Construction of traveling and solitary wave solutions for wave propagation in nonlinear low-pass electrical transmission lines

Aly R. Seadawy; Mujahid Iqbal; Dumitru Baleanu

doi:10.1016/j.jksus.2020.06.011

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Original article

32 (

); 2752-2761

doi:

10.1016/j.jksus.2020.06.011

Construction of traveling and solitary wave solutions for wave propagation in nonlinear low-pass electrical transmission lines

Aly R. Seadawy^{^a,⁎}, Mujahid Iqbal^{^b}, Dumitru Baleanu^{^c,^d,^e}

aMathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

bFaculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China

cDepartment of Mathematics, Cankaya University, Ankara, Turkey

dDepartment of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

eInstitute of Space Sciences, 077125 Magurele, Romania

⁎Corresponding author. Aly742001@yahoo.com (Aly R. Seadawy)

Received: 2019-11-2, Accepted: 2020-6-21, Published: 2020-09-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

In this study, our aim to constructed the traveling and solitary wave solutions for nonlinear evolution equation describe the wave propagation in nonlinear low-pass electrical transmission lines by implemented the modification of mathematical method. We obtained the new and more general solutions in rational, trigonometric, hyperbolic type which represent to kink and anti-kink wave solitons, bright-dark solitons and traveling waves. The physical interpretation of some results demonstrated by graphically with symbolic computation. We are hopefully determined results have numerous applications in optical fiber, geophysics, fluid dynamics, laser optics, engineering, and many other various kinds of applied sciences. The complete investigation prove that proposed technique is more reliable, efficient, straightforward, and powerful to investigate various kinds of nonlinear evolution equations involves in geophysics, fluid dynamics, nonlinear plasma, chemistry, biology, and field of engineering.

Keywords

Nonlinear low-pass electrical transmission lines

Modified Mathematical technique

Solitary wave solutions

Traveling wave solutions

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1 Introduction

The investigations of nonlinear electrical transmission lines and its solitary wave solutions have marvelous applications in communication systems and electronic engineering such as distributing signal of cable television, computer network connection systems, data buses of high speed computer, mobile network systems, connection between transmitters and receivers of radio with their antennas, routing calls in trunk lines between telephone switching centers (Abdoulkary et al., 2013; Arshad et al., 2018; Seadawy, 2017a; Kumar et al., 2018; Aslan, 2016 ). Moreover, nonlinear electrical transmission lines are used as special medium to carrying the alternate current (AC) of frequency of radio (Abdoulkary et al., 2013). The NLETLs are also used to deliver the better way how to observe the nonlinear excitation be have under nonlinear media and modulate the features of exotic of new systems (Jaradat et al., 2018a,b,c,d; Alquran and Jaradat, 2018; Ali et al., 2019; Alquran et al., 2020; Pelap and Faye, 2005 ).

For demonstrated of solitary wave solutions for NLEEs describe the wave propagation in nonlinear low-pass electrical transmission lines (Abdoulkary et al., 2013; Arshad et al., 2018; Seadawy, 2017a; Kumar et al., 2018; Aslan, 2016 ), we consider as:

(1)

\frac{δ^{2} V (x, t)}{δ t^{2}} + α \frac{δ^{2} V^{2} (x, t)}{δ t^{2}} - γ \frac{δ^{2} V^{3} (x, t)}{δ t^{2}} + δ^{2} \frac{δ^{2} V^{2} (x, t)}{δ x^{2}} + \frac{δ^{4}}{12} \frac{δ^{4} V^{2} (x, t)}{δ x^{4}} = 0 .

In Eq. (1), $α, γ, δ$ are the real constants and $V (x, t)$ represents to transmission line voltage. The variables $x, t$ represents to distance for propagation and slow time, respectively. The Eq. (1), is derived by applying the Kirchoff’s laws are seen in Abdoulkary et al. (2013), in this current work, which are omitted due to shortness.

In previous years a huge research work have been done on nonlinear low-pass electrical transmission lines. Many researchers have been determined the various types of solutions for NLETLs by using the different techniques including jacobi elliptic technique, kudryashov technique, auxiliary equation technique, $(\frac{G^{'}}{G})$ -expansion method, extension technique of tanh function, the technique of racatti equation, modification of kudryashov technique, the technique of sine-Gorden and extension of sine-Gorden equation (Abdoulkary et al., 2013; Arshad et al., 2018; Seadawy, 2017a; Kumar et al., 2018; Aslan, 2016 ). The NLETLs are play important role in the investigation of propagation phenomena of electrical solitons, they are travel in nonlinear media of dispersion in the shape of voltage waves.

