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Stable novel and accurate solitary wave solutions of an integrable equation: Qiao model

  • Dexu Zhao , Dianchen Lu , Samir A. Salama and Mostafa M. A. Khater EMAIL logo
Published/Copyright: December 1, 2021
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Open Physics
From the journal Open Physics Volume 19 Issue 1

Abstract

This article investigates the dynamical and physical behavior of the second positive member in a new, utterly integrable hierarchy. This investigation depends on constructing novel analytical and approximate solutions to the Qiao model. The model’s name is after the researcher who derived the mathematical formula of it in 2007. This model possesses a Lax representation and bi-Hamiltonian structure. This study employs the unified and variational iteration (VI) method to obtain analytical and numerical solutions to the considered model. The obtained analytical solutions are used to calculate the necessary conditions for applying the suggested numerical method that makes checking the obtained solutions’ accuracy a valuable option. The obtained solutions are sketched in different techniques to explain more physical and dynamics details of the Qiao model and show the matching between obtained solutions.

Keywords: completely integrable hierarchy; Qiao model; analytical and approximate solutions; stability property

1 Introduction

Recently, many distinct fields such as soil mechanics, nonlinear optics, plasma wave, condensed matter physics, wave propagation in plasma physics, fluid mechanics, population ecology, neural networks, infectious disease epidemiology, thermodynamics, solid-state physics, civil engineering, quantum mechanics, and so on have considered nonlinear partial differential equation (NLPDE) field as a keyword for representing and explaining many of complex phenomena [1,2,3, 4,5,6]. Significantly, this field is able to formulate many complex natural phenomena in mathematical formulas [7,8,9]. The formats of these mathematical models depend on the real experiments to determine the parameters and empirical functions [10,11]. Consequently, these complex phenomena can be explained and simplified, which allows using the models’ properties for improving their applications [12].

Subsequently, a lot of mathematicians and physicists have focused on studying these models by applying some novel and accurate analytical and approximate schemes for evaluating computational solutions [13,14]. Furthermore, some other researchers have been focusing on deriving these schemes, which depends on the following three main icons: ansatz techniques, reduction techniques, and Lie group techniques [15,16,17].

This article handles the second positive member in a new utterly integrable hierarchy using the unified and VI methods to construct novel solitary wave solutions. This model is given by ref. [18]

(1) 2 U t − 1 U q x x x + 1 U q x = 0 .

Using the next wave transformation U ( x , t ) = u ( ζ ) , ζ = x + ω t and replacing q with − r , convert equation (1) into

(2) 2 ω u ′ − ( u r ) ‴ + ( u r ) ′ = 0 .

Replacing u with v 1 r , integrating equation (2) once with respect to ζ , and zero constant of integration, we obtain

(3) 2 ω v λ − v ″ + v = 0 ,

where λ = 1 r . Balancing the terms of equation (3), we obtain n = 2 λ − 1 . Thus, we apply another transformation on equation (3), which is given by v = ν 2 λ − 1 . This transformation converts equation (3) into the following form:

(4) 2 ν 4 ω + ν 2 + 2 ( λ − 3 ) ( ν ′ ) 2 − 2 ( λ − 1 ) ν ν ″ ( λ − 1 ) 2 = 0 .

Balancing the highest order and nonlinear term in equation (4), we obtain n = 1 . Consequently, the general solutions of the Qiao model based on the unified method take the following form:

(5) ν ( ζ ) = a 0 + ∑ i = 1 n ℱ i − 1 ( ζ ) [ a i ℱ ( ζ ) + b i G ( ζ ) ] = a 0 + a 1 ℱ ( ζ ) + b 1 G ( ζ ) ,

where a 0 , a 1 , b 1 are arbitrary constants to be determined through the method’s framework. While ℱ ( ζ ) , G ( ζ ) satisfies the following ordinary differential equations:

