Abstract
In this work, the ()-dimensional nonlinear electrical transmission line equation (NETLE) is investigated by applying three recent technologies, namely, the variational approach, Hamiltonian approach, and energy balance approach. Diverse exact soliton solutions such as the bright, bright-like, kinky bright, bright-dark soliton, and periodic soliton solutions are successfully constructed. The outlines of the different solutions are shown in the form of the 3-D plot with the help of the Wolfram Mathematica. It reveals that the used methods are concise and effective and are expected to provide some inspiration for the study of travelling wave solutions of the PDEs in physics.
1. Introduction
Nonlinear partial differential equations (NLPDEs) appear in mathematics, physics, engineering, and other fields. Many complex phenomena occurring in nature can be described by the NLPDEs. The study of their soliton solutions is of great significance since they can make us more deeply understand the natural phenomena and their internal relations. So far, there are many effective methods available for constructing the soliton solutions such as the exp-function method [1–4], tanh-function method [5–9], (/)-expansion method [10–12], -expansion method [13, 14], extended rational sine-cosine and sinh-cosh methods [15–18], Sardar-subequation method [19–21], and Sine-Gordon expansion method [22, 23] [24–31]. In the current work, we aim to study the ()-dimensional NETLE, which is expressed by [32]
In Eq. (1), , , and are nonzero constants, and represent the longitudinal and transverse distance between two adjacent sections, respectively. Eq. (1) plays an important role in the field of telecommunication and network engineering. Up to now, some effective approaches have been adopted to solve Eq. (1) such as the modified simple equation method [33], Jacobi elliptic function expansion method [34], sine-Gordon expansion method [35], and the Kudryashov method [36]. In recent years, the variational theory-based methods such as the variational approach (VA), Hamiltonian approach (HA), and energy balance approach (EBA) have caught a wide attention for solving the PDEs since they all probe the problem in view of the energy conservation and obtain the solutions by the stationary conditions. Additionally, these methods can help us insight the problem from a physical perspective. Thus, in this work, we aim to seek for the various soliton solutions by means of the variational theory-based methods, which are the VA, HA, and EBA. The rest content of this paper is arranged as follows. In Section 2, the variational principle and Hamiltonian of the studied problem are presented. In Section 3, the various soliton solutions are derived by applying the three methods. The behaviors of the different solutions are presented through the 3-D plot in Section 4. Finally, we get a conclusion in Section 5.
2. Variational Principle and the Hamiltonian
For solving Eq. (1), we introduce the following transformation [36]:
In Eq. (2), and represent the wave number and wave speed, respectively. Applying Eq. (2) to Eq. (1), integrating the results twice with respect to and setting the integrating constants to zero, we get where , , and .
With the aid of the semi-inverse method [37–43], we establish the variational principle of Eq. (3) as
For the convenience of calculation, we reexpress Eq. (4) as where , , and . And there are
Here, is the kinetic energy, and indicates the potential energy. Then, we can get the Hamiltonian as [44, 45]
3. The Solutions
3.1. The VA
The main target of this subsection is to apply the VA to search for the abundant exact solutions of Eq. (1).
3.1.1. The Bright Soliton Solution
Here, we suppose the solution of Eq. (3) is [46]
Taking it into Eq. (5) yields
According to the Ritz-like method (RLM), we take its stationary condition as which leads to
On solving it by the Wolfram Mathematica, we have
Or
So, the solution of Eq. (3) can be obtained as or
In the view of Eq. (2), we have or
3.1.2. The Bright-Like Soliton Solution
The solution of Eq. (3) is assumed as the following form [47]:
Putting above equation into Eq. (5), it gives
Based on the RLM, there is
It results into
Solving above equation, we have or
So, there are or
3.1.3. The Kinky-Bright Soliton
Inspired by the research results obtained in [47], we can also assume that Eq. (3) has the following solution to construct the kinky-bright soliton:
By the same way, we have
Computing its stationary condition with respect to yields
from which, we obtain or
With this, we have or
3.1.4. The Bright-Dark Soliton
Here, we search for a solution of Eq. (3) in the following form [47]:
Substituting Eq.( 33) into Eq. (5), it produces
Making stationary with results in
From Eq. (35), we have or
So, we have or
3.2. The HA
In this section, we aim to use the HA to search for the periodic solution of Eq. (3) as [48–50] where is the amplitude and is the frequency.
Form the obtained Hamiltonian provided by Eq. (7), we construct a new function as
Taking Eq. (40) into above equation, it yields
By the Hamiltonian approach, we set which leads to
So, the periodic soliton solution of Eq. (1) is found as
3.3. The EBA
To use the EBA [50, 51], we first assume the solution of Eq.(3) is
And we have
The energy conservation reveals that the Hamiltonian invariant keep unchanged for the system; so, inserting Eq. (47) into Eq. (7) yields
Then, we substitute Eq. (46) into Eq. (48) and set , and there is
Solving it gives which has a well agreement with Eq. (44). It strongly proves the correctness of the two different methods. Thus, we get the periodic soliton solution of Eq. (1) as
4. Results and Discussion
This section will give the graphical representations and physical interpretation of the obtained solutions in Section 3 by using proper parameters.
For , , , ,, , , and , Figure 1 plots the performance of the bright solitons obtained by Eqs. (16) and (17) within the interval and . Obviously, they have the bright soliton characteristics. .
(a)
(b)
We plot the behaviors of the bright-like soliton solutions given by Eqs. (24) and (25) in Figure 2 by choosing , , , , , ,, and , where the performance are like bright soliton.
(a)
(b)
Solutions of Eqs. (31) and (32) are the kinky-bright solitons. We plot their behaviors in Figure 3 within the interval on and with for , , , , , , , and . As expected, they all have the bright soliton characteristics.
(a)
(b)
Selecting , , , , , , , and , the 3-D plots of the bright-dark solitons given by Eqs. (38) and (39) are presented in Figure 4 within the interval and . It can be observed that the performances are bright-dark soliton.
(a)
(b)
Figure 5 plots the profile of Eqs. (45) and (51) within the interval and by using the parameters as , , , , , , (), , and . It can be seen that the two profiles are all perfect periodic waves. In addition, the two contours are basically the same.
(a)
(b)
5. Conclusion
In this work, diverse soliton solutions of ()-dimensional nonlinear electrical transmission line equation like the bright soliton, bright-like soliton, kinky-bright soliton, bright-dark soliton, and periodic soliton solutions were constructed by using the variational approach, Hamiltonian approach, and energy balance approach. The profiles of the solutions were presented through the 3-D plots via selecting the appropriate parameters by means of the Wolfram Mathematica. The results revealed that the proposed methods were straightforward, simple, and effective, which can be adopted to study the traveling wave theory of the PDEs arising in physics.
Data Availability
The data generated and/or analyzed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.
Conflicts of Interest
This work does not have any conflicts of interest.
Acknowledgments
This work is supported by the Zhuhai Innovation Driven Project (No. ZH0108-1404-170077-P-ZY), the Key Programs of Universities in Henan Province of China (22A140006), the Fundamental Research Funds for the Universities of Henan Province (NSFRF210324), and Program of Henan Polytechnic University (B2018-40).