The nonlinear wave equations are describing the physical process in many areas of science and engineering include fiber optics, geophysics, chemistry, plasma physics, biology, fluid dynamics, laser physics and so on. The NLETLs are the nonlinear partial differential equations. Now a days it is a deals of huge interest to finding the soliton solutions for NLPDEs by implementing the various techniques. To determine the solutions for NLPDEs play key role and deliver the important information to know the applications and its mechanism. Further, this is very keen to know the concept of specific wave called soliton. John Scott Russel first time introduced a solitary wave in the past few decades (Russell, 1844). When solitary wave is discover after that, now its named solitons. Solitions have potential applications in various kinds of areas including soliton dynamics, fluid dynamic, fiber optics, adiabatic dynamic parameters, phenomena of industrial, biomedical problems, engineering and many various kinds of applied sciences to due to their important stability properties. The nonlinear electrical transmission lines are the example in physics. In previous few decades, a lot of researchers and mathematician introduced a various kinds of methods to determine the solitonic solutions for these NLEEs. The name of some important techniques as, Backlund transform method, Hirota bilinear transformation, the Darboux transform method, the technique of exp-function, the jacobian function expansion technique, the trial equation technique, Simple equation technique, modification of fan-Sub equation technique, extend mapping method, the technique of sinh-cosh, the extension of auxiliary equation technique, the extension of direct algebraic technique, modification of extended auxiliary equation technique, Racatti equation mapping method (Juan et al., 2007; Hirota, 1971; Zheng Yi et al., 2006; Naher et al., 2012; Seadawy and Manafian, 2018; Zhang and Xia, 2006; Seadawy and El-Rashidy, 2013; Seadawy and Lu, 2016; Seadawy, 2012; Iqbal et al., 2018a,b; Seadawy et al., 2019a; Iqbal et al., 2019a,b; Seadawy et al., 2020a; Seadawy, 2014a,b; Seadawy, 2015a,b; Seadawy, 2016a,b; Seadawy, 2017b; Seadawy and Wang, 2019; Seadawy and El-Rashidy, 2014; Lu et al., 2018a,b; Seadawy et al., 2020b; Seadawy et al., 2019d,e; Yaro et al., 2019 ). In this current work, our main purpose to find the solutions for NLETLs including bright-dark solitons, kink and anti-kink wave solitons and traveling wave by proposed technique (Seadawy et al., 2019f; Iqbal et al., 2019c; Seadawy et al., 2019b; Iqbal et al., 2020; Seadawy et al., 2019c; Seadawy et al., 2020c ).

This work is organized as, introduction is explain in Section 1. The main feature of proposed method described in Section 2. In Section 3, determined the solutions for NLETLs with proposed technique. In Section 4, compare the obtained results with detail. In Section 5, explain the concluding summery of this article.

2 Algorithm of proposed method

The nonlinear PDEs in general form consider as:

(2)

E (V, V_{t}, V_{x}, V_{tt}, V_{xx}, V_{xt}, \dots \dots) = 0 .

In Eq. (2), Erepresent to polynomial function in $V (x, t)$ and its derivatives. The main features of proposed method explain as:

$Step . 1 :$ We applying the linear transformations on Eq. (2) as:

(3)

V (x, t) = W (ζ), ζ = (κ x + ω t) .

The ODE for Eq. (2) obtain as:

(4)

H (W, W^{'}, W^{″}, W^{‴}, \dots \dots) = 0 .

In Eq. (4), Hrepresent to function of polynomial in $W (ζ)$ and its all derivative.

$Step . 2 :$ The trial solution of Eq. (4), as:

(5)

W (ζ) = \sum_{l = 0}^{m} a_{l} Ψ {(ζ)}^{l} + \sum_{l = - 1}^{- m} b_{- l} Ψ {(ζ)}^{l} + \sum_{l = 2}^{m} c_{2} Ψ {(ζ)}^{l - 2} Ψ' (ζ) + \sum_{l = 1}^{m} d_{l} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{l} .

Here $a_{l}, b_{l}, c_{l}, d_{l}$ are constants which determined in later, $Ψ (ξ)$ and its derivatives satisfy the following auxiliary equation:

(6)

{(\frac{d Ψ}{d ζ})}^{2} = β_{1} Ψ^{2} (ζ) + β_{2} Ψ^{3} (ζ) + β_{3} Ψ^{4} (ζ) .

In Eq. (6), $β_{i}^{,} s$ are represents to real constants which determined to be later.

$Step . 3 :$ We apply the homogeneous method on Eq. (4) to find the value of min Eq. (5).

$Step . 4 :$ Substitute Eq. (5) in Eq. (4), and combine each coefficients of $Ψ^{l} (ζ) Ψ^{' j} (ζ) (l = 1, 2, 3, \dots . n : j = 0, 1)$ , then each coefficient equate to zero and get a system of algebraic equations, after solving these system of equations using any symbolic computation and find the values of constant parameters.

$Step 5 :$ Putting the values of obtain parameters and $Ψ (ζ)$ into Eq. (5), then we find the required solutions for Eq. (2).

3 Application of proposed method

Here we apply proposed technique to determine the solutions for the NLETLs. We consider the traveling wave transformation as:

(7)

V (x, t) = W (ζ), ζ = (κ x + ω t) .

Putting Eq. (7) in Eq. (1), the ODE obtain for Eq. (1), as:

(8)

12 (κ^{2} δ^{2} + ω^{2}) W + 12 α ω^{2} W^{2} - 12 γ ω^{2} W^{3} + δ^{4} κ^{4} W^{″} = 0 .

Balance the nonlinear term and derivative of highest order in Eq. (8), obtain $m = 1$ . General solution for Eq. (8), as:

(9)

W (ξ) = a_{0} + a_{1} Ψ (ζ) + \frac{b_{1}}{Ψ (ζ)} + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} .

Putting Eq. (9) in Eq. (8), and combine each coefficients of $Ψ^{l} (ζ) Ψ^{' j} (ζ) (l = 1, 2, 3, \dots n; j = 0, 1)$ , and equate to each coefficient to zero. We get system of equations. System of equations are solve by using the Mathematica, the values of $a_{0}, a_{1}, b_{1}, d_{1}$ and frequency are obtain as:

Family-I

(10)

\begin{matrix} a_{0} = \frac{α}{3 γ}, a_{1} = - \frac{\sqrt{β_{3}} δ κ \sqrt{- (2 α^{2} + 9 γ)}}{6 \sqrt{6} γ}, b_{1} = 0, d_{1} = - \frac{δ κ \sqrt{- (2 α^{2} + 9 γ)}}{6 \sqrt{6} γ}, \\ β_{1} = - \frac{24 α^{2}}{δ^{2} κ^{2} (2 α^{2} + 9 γ)}, ω = - \frac{3 \sqrt{γ} δ κ}{\sqrt{- 2 α^{2} - 9 γ}} . \end{matrix}