(6) ℱ ′ ( ζ ) = β 1 ℱ ( ζ ) G ( ζ ) , G ′ ( ζ ) = β 2 − β 3 ℱ ( ζ ) + β 1 G ( ζ ) 2 , G ( ξ ) 2 = β 2 + ( β 3 2 − 1 ) ℱ ( ξ ) 2 β 2 − 2 β 3 ℱ ( ξ ) , when β 1 = − 1 , β 4 = − 1 , β 2 ≠ 0 ↦ Case I . G ( ξ ) 2 = β 2 + ( β 3 2 + 1 ) ℱ ( ξ ) 2 β 2 − 2 β 3 ℱ ( ξ ) , when β 1 = − 1 , β 4 = 1 , β 2 ≠ 0 ↦ Case II . G ( ζ ) 2 = − β 2 − ( β 3 2 − 1 ) ℱ ( ζ ) 2 β 2 + 2 β 3 ℱ ( ζ ) , when β 1 = 1 , β 4 = − 1 , β 2 ≠ 0 ↦ Case III ,

where β 2 , β 3 are arbitrary constants to be calculated through the method’s framework.

The rest of the article is organized in the following order: Section 2 investigates analytical and approximate solutions and checks the analytical solutions’ stability characterization. Section 3 demonstrates the results’ novelty and accuracy and the article’s contributions. Section 4 explains the figures’ interpretations. Section 5 gives the conclusion of the whole article.

2 Accurate stable soliton wave solutions of the Qiao model

Here, we apply the unified method, the Hamiltonian system’s characterizations, and the VI method to the nonlinear Qiao model to construct novel, accurate, stable analytical solutions. This investigation is given in the following subsections.

2.1 Abundant wave solutions of Qiao model

Employing the unified method’s framework to equation (5) for constructing novel soliton wave solutions, we obtain the following values of the aforementioned parameters:

Case I: when [ β 1 = − 1 & β 4 = − 1 ].

a 0 = 0 , a 1 = − β 3 2 − 1 4 2 ω , b 1 = 0 , λ = − 3 , β 2 = 4 . a 0 = i 2 2 ω , a 1 = − β 3 2 − 1 8 2 ω , b 1 = i 4 2 ω , λ = − 3 , β 2 = 4 .

Case II: when [ β 1 = − 1 & β 4 = 1 ].

a 0 = 0 , a 1 = − β 3 2 − 1 4 2 ω , b 1 = 0 , λ = − 3 , β 2 = 4 . a 0 = i 2 2 ω , a 1 = − β 3 2 − 1 8 2 ω , b 1 = i 4 2 ω , λ = − 3 , β 2 = 4 .

Case III: when [ β 1 = 1 & β 4 = − 1 ].

a 0 = 0 , a 1 = 1 − β 3 2 4 2 ω , b 1 = 0 , λ = − 3 , β 2 = − 4 . a 0 = i 2 2 ω , a 1 = 1 − β 3 2 8 2 ω , b 1 = i 4 2 ω , λ = − 3 , β 2 = − 4 .

Consequently, the solitary wave solutions of the Qiao model are given in the following formulas:

(7) U 1 ( x , t ) = 1 8 2 3 / 4 β 2 − β 3 2 − 1 ω ( β 3 + cosh ( β 2 ( t ω + x ) ) ) 3 / 2 .

(8) U 2 ( x , t ) = 1 32 2 4 − β 3 2 − 1 β 2 + 2 i β 2 sinh ( β 2 ( t ω + x ) ) + 4 i ( β 3 + cosh ( β 2 ( t ω + x ) ) ) ω ( β 3 + cosh ( β 2 ( t ω + x ) ) ) 3 / 2 .

(9) U 3 ( x , t ) = 1 8 2 3 / 4 β 2 − β 3 2 − 1 ω ( β 3 + sinh ( β 2 ( t ω + x ) ) ) 3 / 2 .

(10) U 4 ( x , t ) = 1 32 2 4 − β 3 2 − 1 β 2 + 4 i ( β 3 + sinh ( β 2 ( t ω + x ) ) ) + 2 i β 2 cosh ( β 2 ( t ω + x ) ) ω ( β 3 + sinh ( β 2 ( t ω + x ) ) ) 3 / 2 .

