On the Exact Solutions of Nonlinear Potential Yu–Toda–Sasa–Fukuyama Equation by Different Methods

Zhu, Qinghao; Qi, Jianming

doi:https://doi.org/10.1155/2022/2179375

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2022 | Article ID 2179375 | https://doi.org/10.1155/2022/2179375

On the Exact Solutions of Nonlinear Potential Yu–Toda–Sasa–Fukuyama Equation by Different Methods

Qinghao Zhu¹and Jianming Qi¹

Academic Editor: Rodica Luca

Received18 Oct 2021

Accepted22 Jan 2022

Published21 Feb 2022

Abstract

In this article, the exact solutions to the potential Yu–Toda–Sasa–Fukuyama equation are successfully examined by the extended complex method and -expansion method. Consequently, we find solutions for three models of Weierstrass elliptic functions, simply periodic functions, and rational function solutions. The obtained results will play an important role in understanding and studying potential Yu–Toda–Sasa–Fukuyama equation. It is observed that the extended complex method and -expansion method are reliable and will be used extensively to seek for exact solutions of any other nonlinear partial differential equations (NPDEs).

1. Introduction

As is known to all, the development of modern natural science is not only linear. Nonlinear science is considered as the most essential forefront of the fundamental understanding of nature. Even nonlinear systems look like counterintuitive, chaotic, or unbelievable, but it is more in line with reality, among which the behavior analysis of energy dynamic system solutions is one of the hot spots in recent research. Especially in biologists science, physicists science, information technology, space science and so on [1], the convenience and stability of society are greatly enhanced.

In recent years, the application of nonlinear science in power systems has become more and more extensive. It provides a large number of reliable references for the operation and planning of the power system. The application of nonlinear equations promoted the development of nonlinear sensitive electronic devices on the load side and grid side of the power system. The stable operation of power system at each level can be effectually protected by exploring the nonlinear phenomena in the case of ferromagnetic resonance overvoltage situation. The harmonic linearization method is proposed to solve the asymmetric nonlinear oscillation problem in the parallel operation of power stations, which will help to select the grid structure reasonably and effectively improve the stability of the grid structure [2]. The high-frequency disturbances that may be encountered can be predicted by constructing a nonlinear prediction model with random disturbances [3]. Nonlinear equations can not only simulate the actual operating conditions of the power system but also simulate possible hazards and play an extremely important role in the stable operation of the power grid [4].

The study and development of nonlinear dynamic systems have been conducted for more than a century. Since the 1970s, the bifurcation theory and chaotic phenomenon of nonlinear dynamics have become a hot topic of nonlinear differential equations. Nonlinear system of equations is used to describe and model the complex nonlinear systems. Recently, in order to obtain various solitary and periodic solutions of nonlinear equations, scientists have come up with many innovative and effective methods.

Much attention has been paid to the exact solutions of nonlinear partial differential equations. Mathematicians and scientists have continuously proposed and improved methods for solving nonlinear differential equations [5, 6]. Julier et al. [7] used the new mean and covariance parameterization, which can be directly transformed by the system equation to predict the transformed mean and covariance. In addition, there are many effective and magical methods to obtain exact solutions of nonlinear differential equations, such as Bäcklund transformation method [8], continuation method [9], Painlevé truncation extension method [10, 11], Hirota bilinear method [12, 13], Exp-Expansion method [14, 15], and so on.

The dimensional potential Yu–Toda–Sasa–Fukuyama equation is a nonlinear partial differential equation which is often used to solve problems in fluid mechanics. In 2014, Hu et al. [16] used three-wave method to obtain some new kink multisoliton solutions about potential Yu–Toda–Sasa–Fukuyama equation. In 2017, for further study of potential Yu–Toda–Sasa–Fukuyama equation, Roshid [17] constructed the exact solution of the equation by using the lump solution method. Zhang and Zong [18] obtained the exact solution by using the F-expansion method. In 2018, Foroutan et al. [19] studied the dimensional potential-YTSF equation by implementing the Hirota bilinear method. In 2019, Zhao and He [20] employed the bilinear method to study the dimensional potential-YTSF equation in fluid dynamics. In this article, we used the extended complex method [21–24] and -expansion method to find the exact solutions of modified dimensional potential-YTSF equation.

Modified dimensional potential-YTSF equation (see [25]) is considered as

Taking the transformation , where , and are arbitrary constants, equation (1) turns into

In (2), is the integration constant. By puttinginto (2) and integrating it, we get

In this paper, the complex method and expansion method are employed by us to seek for the exact solution of (4). These two methods will play a significant role in constructing exact solutions for the nonlinear partial differential equations via the dimensional potential-YTSF equation.