Putting the Eq. (10) in Eq. (9), then the solutions for Eq. (1), obtain as Figs. 1–14 :

(11)

V_{1} (x, t) = \frac{24 α + \frac{\sqrt{6} \sqrt{β_{1}} δ κ \sqrt{- 2 α^{2} - 9 γ} (2 \sqrt{β_{1}} \sqrt{β_{3}} {(∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1)}^{2} + β_{2} ∊ csch {[\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})]}^{2})}{β_{2} (∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1)}}{72 γ},

(12)

V_{2} (x, t) = \frac{24 α + \frac{\sqrt{6} \sqrt{β_{1}} δ κ \sqrt{- 2 α^{2} - 9 γ} ({(η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + ∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}^{2} - 2 ∊ (η \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1))}{(η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) (η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + ∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}}{72 γ},

(13)

\begin{matrix} V_{3} (x, t) = & - \frac{1}{36 γ} (- 12 α + (\sqrt{6} \sqrt{- 2 α^{2} - 9 γ} δ ∊ κ (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1 \\ + \sqrt{6} \sqrt{β_{3}} δ κ \sqrt{- 2 α^{2} - 9 γ} (- \frac{∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} - 1)) \\ - p \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) \sqrt{β_{1}}) / (η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] \\ η \sqrt{p^{2} + 1} + ∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) . \end{matrix}

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (11) represent to solitary wave while ∊=8,ζ0=0.3,κ=2,ω=0.4,α=1,γ=-1,δ=1,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (12) represent to traveling wave while ∊=8,η=-2,ζ0=0.3,κ=2,ω=0.4,α=1,γ=-1,δ=1,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (13) represent to traveling wave while ∊=8,η=-2,ζ0=0.3,κ=2,ω=0.4,α=1,γ=-1,P=0.8,δ=1,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (15) represent to solitary wave while ∊=8,ζ0=0.3,κ=2,ω=0.4,α=1,γ=-1,δ=1,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (16) represent to traveling wave while ∊=8,η=-2,ζ0=0.3,κ=2,ω=0.4,α=1,γ=-1,δ=1,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (17) represent to traveling wave while ∊=8,η=-2,ζ0=0.3,κ=2,ω=0.4,α=1,γ=-1,P=0.8,δ=1,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (20) represent to traveling wave while ∊=-8,η=-2,ζ0=0.3,κ=2,ω=0.4,α=0.5,γ=0.2,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (21) represent to traveling wave while ∊=18,η=-8,ζ0=0.3,κ=2,ω=0.2,α=0.5,γ=0.2,p=1.4,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (23) represent to kink wave soliton while ∊=-8,η=2,ζ0=0.3,κ=4,ω=.8,α=1.5,γ=0.3,δ=0.2,A=1,β1=2,β2=4,β3=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (24) represent to anti-kink wave soliton while ∊=-8,η=2,ζ0=0.3,κ=2,ω=1.4,α=5,γ=2,δ=0.2,p=0.8,A=1,β1=2,β2=4,β3=2,.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (27) represent to traveling wave while ∊=18,η=-8,ζ0=0.3,κ=0.2,ω=4,α=0.3,γ=0.5,δ=0.4,β1=2.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (28) represent to traveling wave while β1=2,β3=4,∊=-8,η=-2,ζ0=0.3,κ=1.2,ω=2,α=3,γ=1,δ=2.4,p=0.9.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (31) represent to traveling wave while β1=2,∊=8,η=-2,ζ0=0.3,κ=2,ω=-2.4,γ=0.2,δ=2.4.

(a) Three dimensional, (b) Two dimensional, (c) Contour plots for Eq. (28) represent to traveling wave while β1=2,β3=4,∊=8,η=-2,ζ0=0.3,κ=1.2,ω=2,γ=0.4,δ=1.5,p=0.6.

Family-II

(14)

\begin{matrix} a_{0} = & \frac{α}{3 γ}, a_{1} = \frac{\sqrt{β_{3}} δ κ \sqrt{- (2 α^{2} + 9 γ)}}{6 \sqrt{6} γ}, b_{1} = 0, d_{1} = \frac{δ κ \sqrt{- (2 α^{2} + 9 γ)}}{6 \sqrt{6} γ}, \\ β_{1} = - \frac{24 α^{2}}{δ^{2} κ^{2} (2 α^{2} + 9 γ)}, ω = \frac{3 \sqrt{γ} δ κ}{\sqrt{- 2 α^{2} - 9 γ}} . \end{matrix}

Putting Eq. (14) in Eq. (9), then solutions for Eq. (1), take as:

(15)

V_{4} (x, t) = - \frac{\frac{\sqrt{6} \sqrt{β_{1}} δ κ \sqrt{- 2 α^{2} - 9 γ} (2 \sqrt{β_{1}} \sqrt{β_{3}} {(∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1)}^{2} + β_{2} ∊ csch {[\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})]}^{2})}{β_{2} (∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1)} - 24 α}{72 γ},

(16)

V_{5} (x, t) = \frac{24 α - \frac{\sqrt{6} \sqrt{β_{1}} δ κ \sqrt{- 2 α^{2} - 9 γ} ({(η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + ∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}^{2} - 2 ∊ (η \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1))}{(η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) (η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + ∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}}{72 γ},