(11) U 5 ( x , t ) = 1 8 2 3 / 4 β 2 1 − β 3 2 ω ( β 3 + cos ( β 2 ( t ω + x ) ) ) 3 / 2 .

(12) U 6 ( x , t ) = 1 32 2 4 1 − β 3 2 β 2 + 2 i β 2 sin ( β 2 ( t ω + x ) ) + 4 i ( β 3 + cos ( β 2 ( t ω + x ) ) ) ω ( β 3 + cos ( β 2 ( t ω + x ) ) ) 3 / 2 .

(13) U 7 ( x , t ) = 1 8 2 3 / 4 β 2 1 − β 3 2 ω ( β 3 + sin ( β 2 ( t ω + x ) ) ) 3 / 2 .

(14) U 8 ( x , t ) = 1 32 2 4 1 − β 3 2 β 2 + 4 i ( β 3 + sin ( β 2 ( t ω + x ) ) ) − 2 i β 2 cos ( β 2 ( t ω + x ) ) ω ( β 3 + sin ( β 2 ( t ω + x ) ) ) 3 / 2 .

2.2 Stability

Studying the stability property of the Qiao model’s solution (1) through the Hamiltonian system’s characterizations with the following values of the constants [ β 2 = 4 , β 3 = 6 ] , we obtain:

T = 1 5,488,000 2 ω 5 / 2 [ i ( − 3,430,000 ω log ( 1 + 12 e 20 ( ω + 1 ) + e 40 ( ω + 1 ) ) + 630 37 log ( 1 + 12 e 20 ( ω + 1 ) + e 40 ( ω + 1 ) ) + 264,600 log ( 1 + 12 e 20 ( ω + 1 ) + e 40 ( ω + 1 ) ) − 1,260 35 tanh − 1 6 + e 20 ( ω + 1 ) 35 − 14,700 1,295 × tanh − 1 6 + e 20 ( ω + 1 ) 35 − 1,260 35 tanh − 1 6 + e 2 ω + 2 35 − 14,700 1,295 tanh − 1 6 + e 2 ω + 2 35 − 5,040 35 ω × tanh − 1 6 + e 2 ω + 20 35 − 19,892 1,295 ω tanh − 1 6 + e 2 ω + 20 35 − 50,400 35 ω tanh − 1 6 + e 20 ω + 2 35 − 198,920 1,295 ω tanh − 1 6 + e 20 ω + 2 35 − 343,000 ω log ( 1 + 12 e 2 ω + 2 + e 4 ω + 4 ) + 264,600 log ( 1 + 12 e 2 ω + 2 + e 4 ω + 4 ) + 343,000 ω log ( 1 + 12 e 2 ω + 20 + e 4 ω + 40 ) − 630 37 log ( 1 + 12 e 2 ω + 20 + e 4 ω + 40 ) − 264,600 log ( 1 + 12 e 2 ω + 20 + e 4 ω + 40 ) + 3,430,000 ω log ( 1 + 12 e 20 ω + 2 + e 40 ω + 4 ) − 630 37 log ( 1 + 12 e 20 ω + 2 + e 40 ω + 4 ) − 264,600 log ( 1 + 12 e 20 ω + 2 + e 40 ω + 4 ) + 343,000 ω log 1 + e 2 ω + 2 6 − 35 − 2,520 35 ω log 1 + e 2 ω + 20 6 − 35 − 9,946 1,295 ω log 1 + e 2 ω + 20 6 − 35 − 343,000 ω log 1 + e 2 ω + 20 6 − 35 − 25,200 35 ω log 1 + e 20 ω + 2 6 − 35