2. Introduction of the Extended Complex Method and Main Result

Because the extended complex method involves the Weierstrass elliptic function, first we give the brief introduction of Weierstrass elliptic function: is a meromorphic function with double periods [26–28] and is defined aswhich satisfies the following equation:where , and , and has the another formula

Next, we will employ the recently established extended complex method [21–24] to seek for traveling wave solutions to the equations mentioned above. Step 1. Insert the traveling wave transform T: into a given PDE and alternate it to the following integer-order ordinary differential equation (ODE): where is a polynomial of and its derivatives. Step 2. Next, we will find out weak condition. To find out the weak condition of equation (8), we can substitute Laurent series into equation (8), and then we can find out the distinct Laurent singular parts as follows: where indicates that the equation has distinct meromorphic solutions and indicates that their multiplicity of the pole at is . Step 3. Take the indeterminate forms into equation (8), respectively, and produce a set of algebraic equations; we get Weierstrass elliptic function solutions, simply periodic solutions, and rational function solutions with a pole at , in which are given by equation (9), , , and equations (12) and (13) have distinct poles of multiplicity . Step 4. On the basis of Step 3, we can easily obtain meromorphic solutions at arbitrary pole. Putting the inverse transform into the solutions , we can easily get all exact solutions in the original FPDE.

Theorem 1. If we employ the extended complex method and suppose , then there are three types solutions of equation (4) listed below:(a)All elliptic solutions are as follows: where . Noting and , where , , here and are arbitrary constants, and is an integral constant.(b)The simply periodic solutions are obtained by where , , and . So, Here is an integral constant.(c)The rational function solutions: where . So, Here is an integral constant.

3. Introduction of -Expansion Method and Main Result

Next, we employ the recently established -expansion method [29, 30] to seek for traveling wave solutions to the equations mentioned above.

Consider the following form of a nonlinear partial differential equation (PDE):where is a polynomial with an unknown function and its derivatives in which nonlinear terms and highest order derivatives are involved. Next the detail steps can be processed as follows: Step 1. Inserting the traveling wave transform into equation (20), the following ordinary differential equation (ODE) is obtained: where is a polynomial of and its derivatives. Step 2. Regarding that, the solution of equation (21) can be expressed by a polynomial in in the following form: where satisfies the second-order linear ordinary differential equation (LODE) as follows: These unknown constants , , are about to be determined. The omitted parts in (22) are also a polynomial in , the degree of which is equal to or less than . The positive integer can be determined by considering the uniform equilibrium between the highest derivative and the nonlinear term in ODE (21). Step 3. Substituting equation (22) into equation (21) and using (23), the left-hand side of equation (21) is converted into another polynomial in . Calculating all the coefficients to zero of the polynomial yields a set of algebraic equations for , , . Step 4. The constants , , can be solved by the system of algebraic equations obtained in Step 3. Because the general solutions of second-order equation (23) are well known, depending on the sign of the discriminant , the exact solutions of the given equation (20) are thus obtained. Here where and are arbitrary constants.

Theorem 2. If we employ the -expansion method, then we obtain the following traveling wave solutions of equation (4): Case 1. If , Here and . and are arbitrary constants. Now, we give three forms of :(1)If , then Because of the rational hyperbolic functions forms, we cannot accumulate the exact expressions of .(2)If , then . Hence,(3)If , then . Hence, where are integral numbers. Case 2. If , Here and . and are arbitrary constants. Almost the same as case 1, it is just a little different form of . If and , forms of are almost the same as case 1. If , then . Hence, where is an integral number. Case 3. If , and . and are arbitrary constants. where is an integral number.

4. Proof of Theorems

4.1. Proof of Theorem 1

Putting (9) into (4), we have , , , , , , , , , , , , where is arbitrary. Hence, (4) satisfies weak condition.

By (12), we have the indeterminant rational style solutions of (4) with pole at that

Substituting into (4), we obtain the following result:where . So,

The rational solutions of (4) are as follows:where . So,

Next we will seek for simply periodic solutions, setting and putting into (4); then,and puttinginto (44), we get thatwhere . Substituting into above relation, we obtain simple periodic solutions (4), and the pole point is located at .

So,

So, all simply periodic solutions of (4) are obtained bywhere , .

So,where are constants.

From (11), we have indeterminant relations of elliptic solutions of (4) and the pole point is located at .

Substituting into the (4), we obtain that

Here . So,where is also an elliptic function.

Thus, all the elliptic solutions for (4) are as follows:

So,where . By using (7), we rewrite it to the formwhere . Then, from (55) and (7), we deduce thatwhere are integral numbers, and here , , where and are arbitrary constants..

The proof is over.