(17)

\begin{matrix} V_{6} (x, t) = & \frac{1}{36 γ} (12 α + (\sqrt{6} \sqrt{- 2 α^{2} - 9 γ} δ ∊ κ (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1 \\ - p \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) \sqrt{β_{1}}) / (η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})] \\ η \sqrt{p^{2} + 1} + ∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]) + \\ \sqrt{6} \sqrt{β_{3}} δ κ \sqrt{- 2 α^{2} - 9 γ} (- \frac{∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} - 1)) . \end{matrix}

Family-III

(18)

\begin{matrix} a_{0} = & 0, a_{1} = - \frac{4 α β_{3}}{3 β_{2} γ}, b_{1} = d_{1} = 0, ω = \frac{\sqrt{\frac{3}{2}} β_{2} \sqrt{γ} δ^{2} κ^{2}}{4 α \sqrt{β_{3}}}, \\ β_{1} = - \frac{9 β_{2}^{2} γ}{8 α^{2} β_{3}} - \frac{12}{δ^{2} κ^{2}} . \end{matrix}

Putting the Eq. (18) in Eq. (9), take solutions for Eq. (1), as:

(19)

V_{7} (x, t) = \frac{4 α β_{1} β_{3} (∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1)}{3 β_{2}^{2} γ},

(20)

V_{8} (x, t) = \frac{2 α \sqrt{\frac{β_{1}}{β_{3}}} β_{3} (\frac{∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} + 1)}{3 β_{2} γ},

(21)

V_{9} (x, t) = - \frac{4 α β_{3} (- \frac{∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} - 1)}{3 β_{2} γ} .

Family-IV

(22)

\begin{matrix} a_{0} = & \frac{3 α^{2} β_{3} γ δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 64 β_{3}) + A}{6 α β_{3} γ^{2} δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 96 β_{3})}, a_{1} = \frac{2 (α^{2} β_{2}^{2} β_{3} γ δ^{4} κ^{4} + A)}{3 α β_{2} γ^{2} δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 96 β_{3})}, b_{1} = d_{1} = 0, \\ ω = - \frac{1}{8} \sqrt{\frac{3 β_{3} γ δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} (5 α^{2} + 18 γ) - 192 β_{3} (2 α^{2} + 9 γ)) - 3 A}{β_{3}^{2} {(2 α^{2} + 9 γ)}^{2}}}, \\ β_{1} = \frac{3 (3 β_{2}^{2} β_{3} γ δ^{4} κ^{4} (α^{2} + 4 γ) (α^{2} + 9 γ) + 64 α^{2} β_{3}^{2} γ δ^{2} κ^{2} (2 α^{2} + 9 γ) + A (α^{2} + 3 γ))}{16 β_{3}^{2} γ δ^{4} κ^{4} {(2 α^{2} + 9 γ)}^{2}}, \end{matrix}

\begin{matrix} where \\ A = \sqrt{3} \sqrt{α^{2} β_{2}^{2} β_{3}^{2} γ^{2} δ^{6} κ^{6} (3 β_{2}^{2} δ^{2} κ^{2} (α^{2} + 4 γ) - 128 β_{3} (2 α^{2} + 9 γ))} . \end{matrix}

Putting the Eq. (22) in Eq. (9), then solutions for Eq. (1), as:

(23)

\begin{matrix} V_{10} (x, t) = & (- 4 A β_{1} β_{3} (∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1) + 3 α^{2} β_{2}^{4} β_{3} γ δ^{4} κ^{4} + \\ β_{2}^{2} (A - 4 α^{2} β_{3}^{2} γ δ^{2} κ^{2} (β_{1} δ^{2} κ^{2} (∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1) + 48)))/ \\ 6 α β_{2}^{2} β_{3} γ^{2} δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 96 β_{3}), \end{matrix}

(24)

V_{11} (x, t) = \frac{\frac{3 α^{2} β_{3} γ δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 64 β_{3}) + A}{β_{3}} - \frac{2 \sqrt{\frac{β_{1}}{β_{3}}} (α^{2} β_{2}^{2} β_{3} γ δ^{4} κ^{4} + A) (\frac{∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} + 1)}{β_{2}}}{6 α γ^{2} δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 96 β_{3})},

(25)

V_{12} (x, t) = \frac{\frac{3 α^{2} β_{3} γ δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 64 β_{3}) + A}{β_{3}} + \frac{4 (α^{2} β_{2}^{2} β_{3} γ δ^{4} κ^{4} + A) (- \frac{∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} - 1)}{β_{2}}}{6 α γ^{2} δ^{2} κ^{2} (β_{2}^{2} δ^{2} κ^{2} - 96 β_{3})} .

Family-V

(26)

\begin{matrix} a_{0} = & \frac{α ω^{2} - \sqrt{ω^{4} (α^{2} + 4 γ) + 4 γ δ^{2} κ^{2} ω^{2}}}{2 γ ω^{2}}, a_{1} = \frac{\sqrt{β_{3}} δ^{2} κ^{2}}{\sqrt{6} \sqrt{γ} ω}, b_{1} = d_{1} = 0, \\ β_{1} = \frac{6 (- α \sqrt{ω^{4} (α^{2} + 4 γ) + 4 γ δ^{2} κ^{2} ω^{2}} + α^{2} ω^{2} + 4 γ (δ^{2} κ^{2} + ω^{2}))}{γ δ^{4} κ^{4}}, \\ β_{2} = \frac{2 \sqrt{\frac{2}{3}} \sqrt{β_{3}} (α ω^{2} - 3 \sqrt{ω^{4} (α^{2} + 4 γ) + 4 γ δ^{2} κ^{2} ω^{2}})}{\sqrt{γ} δ^{2} κ^{2} ω} . \end{matrix}

Putting the Eq. (25) in Eq. (9), then solutions for Eq. (1), as:

(27)