− 99,460 1,295 ω log 1 + e 20 ω + 2 6 − 35 − 3,430,000 ω log 1 + e 20 ω + 2 6 − 35 + 3,430,000 ω log 1 − e 20 ( ω + 1 ) 35 − 6 − 25,200 35 ω log 1 + e 20 ( ω + 1 ) 35 + 6 − 99,460 1,295 ω log 1 + e 20 ( ω + 1 ) 35 + 6 + 3,430,000 ω log 1 + e 20 ( ω + 1 ) 35 + 6 − 2,520 35 ω log 1 + e 2 ω + 2 35 + 6 − 9,946 1,295 ω log 1 + e 2 ω + 2 35 + 6 + 343,000 ω log 1 + e 2 ω + 2 35 + 6 − 343,000 ω log 1 + e 2 ω + 20 35 + 6 − 3,430,000 ω log 1 + e 20 ω + 2 35 + 6 + 171,500 Li 2 e 20 ( ω + 1 ) 35 − 6 + 171,500 × Li 2 e 2 ω + 2 35 − 6 − 1,260 35 Li 2 e 2 ω + 20 35 − 6 − 4,973 1,295 Li 2 e 2 ω + 20 35 − 6 − 171,500 Li 2 e 2 ω + 20 35 − 6 − 1,260 35 Li 2 e 20 ω + 2 35 − 6 − 4,973 1,295 Li 2 e 20 ω + 2 35 − 6 − 171,500 Li 2 e 20 ω + 2 35 − 6 − 1,260 35 × Li 2 − e 20 ( ω + 1 ) 35 + 6 − 4,973 1,295 Li 2 − e 20 ( ω + 1 ) 35 + 6 + 171,500 Li 2 − e 20 ( ω + 1 ) 35 + 6 − 1,260 35 Li 2 − e 2 ω + 2 35 + 6 − 4,973 1,295 Li 2 − e 2 ω + 2 35 + 6 + 171,500 Li 2 − e 2 ω + 2 35 + 6 − 171,500 Li 2 − e 2 ω + 20 35 + 6 − 171,500

× Li 2 − e 20 ω + 2 35 + 6 + 50,400 ω tanh − 1 6 + e 20 ( ω + 1 ) 35 35 + 5,040 ω tanh − 1 6 + e 2 ω + 2 35 35 + 1,260 × tanh − 1 6 + e 2 ω + 20 35 35 + 1,260 tanh − 1 6 + e 20 ω + 2 35 35 + 2,520 ω log 1 + e 2 ω + 2 6 − 35 35 + 25,200 ω log 1 − e 20 ( ω + 1 ) 35 − 6 35 + 2,520 ω log 1 + e 2 ω + 20 35 + 6 35 + 25,200 ω log 1 + e 20 ω + 2 35 + 6 35 + 1,260 Li 2 e 20 ( ω + 1 ) 35 − 6 35 + 1,260 Li 2 e 2 ω + 2 35 − 6 35 + 1,260 Li 2 − e 2 ω + 20 35 + 6 35 + 1,260 Li 2 − e 20 ω + 2 35 + 6 × 35 + 630 log ( 1 + 12 e 2 ω + 2 + e 4 ω + 4 ) 37 + 173,950 e 2 ω + 20 37 1 + 12 e 2 ω + 20 + e 4 ω + 40 + 173,950 e 20 ω + 2 37 1 + 12 e 20 ω + 2 + e 40 ω + 4 + 198,920 ω × tanh − 1 6 + e 20 ( ω + 1 ) 35 1,295 + 19,892 ω tanh − 1 6 + e 2 ω + 2 35 1,295 + 14,700 tanh − 1 6 + e 2 ω + 20 35 1,295 + 14,700 tanh − 1 6 + e 20 ω + 2 35 1,295 + 9,946 ω log 1 + e 2 ω + 2 6 − 35 1,295 + 99,460 ω log 1 − e 20 ( ω + 1 ) 35 − 6 1,295 + 9,946 ω log 1 + e 2 ω + 20 35 + 6 1,295 + 99,460 ω log 1 + e 20 ω + 2 35 + 6 1,295 + 4,973 Li 2 e 20 ( ω + 1 ) 35 − 6 1,295

