Further Results about Seeking for the Exact Solutions of the Nonlinear <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 46.3673 15.2545" width="46.3673pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.017,0,0,-0.017,18.01,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,31.918,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.017,0,0,-0.017,40.154,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Dimensional Jaulent-Miodek Equation

Hu, Jingsen; Qi, Jianming

doi:https://doi.org/10.1155/2021/5258692

Advances in Mathematical Physics

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5258692 | https://doi.org/10.1155/2021/5258692

Further Results about Seeking for the Exact Solutions of the Nonlinear -Dimensional Jaulent-Miodek Equation

Jingsen Hu¹and Jianming Qi¹

Academic Editor: Wen-Xiu Ma

Received21 May 2021

Revised25 Jun 2021

Accepted10 Jul 2021

Published29 Jul 2021

Abstract

Nonlinear science is a great revolution of modern natural science. As a result of its rise, the various branches of subjects characterized by nonlinearity have been developed vigorously. In particular, more attention to acquiring the exact solutions of a wide variety of nonlinear equations has been paid by people. In this paper, three methods for solving the exact solutions of the nonlinear -dimensional Jaulent-Miodek equation are introduced in detail. First of all, the exact solutions of this nonlinear equation are obtained by using the exp-expansion method, method, and sine-cosine method. Secondly, the relevant results are verified and simulated by using Maple software. Finally, the advantages and disadvantages of the above three methods listed in the paper are analyzed, and the conclusion was drawn by us. These methods are straightforward and concise in very easier ways.

1. Introduction

Since the 20th century, with the integration of various disciplines, people have gradually discovered nonlinear phenomena which exist in different disciplines but have common characteristics. The resulting nonlinear science is quickly and widely used in medical diagnosis, information technology, electrical engineering, and other fields, which greatly promotes the progress and development of human society [1].

Nonlinear science has laid a solid theoretical foundation for the safety, reliability, and stability of power systems [2–4]. By studying the nonlinear differential equations that appear in the ferromagnetic resonance overvoltage phenomenon, the safe operation of power grids at all levels can be effectively protected [5]. Aimed at the problem of asymmetric nonlinear oscillation in parallel operation of power stations, the harmonic linearization method is put forward, which is helpful to reasonably select the power grid structure and effectively improve the stability of power grid structure [6]. By constructing the nonlinear system model with random disturbance, the high-frequency faults of the power grid caused by disturbance can be greatly reduced [4, 7, 8]. According to the nonlinear characteristics of the actual power system, a reliable and stable nonlinear power grid structure model is constructed, and the exact solution of the equation in the structure model is obtained, which plays a decisive role in the stable operation of the electrified wire netting [9, 10].

Study on the exact solutions of a wide variety of nonlinear equations can not only help people discover new nonlinear phenomena and their laws but also help ensure the reliability and stability of nonlinear calculations. Therefore, more and more scholars have proposed different methods for studying the exact solutions of a wide variety of nonlinear equations. Gardner et al. [11] used the backscattering argument of the Schrödinger equation to propose the inverse scattering transformation method. Wang and Li [12] used the principle of homogeneous balance to derive the nonlinear transformation of a great deal of nonlinear PDE and obtained the exact solution of the equation. Yu and Sun [13] presented a new -dimensional KP-like nonlinear partial differential equation and construct the lump solutions, rationally localized in all directions in the space, to its two-dimensionally reduced cases. The method is based on a generalized bilinear differential equation, so we can build the lump solutions to the presented KP-like equation from special polynomial solutions to the aforementioned generalized bilinear equation which is helpful to improve the research efficiency. Yu and Sun [14] introduced the -dimensional Kadomtsev-Petviashvili-Boussinesq-like equation. Based on this constructed bilinear Bäcklund transformation, some classes of exponential and rational traveling wave solutions with arbitrary wave numbers are presented. It provides a new idea for the follow-up research. Matinfar and Hosseini [15] formally studied the dynamic behavior of the solution obtained in the presence of linear and nonlinear effects. The research process is clear and interesting. Akinyemi et al. [16] studied singular solitons of the -dimensional improved nonlinear Schrödinger equation with spatiotemporal dispersions, group velocity dispersions, and power law nonlinearity by using the Kudryashov method, a complex transformation as well as symbolic computations. The results show that the suggested approach is incredibly accurate and powerful and can be used to explore solitons and other forms of solutions of nonlinear wave models in applied sciences and engineering. In addition, there are many effective methods for solving exact solutions of nonlinear differential equations, for instance, Bäcklund transformation method [17], Painlevé truncation extension method [18, 19], continuation method [20], trial equation method [21], and Hirota bilinear method [22–25].