V_{13} (x, t) = \frac{α ω^{2} - \sqrt{ω^{4} (α^{2} + 4 γ) + 4 γ δ^{2} κ^{2} ω^{2}}}{2 γ ω^{2}} - \frac{β_{1} \sqrt{β_{3}} δ^{2} κ^{2} (∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1)}{\sqrt{6} β_{2} \sqrt{γ} ω},

(28)

V_{14} (x, t) = - \frac{6 \sqrt{ω^{4} (α^{2} + 4 γ) + 4 γ δ^{2} κ^{2} ω^{2}} - 6 α ω^{2} + \sqrt{6} \sqrt{β_{1}} \sqrt{γ} δ^{2} κ^{2} ω (\frac{∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]}{η + \cosh (\sqrt{β_{1}} (κ x + ω t + ζ_{0}))} + 1)}{12 γ ω^{2}},

(29)

V_{15} (x, t) = \frac{- 3 \sqrt{ω^{4} (α^{2} + 4 γ) + 4 γ δ^{2} κ^{2} ω^{2}} + 3 α ω^{2} + \sqrt{6} \sqrt{β_{3}} \sqrt{γ} δ^{2} κ^{2} ω (- \frac{∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} - 1)}{6 γ ω^{2}} .

Family-VI

(30)

\begin{matrix} a_{0} = & 0, a_{1} = \frac{\sqrt{β_{3}} δ^{2} κ^{2}}{\sqrt{6} \sqrt{γ} ω}, b_{1} = d_{1} = 0, β_{1} = - \frac{12 (δ^{2} κ^{2} + ω^{2})}{δ^{4} κ^{4}}, \\ β_{2} = - \frac{4 \sqrt{\frac{2}{3}} α \sqrt{β_{3}} ω}{\sqrt{γ} δ^{2} κ^{2}} . \end{matrix}

Putting the Eq. (29) in Eq. (9), then solutions for Eq. (1), as:

(31)

V_{16} (x, t) = - \frac{β_{1} \sqrt{β_{3}} δ^{2} κ^{2} (∊ \coth [\frac{1}{2} \sqrt{β_{1}} (κ x + ω t + ζ_{0})] + 1)}{\sqrt{6} β_{2} \sqrt{γ} ω},

(32)

V_{17} (x, t) = - \frac{\sqrt{β_{1}} δ^{2} κ^{2} (\frac{∊ \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} + 1)}{2 \sqrt{6} \sqrt{γ} ω},

(33)

V_{18} (x, t) = \frac{\sqrt{β_{3}} δ^{2} κ^{2} (- \frac{∊ (p + \sinh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (κ x + ω t + ζ_{0})]} - 1)}{\sqrt{6} \sqrt{γ} ω} .

4 Results and discussion

Here we explain and compare our obtained results with other which already have been determined before this work in previous literature by other various methods. In the current study, we have constructed new and more general results. The main features of this work is the general solution Eq. (5) with range of four parameters and having various structure. The constant parameters values $a_{l}, b_{l}, c_{l}, d_{l}$ are combine by any symbolic computation then Eq. (6) has various kinds of solutions. We obtained new and more general results by the proposed method.

Now we make compare the obtained results with other methods. In past research many scholars have been found different kinds of solutions for NLETLs equation e.g., hyperbolic, elliptic, trigonometric, rational type including kink and anti-kink wave solitons, singular and combined bright-dark solitions, bell shaped, periodic wave by implemented the jacobi elliptic technique, kudryashov technique, auxiliary equation technique, $(\frac{G^{'}}{G})$ -expansion method, extension technique of tanh function, the technique of racatti equation, modification of kudryashov technique, the technique of sine-Gorden and extension of sine-Gorden equation (Abdoulkary et al., 2013; Arshad et al., 2018; Seadawy, 2017a; Kumar et al., 2018; Aslan, 2016 ).

From the above detail discussion and comparison conclude that our constructed results are new and more general which have been not determined in the past literature. The complete investigation prove that proposed technique is more reliable, efficient, straightforward, and powerful to investigate various kinds of nonlinear evolution equations.

5 Conclusion

In this study, we proposed successfully modified mathematical method on NLETLs equation and demonstrated new results. The obtained results are new and more general like rational, trigonometric, hyperbolic type including kink and anti-kink wave, bright-dark solitons and traveling waves. The physical interpretation of some obtained results demonstrated by graphically with symbolic computation. The complete investigation prove that proposed technique is more reliable, efficient, straightforward, and powerful to investigate various kinds of nonlinear evolution equations involves in geophysics, fluid dynamics, nonlinear plasma, chemistry, biology, and field of engineering.We are hopefully determined results have numerous applications in optical fiber, geophysics, fluid dynamics, laser optics, engineering, and many other various kinds of applied sciences.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