(15) + 4,973 Li 2 e 2 ω + 2 35 − 6 1,295 + 4,973 Li 2 − e 2 ω + 20 35 + 6 1,295 + 4,973 Li 2 − e 20 ω + 2 35 + 6 1,295 − 173,950 37 e 20 ( ω + 1 ) 1 + 12 e 20 ( ω + 1 ) + e 40 ( ω + 1 ) + 1,073,100 e 20 ( ω + 1 ) 1 + 12 e 20 ( ω + 1 ) + e 40 ( ω + 1 ) + 178,850 1 + 12 e 20 ( ω + 1 ) + e 40 ( ω + 1 ) − 173,950 37 e 2 ω + 2 1 + 12 e 2 ω + 2 + e 4 ω + 4 + 1,073,100 e 2 ω + 2 1 + 12 e 2 ω + 2 + e 4 ω + 4 + 178,850 1 + 12 e 2 ω + 2 + e 4 ω + 4 − 1,073,100 e 2 ω + 20 1 + 12 e 2 ω + 20 + e 4 ω + 40 − 178,850 1 + 12 e 2 ω + 20 + e 4 ω + 40 − 1,073,100 e 20 ω + 2 1 + 12 e 20 ω + 2 + e 40 ω + 4 − 178,850 1 + 12 e 20 ω + 2 + e 40 ω + 4 ,

where T is the Hamiltonian system’s momentum. Thus, we have

(16) ∂ T ∂ ω ω = − 9 = 0.00142019356 > 0 .

Consequently, the investigated solution (8) is the stable solution in the following interval x ∈ [ 1 , 10 ] , t ∈ [ 1 , 10 ] . Employing the aforementioned steps for the other obtained solutions demonstrates their stability characterization.

2.3 Approximate solutions

Here, the approximate solution of the investigated model is constructed by applying the variational iterational (VI) method. The method’s head lines are given as follows. The NLPDE can be given in the following formula:

(17) ℒ U ( x , t ) + N U ( x , t ) = G ( x , t ) ,

where ℒ , N , G ( x , t ) represent, respectively, a linear operator, a nonlinear operator, and a known differential function. The correction functional in the VI method’s framework is constructed in the next formula:

(18) U n + 1 ( x , t ) = U n ( x , t ) + ∫ 0 t λ { L U n ( x , s ) + N U ˜ n ( x , s ) − G ( x , s ) } d s ,

where λ , U n ( x , t ) , U ˜ n ( x , t ) are, respectively, a general Lagrange multiplier that can be identified optimally via variational theory, the n th approximate solution, and considered a restricted variation. On the other hand, this term ∫ 0 t λ { L U n ( x , s ) + N U ˜ n ( x , s ) − G ( x , s ) } d s is called the correction.

Handling Qiao model by the VIM, we obtain the following approximate solution:

(19) U Approx. ( x , t ) = 1 32 2 4 [ 2 Ξ sech 2 ( x ) + ( sinh ( x ) Ξ − cosh ( x ) Ξ ) sech ( x ) + ( cosh ( x ) Ξ + sinh ( x ) Ξ ) sech ( x ) + 2 Ξ + 1 ( Ξ ) 5 [ sech 7 ( x ) ( 768 t cosh ( 5 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 + 768 t sinh ( 5 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 ) ] + sech 7 ( x ) ( Ξ ) 5 [ 576 t cosh ( 5 x ) 2 Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 + 576 t sinh ( 5 x ) 2 Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 ] + sech 7 ( x ) ( Ξ ) 5 [ − 768 t cosh ( 3 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 − 768 t sinh ( 3 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 ] + sech 7 ( x ) ( Ξ ) 5 [ − 576 2 t sinh ( 3 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 − 576 t cosh ( 3 x ) 2 Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 ] + sech 6 ( x ) ( Ξ ) 5 [ − 1,920 t cosh ( 4 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 − 1,920 t sinh ( 4 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 ] + sech 6 ( x ) ( Ξ ) 5 [ − 768 2 t sinh ( 4 x ) Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 − 768 t cosh ( 4 x ) 2 Ξ ( ( Ξ ) 3 / 2 ) 2 / 3 ] ] + ⋯ ,

where Ξ = 2 sech 2 ( x ) + 2 tanh ( x ) + 2 .

3 Results and discussion

Here, we discuss the obtained solutions in detail to show the article’s contributions. Also, we compare our obtained results with recently published articles that have investigated the same studied model. Finally, we show our obtained solutions’ accuracy by studying the above-shown tables. We have obtained some solutions in different formats such as hyperbolic and trigonometric. These solutions represent bright soliton (7), dark soliton (8), formal soliton (9), singular dark soliton (10), and periodic wave solutions (11), (12), (13), and (14).