In 1976, Jaulent and Miodek [26] proposed the coupled Jaulent-Miodek equation when studying the relationship between the nonlinear evolution equation and the energy-dependent Schrödinger potential. In 2007, Feng and Li [27] used the theory of the plane dynamic system to study the existence of solitary waves and periodic waves of the coupled Jaulent-Miodek equation and obtained all possible explicit expressions. In 2009, Wazwaz [28] obtained multiple kink solutions and multiple singular kink solutions by studying the -dimensional nonlinear evolution equation generated by the Jaulent-Miodek hierarchy. In 2013, Cai et al. [29] used the CK direct method to obtain new exact multiple kink and singular kink solutions through the symmetric transformation of the -dimensional Jaulent-Miodek equation. In 2014, Ma et al. [30] used the auxiliary equation method and the Clarkson-Kruskal direct method to study -dimensional nonlinear model systems and obtained some new exact solutions. In 2015, Matinfar et al. [31] studied some -dimensional integrable models generated by the Jaulent-Miodek hierarchy and deduced the quasiperiodic solution of the -dimensional integrable model. In 2016, Li et al. [32] obtained the classical Lie symmetry of the extended -dimensional Jaulent-Miodek equation through the direct symmetry method and obtained a large number of new exact solutions by solving the reduced equation. In 2017, Wei et al. [33] studied the -dimensional Jaulent-Miodek equation using the homogeneous balance method and obtained four new solitary wave solutions of the equation. In 2018, Motsepa et al. [34] derived the conservation law and obtained the traveling wave solutions of the -dimensional Jaulent-Miodek equation. In 2018, Gu et al. [35] used the complex method to seek for the exact solutions of the -dimensional Jaulent-Miodek equation. In 2019, Shehata and Zahran [36] used the Jaulent-Miodek equation to construct the exact traveling wave solutions of parametric equations with different methods. In 2020, Kaewta et al. [37] transformed the -dimensional Jaulent-Miodek equation into a fourth-order partial differential equation and then obtained the exact solution which is simple and reliable.

The -dimensional Jaulent-Miodek equation (see [35, 38, 39]) is considered: In equation (1), is constant ()

Substituting traveling wave transform into equation (1) and integrating it deduce In equation (3), , where , , and are constants and is the integral constant. By setting , equation (3) becomes

For the nonlinear -dimensional Jaulent-Miodek equation, this paper adopts the exp-expansion method, method, and sine-cosine method to obtain the exact traveling wave solutions in turn.

2. Introduction of the -Expansion Method and Main Result

Discuss a nonlinear partial differential equation (PDE) of the following form: In equation (5), for the polynomial containing the nonlinear term, the highest order polynomial, and the unknown function , the following steps can be followed.

Step 1. By substituting traveling wave transform and putting them into equation (5), one can get the new equation, which is an ordinary differential equation (ODE): in equation (6) is both a polynomial of and a polynomial of the derivatives.

Step 2. The traveling wave solutions of equation (6) are shown as indicated below: Equation (7) contains a constant term , and it will be discussed later; at the same time, and satisfy the ODE which will get

Next, different forms of equation (8) are shown under different conditions.

If , then

In the above equations, , , and are undetermined constants, and the constant can take any value. We consider the homogeneous balance between nonlinear terms and highest order derivatives of equation (6), and then, we can get the positive integer .

Step 3. Through simultaneous equations (6) and (7), the polynomial about can be obtained. Calculating all the coefficients of the same power of to zero, we can get a set of simultaneous equations.

Theorem 1. Based on the -expansion method, we can get fourteen kinds of solutions to equation (3).
If , , where , , and , , , and are integral constants.