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Alquran Marwan, Jaradat Imad. A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application. Nonlinear Dyn.. 2018;91(4):2389-2395.
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Alquran Marwan, Jaradat Imad, Abdel-Muhsen Ruwa. Embedding (3+1)-dimensional diffusion, telegraph, and Burgers’ equations into fractal 2D and 3D spaces: an analytical study. J. King Saud Univ.-Sci.. 2020;32(1):349-355.
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Hirota R.. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett.. 1971;27(18):1456-1458.
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Iqbal M., Seadawy A.R., Lu D.. Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods. Modern Phys. Lett. A. 2018;33(32):1850183.
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Jaradat I., Al-Dolat M., Al-Zoubi K., Alquran M.. Theory and applications of a more general form for fractional power series expansion. Chaos Solitons Fract.. 2018;108:107-110.
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Jaradat Imad, Alquran Marwan, Al-Dolat Mohammad. Analytic solution of homogeneous time-invariant fractional IVP. Adv. Differ. Equ.. 2018;2018:143.
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Jaradat Imad, Alquran Marwan, Abdel-Muhsen Ruwa. An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering. Nonlinear Dyn.. 2018;93(4):1911-1922.
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Lu D., Seadawy A.R., Iqbal M.. Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations. Open Phys.. 2018;16(1):896-909.
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Pelap F.B., Faye M.. Solitonlike excitations in a one dimensional electrical transmission line. J. Math. Phys.. 2005;46 033502-1
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Russell, J.S., 1844. Report on waves. In: Report of the Four-teenth Meeting of the British Association for the Advancement of Science. pp. 311–390.
Seadawy A.R.. The solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method. Appl. Math. Sci.. 2012;6(82):4081-4090.
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Seadawy A.R.. Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl.. 2014;67(1):172-180.
[Google Scholar]
Seadawy A.R.. Stability analysis for two-dimensional ion-acoustic waves in quantum plasmas. Phys. Plasmas. 2014;21(5):052107.
[Google Scholar]
Seadawy A.R.. Approximation solutions of derivative nonlinear Schrodinger equation with computational applications by variational method. Eur. Phys. J. Plus. 2015;130(9):182.
[Google Scholar]
Seadawy, A.R., 2015 Nonlinear wave solutions of the three dimensional Zakharov-Kuznetsov Burgers equation in dusty plasma. Physica A S0378437115006354.
Seadawy A.R.. Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma. Math. Methods Appl. Sci. 2016
[Google Scholar]
Seadawy A.R.. Three dimensional nonlinear modified Zakharov Kuznetsov equation of ion acoustic waves in a magnetized plasma. Comput. Math. Appl.. 2016;71(1):201-212.
[Google Scholar]
Seadawy A.R.. Modulation instability analysis for the generalized derivative higher order nonlinear Schrdinger equation and its the bright and dark soliton solutions. J. Electromagn. Waves Appl.. 2017;31(14):1353-1362.
[Google Scholar]
Seadawy A.R.. Solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili dynamic equation in dust-acoustic plasmas. Pramana. 2017;89(3):49.
[Google Scholar]
Seadawy A.R., El-Rashidy K.. Traveling wave solutions for some coupled nonlinear evolution equations. Math. Comput. Modell.. 2013;57(5–6):1371-1379.
[Google Scholar]
Seadawy A.R., El-Rashidy K.. Water wave solutions of the coupled system Zakharov-Kuznetsov and generalized coupled KdV equations. Scientific World J. 2014:1-6.
[Google Scholar]
Seadawy A.R., Lu D.. Ion acoustic solitary wave solutions of three-dimensional nonlinear extended ZakharovKuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results Phys.. 2016;6:590-593.
[Google Scholar]
Seadawy A.R., Manafian J.. New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod. Results Phys.. 2018;8
[Google Scholar]
Seadawy A.R., Wang J.. Three-dimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized dusty plasma via acoustic solitary wave solutions. Braz. J. Phys.. 2019;49(1):67-78.
[Google Scholar]
Seadawy A.R., Lu D., Iqbal M.. Application of mathematical methods on the system of dynamical equations for the ion sound and Langmuir waves. Pramana. 2019;93(1):10.
[Google Scholar]
Seadawy A.R., Iqbal M., Lu D.. Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. J. Taibah Univ. Sci.. 2019;13(1):1060-1072.
[Google Scholar]
Seadawy A.R., Iqbal M., Lu D.. Ion-acoustic solitary wave solutions of nonlinear damped Korteweg-de Vries and damped modified Korteweg-de Vries dynamical equations. Indian J. Phys. 2019
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Seadawy, A.R., Iqbal, M., Lu, D., 2019. Construction of soliton solutions of the modify unstable nonlinear Schrödinger dynamical equation in fiber optics. Indian J. Phys. 1–10.https://doi.org/10.1007/s12648-019-01532-5.
Seadawy A.R., Iqbal M., Lu D.. Analytical methods via bright-dark solitons and solitary wave solutions of the higher-order nonlinear Schrödinger equation with fourth-order dispersion. Modern Phys. Lett. B. 2019;1950443
[Google Scholar]
Seadawy A.R., Iqbal M., Lu D.. Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput. Math. Appl.. 2019;78:3620-3632.
[Google Scholar]
Seadawy A.R., Iqbal M., Lu D.. Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg-de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Physica A. 2020;544 123560
[Google Scholar]
Seadawy A.R., Iqbal M., Lu D.. Propagation of long-wave with dissipation and dispersion in nonlinear media via generalized Kadomtsive-Petviashvili modified equal width-Burgers equation. Indian J. Phys.. 2020;94(5):675-687.
[Google Scholar]
Seadawy A.R., Iqbal M., Lu D.. The nonlinear diffusion reaction dynamical system with quadratic and cubic nonlinearities with analytical investigations. Int. J. Modern Phys. B. 2020;34(9):2050085.
[Google Scholar]
Yaro D., Seadawy A.R., Lu D., Apeanti W.O., Akuamoah S.W.. Dispersive wave solutions of the nonlinear fractional Zakhorov-Kuznetsov-Benjamin-Bona-Mahony equation and fractional symmetric regularized long wave equation. Results Phys.. 2019;12:1971-1979.
[Google Scholar]
Zhang S., Xia T.C.. A further improved extended Fan sub-equation method and its application to the (3+1)-dimensional KadomstevPetviashvili equation. Phys. Lett. A. 2006;356(2):119-123.
[Google Scholar]
Zheng Yi M., Hua G.S., Zheng C.L.. Multisoliton excitations for the Kadomtsev-Petviashvili equation. Z. Naturforschung A. 2006;61(1–2):32-38.
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Introduction