Second, the result’s novelty can be explained by comparing our solutions with those that have been published in ref. [19], where Pan and Liu have derived the traveling wave systems of the Qiao model. In that article, they have obtained just two solutions (31) and (32). Both solutions [(31), (32) in [19]] are similar to our solutions (7) and (9) when − p + q = 3 2 , ω = 1 1,296 c 2 p 2 ( p + q ) 2 . All our other solutions are novel solutions.

Finally, the results, accuracy can be tested by studying Tables 1, and 2, where these tables show the value of approximate, analytical, and absolute errors between both of them. These two tables show the comparison of both solutions, which refers to the solutions’ accuracy. Furthermore, Figure 5 shows the absolute error between analytical and approximate solutions when t ∈ { 2 , 4 , 6 , 8 } in two different zooms for more explanation of the table’s values.

Table 1

Analytical and approximate solutions of the investigated model through different values of t ∈ { 2 , 4 , 6 , 8 , 10 }

t = 2 t = 4 t = 6 t = 8 t = 10
Ex. Approx. Ex. Approx. Ex. Approx. Ex. Approx. Ex. Approx.
0 2.98 × 1 0 − 11 − 1.9826002 1.12 × 1 0 21 − 4.130979 0 − 6.2793579 0 − 8.4277367 0 − 10.576116
2 1.2 × 1 0 8 0.2314726 4.53 × 1 0 19 0.2505121 0 0.2695517 0 0.2885913 0 0.3076309
4 4.84 × 1 0 6 0.2116478 1.83 × 1 0 16 0.2130278 0 0.2144078 0 0.2157878 0 0.2171677
6 0.001872 0.2102506 7.38 × 1 0 14 0.2102763 2.94 × 1 0 24 0.2103019 0 0.2103276 0 0.2103533
8 0.165779 0.2102246 2.98 × 1 0 11 0.2102251 1.12 × 1 0 21 0.2102255 0 0.210226 0 0.2102265
10 0.212433 0.2102241 1.2 × 1 0 8 0.2102241 4.53 × 1 0 19 0.2102241 0 0.2102241 0 0.2102241
12 0.210268 0.2102241 4.84 × 1 0 6 0.2102241 1.83 × 1 0 16 0.2102241 0 0.2102241 0 0.2102241
14 0.210225 0.2102241 0.001872 0.2102241 7.38 × 1 0 14 0.2102241 2.94 × 1 0 24 0.2102241 0 0.2102241
16 0.210224 0.2102241 0.165779 0.2102241 2.98 × 1 0 11 0.2102241 1.12 × 1 0 21 0.2102241 0 0.2102241
18 0.210224 0.2102241 0.212433 0.2102241 1.2 × 1 0 8 0.2102241 4.53 × 1 0 19 0.2102241 0 0.2102241
20 0.210224 0.2102241 0.210268 0.2102241 4.84 × 1 0 6 0.2102241 1.83 × 1 0 16 0.2102241 0 0.2102241
22 0.210224 0.2102241 0.210225 0.2102241 0.001872 0.2102241 7.38 × 1 0 14 0.2102241 2.94 × 1 0 24 0.2102241
24 0.210224 0.2102241 0.210224 0.2102241 0.165779 0.2102241 2.98 × 1 0 11 0.2102241 1.12 × 1 0 21 0.2102241
26 0.210224 0.2102241 0.210224 0.2102241 0.212433 0.2102241 1.2 × 1 0 8 0.2102241 4.53 × 1 0 19 0.2102241
28 0.210224 0.2102241 0.210224 0.2102241 0.210268 0.2102241 4.84 × 1 0 6 0.2102241 1.83 × 1 0 16 0.2102241
30 0.210224 0.2102241 0.210224 0.2102241 0.210225 0.2102241 0.001872 0.2102241 7.38 × 1 0 14 0.2102241
32 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.165779 0.2102241 2.98 × 1 0 11 0.2102241
34 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.212433 0.2102241 1.2 × 1 0 8 0.2102241
36 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210268 0.2102241 4.84 × 1 0 6 0.2102241
38 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210225 0.2102241 0.001872 0.2102241
40 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.165779 0.2102241
42 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.212433 0.2102241
44 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210268 0.2102241
46 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210225 0.2102241
48 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241
50 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241
52 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241
54 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241
56 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241
58 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241
60 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241 0.210224 0.2102241
Table 2

