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Stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation

The low-pass nonlinear electrical transmission lines and the Cahn-Allen equation are important nonlinear model equations to figure out different tangible systems, namely, electrical engineering, fluid dynamics etc. The contrivance of this study is to introduce advanced Bernoulli sub-equation function method to search for stable and effective solitary solutions of the described wave equations. Stable solitary solutions are reported as an integration of exponential functions, hyperbolic functions, etc., and the graphical implications for specific values of the corresponding parameters are explained in the solutions obtained in order to uncover the inmost structure of the tangible phenomena. It is establish that the IBSEF method is reliable, contented and might be used in further works to found ample novel soliton solutions for other types of NLEEs arising in physical science and engineering.


Introduction
Research on NLEEs and related topics increasing day by day and is become a subject of extensive interest in several spheres of physical and mathematical sciences, explicitly, in plasma physics, signal processing, control theory, fluid mechanics, mechanical engineering, etc. Closed-form analytical solutions convey a significant contribution in describing these phenomena simply, more accurately, and explicitly. Even a trivial closed-form solution that does not have a realistic physical implication can be used as a test problem to confirm the reliability and to assess errors of various numerical, approximate analytical and asymptotic, approaches. The exact solutions can also serve as the basis for the excellence and precision of computer algebra software in solving NLEEs. Therefore, a number of important methods for the explicit and detailed stable soliton solutions of nonlinear physical models have currently been developed with the aid of Matlab, Mathematica, etc., such as, the differential transformation [1] method, the Hirota's bilinear method [2,3,4,5], the approach of modified simple equation [6,7], the F-expansion method [8], the Exp-function [9] method, the modified exponential-function method [10], the ( ′ ∕ )-expansion [11,12,13] scheme, the improved ( ′ ∕ )-expansion method [14], the rational ( ′ ∕ )-method [15], the extended trial equation method [16], the improved tan ( ( )∕2)-expansion method [17], the first integral soliton solutions linked with arbitrary free parameters to the earlier stated equations over the IBSEF method banding together of exponential function and hyperbolic function. The general solution established some typical solutions and some advanced solutions for the distinct values of the constraints. It is noteworthy that some distinguished solutions explicitly, the bell-shape soliton, kink, singular kink, spike type waves have developed making use the wide-spectrum general solution. The acquired solutions might play an vital role in analyzing transmission lines in electrical engineering, dynamical processes of non-chemical nature, like neutron diffusion theory, geology, biology and ecology.

Analysis of the IBSEF method
The improved Bernoulli sub-equation function (IBSEF) method was initiated from the generalized Bernoulli sub-equation function method, the steps of which are laid out in the below: Step I. Let us presume the following NLEE where 1 , 2 , 3 are the wave numbers and is the wave speed. The wave variable (2.2) translates the NLEE into an ordinary nonlinear differential equation as follows: Step II. The solution is presented as follows, pursuing the IBSEF method: Let us suppose the general form of improve Bernoulli differential equation in accordance with the Bernoulli theory as underneath: where ( ) is the solution of (2.5).
Inserting the improved Bernoulli equation and the solution (2.4) into the equation (2.3) yields a polynomial equation Ω ( ) of as follows: By using the homogeneous principle on equation (2.3) and associate with equation (2.5), this procedure provides the values of the free constants , and .
Step III. We attain a system of equation through equating the coefficients of Ω(H) to zero; and determine the ultimate values of 0 , … , and 0 , 1 , … , by solving the system.
Step IV. The solutions of equation (2.5) associate with the relation to the parameters and . Depending on the relationship between the parameters and couple of solutions are found as follows: We secure the exact solutions to equation (2.3) by considering entire system for polynomial of ( ) along with the value arranged in (2.7). This finishes the establishment of the solutions. For the better explanation of the results acquired in this technique, we depict threedimensional figures of the analytical solutions by indicating suitable values of unknown parameters.

Construction of solutions
In this section, we construct some fresh and broad-ranging closed form stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation taking supremacy of the IBSEF method.

