

Figure 3. The 3D profiles of solution

u_{s, 1} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 1, t_{0} = 1

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 1, t_{0} = - 1

Figure 4. The 2D profiles of solution

u_{s, 1} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 1, t_{0} = 0

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 1, t_{0} = 0

Figure 4. The 2D profiles of solution

u_{s, 1} (x + ν t)

of Equation (1) corresponding to (i)

x_{0} = 1, t_{0} = 0

, (ii)

x_{0} = 0, t_{0} = 0

, (iii)

x_{0} = - 1, t_{0} = 0

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Ye, F.; Tian, J.; Zhang, X.; Jiang, C.; Ouyang, T.; Gu, Y. All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method. Axioms 2022, 11, 330. https://doi.org/10.3390/axioms11070330

AMA Style

Ye F, Tian J, Zhang X, Jiang C, Ouyang T, Gu Y. All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method. Axioms. 2022; 11(7):330. https://doi.org/10.3390/axioms11070330

Chicago/Turabian Style

Ye, Feng, Jian Tian, Xiaoting Zhang, Chunling Jiang, Tong Ouyang, and Yongyi Gu. 2022. "All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method" Axioms 11, no. 7: 330. https://doi.org/10.3390/axioms11070330

APA Style

Ye, F., Tian, J., Zhang, X., Jiang, C., Ouyang, T., & Gu, Y. (2022). All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method. Axioms, 11(7), 330. https://doi.org/10.3390/axioms11070330

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All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

Abstract

1. Introduction and Main Results

2. Preliminary Lemmas and the Complex Method

3. Proof of Theorem 1

4. Computer Simulations

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Appendix A

References

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