Algorithm of proposed method

Application of proposed method

Results and discussion

Conclusion

Declaration of Competing Interest

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[1] Abdoulkary S., Beda T., Dafounamssou O., Tafo E.W., Mohamadou A.. Dynamics of solitary pulses in the nonlinear low-pass electrical transmission lines through the auxiliary equation method. J. Mod. Phys. Appl.. 2013;2:69-87.
[Google Scholar]

[2] Ali Mohammed, Alquran Marwan, Jaradat Imad. Asymptotic-sequentially solution style for the generalized Caputo time-fractional Newell-Whitehead-Segel system. Adv. Diff. Eqs.. 2019;2019:70.
[Google Scholar]

[3] Alquran Marwan, Jaradat Imad. A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application. Nonlinear Dyn.. 2018;91(4):2389-2395.
[Google Scholar]

[4] Alquran Marwan, Jaradat Imad, Abdel-Muhsen Ruwa. Embedding (3+1)-dimensional diffusion, telegraph, and Burgers’ equations into fractal 2D and 3D spaces: an analytical study. J. King Saud Univ.-Sci.. 2020;32(1):349-355.
[Google Scholar]

[5] Arshad M., Seadawy A.R., Lu D.. Modulation stability and dispersive optical soliton solutions of higher order nonlinear schrodinger equation and its applications in mono-mode optical fibers. Superlattices Microstruct.. 2018;113:419-429.
[Google Scholar]

[6] Aslan I.. Exact solutions for a local fractional DDE associated with a nonlinear transmission line. Commun. Theor. Phys.. 2016;66(3):315-320.
[Google Scholar]

[7] Hirota R.. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett.. 1971;27(18):1456-1458.
[Google Scholar]

[8] Iqbal M., Seadawy A.R., Lu D.. Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods. Modern Phys. Lett. A. 2018;33(32):1850183.
[Google Scholar]

[9] Iqbal M., Seadawy A.R., Lu D.. Dispersive solitary wave solutions of nonlinear further modified Kortewege-de Vries dynamical equation in a unmagnetized dusty plasma via mathematical methods. Modern Phys. Lett. A. 2018;33(37):1850217.
[Google Scholar]

[10] Iqbal M., Seadawy A.R., Lu D.. Applications of nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod and new solitary wave solutions. Modern Phys. Lett. B. 2019;33(18):1950210.
[Google Scholar]

[11] Iqbal M., Seadawy A.R., Lu D., Xia X.. Construction of a weakly nonlinear dispersion solitary wave solution for the Zakharov-Kuznetsov-modified equal width dynamical equation. Indian J. Phys. 2019:1-10.
[CrossRef] [Google Scholar]

[12] Iqbal M., Seadawy A.R., Lu D., Xia X.. Construction of bright-dark solitons and ion-acoustic solitary wave solutions of dynamical system of nonlinear wave propagation. Modern Phys. Lett. A. 2019;34:1950309.
[Google Scholar]

[13] Iqbal M., Seadawy A.R., Khalil O.H., Lu D.. Propagation of long internal waves in density stratified ocean for the (2+1)-dimensional nonlinear Nizhnik-Novikov-Vesselov dynamical equation. Results Phys.. 2020;16 102838
[Google Scholar]

[14] Jaradat Imad, Alquran Marwan, Al-Khaled Kamel. An analytical study of physical models with inherited temporal and spatial memory. Eur. Phys. J. Plus.. 2018;133:162.
[Google Scholar]

[15] Jaradat I., Al-Dolat M., Al-Zoubi K., Alquran M.. Theory and applications of a more general form for fractional power series expansion. Chaos Solitons Fract.. 2018;108:107-110.
[Google Scholar]

[16] Jaradat Imad, Alquran Marwan, Al-Dolat Mohammad. Analytic solution of homogeneous time-invariant fractional IVP. Adv. Differ. Equ.. 2018;2018:143.
[Google Scholar]

[17] Jaradat Imad, Alquran Marwan, Abdel-Muhsen Ruwa. An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering. Nonlinear Dyn.. 2018;93(4):1911-1922.
[Google Scholar]

[18] Juan L., Xu T., Meng X.H., Zhang Y.X., Zhang H.Q.. Lax pair, Backlund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation. J. Math. Anal. Appl.. 2007;336(2):1443-1455.
[Google Scholar]

[19] Kumar D., Seadawy A.R., Haque M.R.. Multiple soliton solutions of the nonlinear partial differential equations describing the wave propagation in nonlinear lowpass electrical transmission lines. Chaos Solitons Fract.. 2018;115:62-76.
[Google Scholar]

[20] Lu D., Seadawy A.R., Iqbal M.. Mathematical method via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications. Results Phys.. 2018;11:1161-1171.
[Google Scholar]

[21] Lu D., Seadawy A.R., Iqbal M.. Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations. Open Phys.. 2018;16(1):896-909.
[Google Scholar]

[22] Naher H., Abdullah F.A., Akbar M.A.. New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method. J. Appl. Math.. 2012;2:1-7.
[Google Scholar]

[23] Pelap F.B., Faye M.. Solitonlike excitations in a one dimensional electrical transmission line. J. Math. Phys.. 2005;46 033502-1
[Google Scholar]

[24] Russell, J.S., 1844. Report on waves. In: Report of the Four-teenth Meeting of the British Association for the Advancement of Science. pp. 311–390.