4.2. Proof of Theorem 2

Considering the homogeneous equilibrium between and in (4), we get . So, we can set the solution of (4) as follows:

here are constants to be determined later.

Next, we will use (58) and (23) together and collect all terms of the same power as .

First, from (58), we get

Substituting (23) into (59), we obtainand then we deduce

Now, we put (58), (60), and (61) into (4) and collect all terms with the same power of together:

For the function with the same homomorphism power, its undetermined coefficients are extracted and set to zero to obtain the following equations:

Solving the above simultaneous equations, we get

By using (64), (58) is written as

Replacing the general solution for (23) and (65) with (4), we obtain the following exact solution for (4): Case 1. If , where and . and are arbitrary constants. Now, we give three forms of .(1)If , then . Because of the rational hyperbolic functions forms, we cannot accumulate the exact expressions of .(2)If , then . Hence,(3)If , then . Hence, where are integral numbers. Case 2. If , where and . and are arbitrary constants. Almost the same as case 1, it is just a little different form of . If and , forms of are the same as case 1. If , then . Hence, where is an integral number. Case 3. If , and . and are arbitrary constants. where is an integral number.

5. Comparison

In this paper, about equation (4), we consider the case where integral constant is not zero. This point is different from reference [31]. Obviously, we consider a much more complex scenario. By the -expansion method and the extended complex method, we found various forms for solutions of potential Yu–Toda–Sasa–Fukuyama equation. Using the extended complex method, we found elliptic function solutions, hyperbolic function solutions, and rational solutions. We also can find more solutions by the -expansion method including hyperbolic function solutions and rational solutions. However, only by applying extended complex methods, we can obtain elliptic function solutions. If the appropriate parameters are selected, both methods obtain the same rational solution. From the perspective of the calculation process, the extended complex method is more concise and clear. The -expansion method is relatively complex, but more formal solutions can be obtained. Both methods have their own characteristics.

We firmly believe that the extended complex and -expansion methods proposed in this paper can be more efficiently applied to find solutions to other nonlinear evolution equations. Both methods are effective tools to obtain exact solutions of the nonlinear evolution equations.

6. Computer Simulations

In this section, we attempt to interpret the results obtained by two different methods by computer simulated images and further analyze the properties of , , and in the dimensional potential-YTSF equation.

Figure 1 shows the 3D images of by considering the values , , , , , , , and ; takes three different values: , , and . Three graphs demonstrate poles of solution on the domain.

(a)

(b)

(c)

Figure 2 shows the 3D images of by considering the values , , , , , , and ; takes the following three different values: , , and . However, the images change dramatically accompanied with different .

(a)

(b)

(c)

Figure 3 shows the 3D images of by considering the values , , , , , , and ; takes the following three different values: , , and . Three graphs show the singularities on the domain.

(a)

(b)

(c)

Presented profiles of the above 3D graphs will help to understand wave function fluctuations dynamics to a greater extent.

Figure 4 shows the 2D images of by taking the values , , , , , , , , and ; takes the following three different values: , , and .

(a)

(b)

(c)

Figure 5 shows the 2D images of by taking the values , , , , , , ; takes the following three different values: , , and .

(a)

(b)

(c)

Figure 6 shows the 2D images of by taking the values , , , , , , , and ; takes the following three different values: , , and .

(a)

(b)

(c)

The above 2D images all intuitively and clearly reflect the high frequencies and amplitude of function fluctuations.

7. Conclusions and Future Outlook of the Study

By traveling wave transformation, many of the PDEs can be converted into nonlinear differential equations similar to equation (4). In this paper, we employed the extended complex method and the -expansion method to study the exact solutions about the potential Yu–Toda–Sasa–Fukuyama equation. By traveling wave transformation, we can reduce the dimensions of the PDEs to nonlinear evolution differential equations related to mathematical physics and engineering.

The results in the full text strongly demonstrate that the above methods are very effective and powerful in solving the exact solutions of the nonlinear evolution equations. We can apply these ideas and methods to other nonlinear evolution equations. Our work shows that there is a simple periodic solution, , which does not continuously degenerate successively to the elliptic function solutions.

In the future, we may consider the unified -expansion method [32] and generalized direct algebraic method [33] to further investigate the potential Yu–Toda–Sasa–Fukuyama equation. In 2021, Behera et al. [34] used the modified -expansion method for investigating the exact solutions of the traveling waves about Drinfeld–Sokolov–Wilson (DSW) equation and Benjamin–Ono (BO) equation. This is also a new research direction that we will focus on and have great interest in the future.

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

This study was supported by Leading Academic Applied Mathematical of Shanghai Dianji University (16JCXK02).

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Copyright © 2022 Qinghao Zhu 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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