If , , where , , and , , , and are integral constants.

If , , where , , and and are integral constants.

3. Introduction of the Method and Main Result

The canonical method proposed by [40, 41] includes the following steps.

Step 1. The method takes the form of introducing a new independent variable:

The derivatives of the independent variable change as follows: where other derivatives of this kind can be obtained in a similar way.

Step 2. The premise of using the method is to assume that the solution has the form where is a positive integer and its value needs to be determined here. Then, by substituting equations (31) and (32) into the simplified ODE, we can get equations about the powers.

Step 3. Through balancing the highest order of linear and nonlinear terms, we can get parameter . By equating the power coefficients of in the obtained equations, a set of algebraic equations about , , and can be obtained. After the above parameters are determined, an analytical solution can be obtained by using equation (33) under closed condition.

Theorem 2. By employing the method, we found that there will be two forms of solutions of (3). where and are integral constants.

4. Introduction of the Sine-Cosine Method and Main Result

It is proven that the method of sine-cosine is effective in dealing with compaction problems. Next, the report describes this method briefly below, as the details have been shown in [42].

The sine-cosine method applies to the following form: or another form otherwise, it is zero. The values of , , and parameters were measured. In equation (5), for the polynomial containing the nonlinear term, the highest order polynomial, and the unknown function , the following steps can be followed.

Step 1. By substituting traveling wave transform into equation (5), it is transformed into ordinary differential equation (6). in equation (6) is both a polynomial of and a polynomial of the derivatives.

Step 2. The traveling wave solutions of equation (6) are shown as indicated below: or another form otherwise, it is zero. The values of , , and parameters were measured.

Step 3. For the cosine or sine functions of the same homomorphic power, extract its undetermined coefficients and set them to zero, so we have simultaneous equations. The determination of the solutions of equation (5) can be obtained by obtaining the values of , , and .

Theorem 3. Based on the sine-cosine method, the four forms of all meromorphic solutions of equation (3) are shown as indicated below. (1)The cosine method is used to solve equation (3) as follows: where and are integral constants.(2)The sine method is used to solve equation (3) as follows: where and are integral constants.

5. Proof of Theorems

5.1. Proof of Theorem 1

By homogeneous equilibrium of and in equation (4), we can get

In the above formula, and are undetermined constants. Moreover, satisfies equation (8), where the constants and can be arbitrarily taken.

From equation (42), we insert , , and into equation (4). By calculating all the coefficients of the same power of to zero, the following equations can be obtained:

Simultaneously, the above equations can be obtained:

Substitute the result of the above formula into (42) to get

We apply equations (9)–(15) into equations (45) and (46), respectively; the traveling wave solutions of the -dimensional Jaulent-Miodek equations are shown as indicated below.

If , , where , , and , , , and are integral constants.

If , , where , , and and are integral constants.

5.2. Proof of Theorem 2

The method takes the form of introducing a new independent variable:

The derivatives of the independent variable change as follows:

For the nonlinear -dimensional Jaulent-Miodek equation,

Balancing the highest order of linear and nonlinear terms, we get It is derived that ; therefore,

In the above formula, where , , and appearing in the above are all undetermined constants; substituting (62) and (63) into (64), we get

Substituting (66) and (67) into (68), we get

In equation (69), , , , , and are all constants and is the integral constant. Extract the coefficients before all the powers of , and set each coefficient to zero, so as to obtain the simultaneous equations about the undetermined coefficients , , and :

Solving the above simultaneous equations can get

Substituting , , and into equation (68), we can get solutions for two cases: where and are integral constants.

5.3. Proof of Theorem 3

The sine-cosine method applies to the following form: or another form otherwise, it is zero; the values of , , and parameters were measured.

We substitute into equation (4) and obtain

For the cosine functions of the same homomorphic power, extract its undetermined coefficients and set them to zero, and the following equations can be obtained:

Solving the above simultaneous equations can get

In the end, we get the result where and are integral constants.