[25] Seadawy A.R.. The solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method. Appl. Math. Sci.. 2012;6(82):4081-4090.
[Google Scholar]

[26] Seadawy A.R.. Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl.. 2014;67(1):172-180.
[Google Scholar]

[27] Seadawy A.R.. Stability analysis for two-dimensional ion-acoustic waves in quantum plasmas. Phys. Plasmas. 2014;21(5):052107.
[Google Scholar]

[28] Seadawy A.R.. Approximation solutions of derivative nonlinear Schrodinger equation with computational applications by variational method. Eur. Phys. J. Plus. 2015;130(9):182.
[Google Scholar]

[29] Seadawy, A.R., 2015 Nonlinear wave solutions of the three dimensional Zakharov-Kuznetsov Burgers equation in dusty plasma. Physica A S0378437115006354.

[30] Seadawy A.R.. Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma. Math. Methods Appl. Sci. 2016
[Google Scholar]

[31] Seadawy A.R.. Three dimensional nonlinear modified Zakharov Kuznetsov equation of ion acoustic waves in a magnetized plasma. Comput. Math. Appl.. 2016;71(1):201-212.
[Google Scholar]

[32] Seadawy A.R.. Modulation instability analysis for the generalized derivative higher order nonlinear Schrdinger equation and its the bright and dark soliton solutions. J. Electromagn. Waves Appl.. 2017;31(14):1353-1362.
[Google Scholar]

[33] Seadawy A.R.. Solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili dynamic equation in dust-acoustic plasmas. Pramana. 2017;89(3):49.
[Google Scholar]

[34] Seadawy A.R., El-Rashidy K.. Traveling wave solutions for some coupled nonlinear evolution equations. Math. Comput. Modell.. 2013;57(5–6):1371-1379.
[Google Scholar]

[35] Seadawy A.R., El-Rashidy K.. Water wave solutions of the coupled system Zakharov-Kuznetsov and generalized coupled KdV equations. Scientific World J. 2014:1-6.
[Google Scholar]

[36] Seadawy A.R., Lu D.. Ion acoustic solitary wave solutions of three-dimensional nonlinear extended ZakharovKuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results Phys.. 2016;6:590-593.
[Google Scholar]

[37] Seadawy A.R., Manafian J.. New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod. Results Phys.. 2018;8
[Google Scholar]

[38] Seadawy A.R., Wang J.. Three-dimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized dusty plasma via acoustic solitary wave solutions. Braz. J. Phys.. 2019;49(1):67-78.
[Google Scholar]

[39] Seadawy A.R., Lu D., Iqbal M.. Application of mathematical methods on the system of dynamical equations for the ion sound and Langmuir waves. Pramana. 2019;93(1):10.
[Google Scholar]

[40] Seadawy A.R., Iqbal M., Lu D.. Nonlinear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. J. Taibah Univ. Sci.. 2019;13(1):1060-1072.
[Google Scholar]

[41] Seadawy A.R., Iqbal M., Lu D.. Ion-acoustic solitary wave solutions of nonlinear damped Korteweg-de Vries and damped modified Korteweg-de Vries dynamical equations. Indian J. Phys. 2019
[CrossRef] [Google Scholar]

[42] Seadawy, A.R., Iqbal, M., Lu, D., 2019. Construction of soliton solutions of the modify unstable nonlinear Schrödinger dynamical equation in fiber optics. Indian J. Phys. 1–10.https://doi.org/10.1007/s12648-019-01532-5.

[43] Seadawy A.R., Iqbal M., Lu D.. Analytical methods via bright-dark solitons and solitary wave solutions of the higher-order nonlinear Schrödinger equation with fourth-order dispersion. Modern Phys. Lett. B. 2019;1950443
[Google Scholar]

[44] Seadawy A.R., Iqbal M., Lu D.. Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput. Math. Appl.. 2019;78:3620-3632.
[Google Scholar]

[45] Seadawy A.R., Iqbal M., Lu D.. Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg-de Vries equation arising in ion-acoustic wave in an unmagnetized collisional dusty plasma. Physica A. 2020;544 123560
[Google Scholar]

[46] Seadawy A.R., Iqbal M., Lu D.. Propagation of long-wave with dissipation and dispersion in nonlinear media via generalized Kadomtsive-Petviashvili modified equal width-Burgers equation. Indian J. Phys.. 2020;94(5):675-687.
[Google Scholar]

[47] Seadawy A.R., Iqbal M., Lu D.. The nonlinear diffusion reaction dynamical system with quadratic and cubic nonlinearities with analytical investigations. Int. J. Modern Phys. B. 2020;34(9):2050085.
[Google Scholar]

[48] Yaro D., Seadawy A.R., Lu D., Apeanti W.O., Akuamoah S.W.. Dispersive wave solutions of the nonlinear fractional Zakhorov-Kuznetsov-Benjamin-Bona-Mahony equation and fractional symmetric regularized long wave equation. Results Phys.. 2019;12:1971-1979.
[Google Scholar]

[49] Zhang S., Xia T.C.. A further improved extended Fan sub-equation method and its application to the (3+1)-dimensional KadomstevPetviashvili equation. Phys. Lett. A. 2006;356(2):119-123.
[Google Scholar]

[50] Zheng Yi M., Hua G.S., Zheng C.L.. Multisoliton excitations for the Kadomtsev-Petviashvili equation. Z. Naturforschung A. 2006;61(1–2):32-38.
[Google Scholar]

Construction of traveling and solitary wave solutions for wave propagation in nonlinear low-pass electrical transmission lines

Abstract

Keywords

Nonlinear low-pass electrical transmission lines

Modified Mathematical technique

Solitary wave solutions

Traveling wave solutions

1 Introduction

2 Algorithm of proposed method

3 Application of proposed method

4 Results and discussion

5 Conclusion

Declaration of Competing Interest

References

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