Absolute values of error between analytical and approximate solutions that have been obtained by unified method and VI method through different values of t ∈ { 2 , 4 , 6 , 8 , 10 }

t = 2 t = 4 t = 6 t = 8 t = 10
0 1.982600202 4.130979035 6.279357868 8.4277367 10.57611553
2 0.231472548 0.250512137 0.269551713 0.28859129 0.307630867
4 0.211642993 0.213027812 0.214407789 0.215787767 0.217167744
6 0.208378366 0.210276254 0.210301927 0.210327601 0.210353275
8 0.044445958 0.210225059 0.21022553 0.210226 0.21022647
10 0.00220887 0.210224109 0.21022413 0.210224139 0.210224147
12 4.37535 × 1 0 5 0.210219263 0.210224104 0.210224104 0.210224105
14 8.02511 × 1 0 7 0.20835189 0.210224104 0.210224104 0.210224104
16 1.46989 × 1 0 8 0.044445473 0.210224104 0.210224104 0.210224104
18 2.6922 × 1 0 10 0.002208879 0.210224092 0.210224104 0.210224104
20 4.93092 × 1 0 12 4.37536 × 1 0 5 0.210219262 0.210224104 0.210224104
22 9.02611 × 1 0 14 8.02514 × 1 0 7 0.20835189 0.210224104 0.210224104
24 1.63758 × 1 0 15 1.46989 × 1 0 8 0.044445473 0.210224104 0.210224104
26 5.55112 × 1 0 17 2.69221 × 1 0 10 0.002208879 0.210224092 0.210224104
28 2.77556 × 1 0 17 4.93092 × 1 0 12 4.37536 × 1 0 5 0.210219262 0.210224104
30 2.77556 × 1 0 17 9.02611 × 1 0 14 8.02514 × 1 0 7 0.20835189 0.210224104
32 2.77556 × 1 0 17 1.63758 × 1 0 15 1.46989 × 1 0 8 0.044445473 0.210224104
34 2.77556 × 1 0 17 5.55112 × 1 0 17 2.69221 × 1 0 10 0.002208879 0.210224092
36 − 5.55112 × 1 0 17 2.77556 × 1 0 17 4.93092 × 1 0 12 4.37536 × 1 0 5 0.210219262
38 − 5.55112 × 1 0 17 2.77556 × 1 0 17 9.02611 × 1 0 14 8.02514 × 1 0 7 0.20835189
40 2.77556 × 1 0 17 2.77556 × 1 0 17 1.63758 × 1 0 15 1.46989 × 1 0 8 0.044445473
42 2.77556 × 1 0 17 2.77556 × 1 0 17 5.55112 × 1 0 17 2.69221 × 1 0 10 0.002208879
44 − 5.55112 × 1 0 17 − 5.55112 × 1 0 17 2.77556 × 1 0 17 4.93092 × 1 0 12 4.37536 × 1 0 5
46 2.77556 × 1 0 17 − 5.55112 × 1 0 17 2.77556 × 1 0 17 9.02611 × 1 0 14 8.02514 × 1 0 7
48 2.77556 × 1 0 17 2.77556 × 1 0 17 2.77556 × 1 0 17 1.63758 × 1 0 15 1.46989 × 1 0 8
50 2.77556 × 1 0 17 2.77556 × 1 0 17 2.77556 × 1 0 17 5.55112 × 1 0 17 2.69221 × 1 0 10
52 2.77556 × 1 0 17 − 5.55112 × 1 0 17 − 5.55112 × 1 0 17 2.77556 × 1 0 17 4.93092 × 1 0 12
54 2.77556 × 1 0 17 2.77556 × 1 0 17 − 5.55112 × 1 0 17 2.77556 × 1 0 17 9.02611 × 1 0 14
56 2.77556 × 1 0 17 2.77556 × 1 0 17 2.77556 × 1 0 17 2.77556 × 1 0 17 1.63758 × 1 0 15
58 2.77556 × 1 0 17 2.77556 × 1 0 17 2.77556 × 1 0 17 2.77556 × 1 0 17 5.55112 × 1 0 17
60 2.77556 × 1 0 17 2.77556 × 1 0 17 − 5.55112 × 1 0 17 − 5.55112 × 1 0 17 2.77556 × 1 0 17