The low-pass nonlinear electrical transmission lines
In this module, we have extracted the general and detailed closed form stable wave solutions to the low-pass nonlinear electrical transmission lines [53]: where ( , ) is the voltage and , and are free constants in the transmission line. Also, and illustrated as the transmission distance and slow time respectively. In order to examine the equation (3.1.1), we relate the following wave transformation: where is the wave velocity in the transmission line. Substituting (3.1.2) into (3.1.1) and after twice integrating, we reach to the subsequent NODE as follows: For the highest order linear term ( ) , the ultimate exponent of is + − − 1 and for the highest degree nonlinear term , the ultimate exponent of is ( − ) + 1. Since, we are searching for soliton solutions; therefore, the exponent of linear term ( ) and the exponent of nonlinear term are identical. Therefore, balancing these two exponents, we establish + − − 1 = ( − ) + 1. In equation (3.1.3), the linear term of highest order is ′′ (i.e. = 2) and the highest degree nonlinear term is 3 (i.e. = 3). Thus, from the above implication, for positive integers , and , we found the following relationship: Here and are free parameters, therefore choosing = 1 and = 3, gives = 3. Therefore, the trial solution to equation (3.1.3) gives the general solution: where will be estimated from the improved Bernoulli equation. Setting the solution (3.1.4), alongside with (2.5) into (3.1.3), yields a polynomial in . We accomplish the value of the constants by unraveling the system of the algebraic equations.
Case 1(a): When ≠ Substituting the values of the parameters assembled in (3.1.5) along with (2.8) into solution (3.1.4), we attain the exponential function solution to the low-pass nonlinear electrical transmission line given in the subsequent: ) .
After simplified, solution (3.1.8) can be marked out in the form Since ≠ 0, ≠ 0 are the Bernoulli parameters and is a real constant, one may openly choose their values. Thus, if we choose = 2 , = 1, = 1, then the solution (3.1.9) becomes On the other hand, if we choose = 1, = 1 and = 5, from solution (3.1.9) we establish the underlying solution as a combination of sinh and cosh functions In a new way, if we assign = , = 4 and = 7, from solution (3.1.9) we ascertain the following solution as a linear combination of sinh and cosh functions We establish some known solutions nearby in the literature by considering = − into solution (3.1.9), we achieve the kink shape soliton solution given in the following: Alternatively, setting = , solution (3.1.9) generates the singular kink soliton laid out in the next: .
In a similar manner, for = 1 and using = d Based on the different values of and arrange many different closed form stable wave solutions to equation (3.1.1) but for succinctness and to avoid the repetition, the other solutions have not been noted here. Therefore, it is seen that from a typical and broad-ranging solution formula, it is possible to determine numerous solutions by setting different values of parameters, some of which are accessible in the literature and some are new. This feature has made the method sublime. We observe that, analogous solutions can be found for the values of parameters accumulated in (3.1.6) and (3.1.7) and thus these solutions have not been described.

The Cahn-Allen equation
The Cahn-Allen equation is of the form [55]: Using the wave variable

2.2)
wherein is the speed of the traveling wave which is to be calculated later, the Cahn-Allen equation (3.

2.1) is transformed into the following ODE
Since, in section 3.1, the relation among , , and is discussed in detail, in this section, it is not discussed elaborately again. In equation (3.2.3), the highest order derivative term is ′′ and the nonlinear term of the maximum order is 3 . Thus, we obtain a relationship for , and by balancing the maximal order linear and nonlinear terms ascend simultaneously in equation

Case 1(a): When ≠
Inserting the values of the parameters arranged in (3.2.5) together with (2.8) into solution (3.2.4) gives the exponential function solution as follows: After simplifying, from solution (3.2.9), we derive ) .

Case 1(b): When =
Replacing the values of the parameters settled in (3.2.5) with the help of (2.9) into solution (3.2.3) gives ) .
Since is a real number, we may choose the value of = −3 and = 1, then we get from solution (3.2.15) as below . ( .
we ascertain . (3.2.18) We will attain more solutions comparable to the solutions (3.2.5) for the values of the parameters noted in (3.2.6), (3.2.7) and (3.2.8). In order to obviate repetition, these solutions have been passed over. It is noteworthy that the general solutions of the Cohen-Allen equation here are effective, efficient and have not been established in previous studies. The solutions obtained may be feasible to examine the propagation of one-way waves in nonlinear media with the relative one-particle theory of dispersion.

Results and discussion and graphical representations
The attained stable traveling wave solutions are presented in the diagrams and speculated the nature of illustrated waves for different values of the constraints through the medium of software Mathematica.

The low-pass nonlinear electrical transmission lines
In this part, we have portrayed the graphical representations of the derived results of the nonlinear low-pass electrical transmission lines for miscellaneous values of the parameters. The graphs of the solution are shown as below.
Considering negative sign of solution (3.1.14) embodies the singular kink shape soliton presented in Fig. 2a     It is noticed that in this study, we establish different type of soliton, such as, kink, singular kink, compressed kink shape, some special shape waves to the low-pass nonlinear electrical transmission lines for diverse values of the constraints.

Figures of the solutions of the Cahn-Allen equation
In this sub-section, for the assorted values of the free constants of the ascertained solutions to the Cahn-Allen equation depict the graphical presentations as below.
The solution (3.2.13) represents singular kink soliton and specified in Fig. 4a and we accomplish kink shape wave for solution (3.2.14) and indicated in Fig. 4b, plotted within the interval −8 ≤ , ≤ 8.
From the above analysis, it is observed that we ascertain kink, singular kink, singular spike like soliton and other type solitons. Although similar figures are available for all the other solutions, for the simplicity we have not appended here.

Conclusions
In this study, the IBSEF method has effectively been contrivance to establish the stable solitary wave solutions to the nonlinear low-pass   electrical transmission lines and the Cahn-Allen equation. The novelty of this article is that we have derived some further general wave solutions as the linear combination of exponential and hyperbolic functions or disjointedly which are not conveyed in the prior literature and the fresh results are associated with several free parameters. The significances of the attained solutions associate with free parameters might be essential to interpret the complicated physical phenomena. This analysis shows that the proposed scheme is effective and can be applied for other NLEEs in mathematical physics and engineering. This method can be extended to examine other kind of problems with the aid of computer algebra, like Mathematica and this is our future mission.

Author contribution statement
Md. Ekramul Islam: Conceived and designed the analysis; Analyzed and interpreted the data; Wrote the paper. H. K. Barman, M. Ali Akbar: Analyzed and interpreted the data; Wrote the paper.

Funding statement
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement
No data was used for the research described in the article.

Declaration of interests statement
The authors declare no conflict of interest.

Additional information
No additional information is available for this paper.