We substitute into equation (4) and obtain

For the sine functions of the same homomorphic power, extract its undetermined coefficients and set them to zero, and the following equations can be obtained:

Solving the above simultaneous equations can get

In the end, we get the result: where and are integral constants.

6. Computer Simulations

In this section, we use computer simulation images to explain the results obtained according to three different methods and further analyze the nature of the simple periodic solutions , , and in the -dimensional Jaulent-Miodek equation.

Figure 1 shows the three-dimensional images of by considering the values , , , , , , , , , and ; from Figures 1(a)–1(c), takes the following three different values: , , and . Figures 1(a)–1(c) demonstrate discontinuities of solution on the domain.

(a)

(b)

(c)

Figure 2 shows the three-dimensional images of by considering the values , , , , , , and ; from Figures 2(a)–2(c), takes the following three different values: , , and . Figures 2(a)–2(c) describe the kink wave and parabola solution to the nonlinear evolution equation (3).

(a)

(b)

(c)

Figure 3 shows the three-dimensional images of by considering the values , , , , , , and ; from Figures 3(a)–3(c), takes the following three different values: , , and . Figures 3(a)–3(c) show the singularities on the domain. However, when , the image changes dramatically.

(a)

(b)

(c)

Figure 4 shows the two-dimensional images of by considering the values , , , , , , , , , and ; from Figures 4(a)–4(c), takes the following three different values: , , and .

(a)

(b)

(c)

Figure 5 shows the two-dimensional images of by considering the values , , , , , , and ; from Figures 5(a)–5(c), takes the following three different values: , , and .

(a)

(b)

(c)

Figure 6 shows the two-dimensional images of by considering the values , , , , , , and ; from Figures 6(a)–6(c), takes the following three different values: , , and .

(a)

(b)

(c)

The above two-dimensional images all intuitively and accurately reflect the periodicity of function fluctuations.

7. Discussions and Conclusions

By the exp-expansion method, we have obtained fourteen forms of solutions to the Jaulent-Miodek equation. Utilizing the method, we obtained four forms of solutions of the Jaulent-Miodek equation. In the end, utilizing the sine-cosine method, we obtained two forms of solutions to the Jaulent-Miodek equation. Combining the existing results found in this paper, more results are easily obtained by the exp-expansion method.

In [35], the authors adopt this complex method to seek for analytic solution of equation (3); the elliptic function solution is obtained by the complex method. From our paper and [35], it is easy to see that by utilizing the exp-expansion method and complex method, both can obtain the hyperbolic function solutions for the Jaulent-Miodek equation. Unfortunately, utilizing the sine-cosine approach, we find only the trigonometric function solutions for the Jaulent-Miodek equation.

In [37], the authors used both methods, namely, the -expansion method and the Jacobi elliptic equation method to calculate and analyze the explicit exact traveling wave solutions of the -dimensional Jaulent-Miodek equation as given by equation (1). By applying the expansion method to the equation, the solutions were obtained by the author including the triangular hyperbolic function solutions and rational function solutions. Implemented by the Jacobi elliptic equation method (analytical equation (1)), three sets of parameter values were reached. Every set has produced fifteen exact terms for the solution of Jacobi elliptic function solutions. By comparing the methods to [37], the methods in our article seem more concise and straightforward.

All three methods in this paper have their own characteristics. We trust that the above method presented here and the methods in [35, 37] together can be applied more effectively to find out solutions for nonlinear evolution equations currently and henceforth. The above-mentioned methods are very helpful tools for unearthing exact solutions to the nonlinear evolution equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors typed, read, and approved the final manuscript.

Acknowledgments

The work presented in this paper is supported by the Plateau Disciplines in Shanghai. Also, this work was supported by the Leading Academic Applied Mathematical of Shanghai Dianji University (16JCXK02) and Humanity and Social Science Youth Foundation of Ministry of Education (18YJC630120). The work also presented in this paper is supported by the Shanghai Multi-Direction Dieforging Engineering Technology Research Center (No. 20DZ2253200) and supported by the Science and Technology Support Project in the Biomedical Field of “Science and Technology Innovation Action Plan” of Shanghai in 2019 (No. 19441914900).

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Copyright © 2021 Jingsen Hu and Jianming Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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