4 Figures’ interpretation

This part is responsible for explaining the above-shown sketches and their physical interpretation of each. Generally, these figures illustrate the perspective view, the wave propagation pattern along x -axis, and the overhead view of the tested solution. This explanation is given in the following form.

  • Figure 1 explains bright soliton wave of equation (7) when β 2 = 4 , β 3 = 2 in three-dimensional (3D), two-dimensional (2D), and contour plots.

  • Figure 2 clarifies dark soliton wave of equation (8) when β 2 = 4 , β 3 = 6 in 3D, 2D, and contour plots.

  • Figure 3 shows combined formal soliton wave of equation (19) in 3-D, 2-D, and contour plots.

  • Figure 4 demonstrates the matching between analytical and approximate solutions based on the data shown in Table 1.

  • Figure 5 represents the absolute error through the values shown in Table 2 in 2D plot.

Figure 1 3D (a), 2D (b), and density (c) elucidations of equation (7).
Figure 1

3D (a), 2D (b), and density (c) elucidations of equation (7).

Figure 2 3D (a), 2D (b), and density (c) elucidations of equation (8).
Figure 2

3D (a), 2D (b), and density (c) elucidations of equation (8).

Figure 3 3D (a), 2D (b), and density (c) elucidations of equation (19).
Figure 3

3D (a), 2D (b), and density (c) elucidations of equation (19).

Figure 4 (a) 2D, (b) distribution plots for the obtained analytical wave solutions with different values of t t , (c) 2D, (d) distribution/Rug plots for the obtained approximate wave solutions with different values of t t , (e), (f) 2D plots of analytical and approximate in two different zoom when t ∈ { 2 , 4 , 6 , 8 , 10 } t\in \left\{2,4,6,8,10\right\} .
Figure 4

(a) 2D, (b) distribution plots for the obtained analytical wave solutions with different values of t , (c) 2D, (d) distribution/Rug plots for the obtained approximate wave solutions with different values of t , (e), (f) 2D plots of analytical and approximate in two different zoom when t ∈ { 2 , 4 , 6 , 8 , 10 } .

Figure 5 (a) Zoom 2D plot for the absolute error and (b) regular 2D plot for the absolute error based on the values shown in Table 2.
Figure 5

(a) Zoom 2D plot for the absolute error and (b) regular 2D plot for the absolute error based on the values shown in Table 2.

5 Conclusion

This article has successfully investigated the analytical and approximate solutions of the Qiao model. The unified and VI methods have been applied, and many novel analytical solutions have been obtained. Additionally, the approximate solution has been obtained. The obtained solutions have been demonstrated through some different figures in 2D, 3D, and contour plots. The stability property of the results has been checked by using the Hamiltonian system’s properties. Finally, all obtained solutions have been checked by putting them back into the original model using Mathematica 12.

  1. Funding information: The authors greatly thank Taif University for providing fund for this work through Taif University Researchers Supporting Project number (TURSP-2020/52), Taif University, Taif, Saudi Arabia.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: This work does not have any conflicts of interest.

  4. Data availability statement: The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Received: 2021-10-14
Revised: 2021-10-22
Accepted: 2021-10-24
Published Online: 2021-12-01

© 2021 Dexu Zhao et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2021-0078
Keywords for this article
completely integrable hierarchy; Qiao model; analytical and approximate solutions; stability property
Creative Commons
BY 4.0

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