Full article: Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity

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Abstract

In this research, we constructed the exact travelling and solitary wave solutions of the Kudryashov–Sinelshchikov (KS) equation by implementing the modified mathematical method. The KS equation describe the phenomena of pressure waves in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. Our new obtained solutions in the shape of hyperbolic, trigonometric, elliptic functions including dark, bright, singular, combined, kink wave solitons, travelling wave, solitary wave and periodic wave. We showed the physical interpretation of obtained solutions by three-dimensional graphically. These new constructed solutions play vital role in mathematical physics, optical fiber, plasma physics and other various branches of applied sciences.

Keywords:

1. Introduction

Kudryashov and Sinelshchikov (KS) in (2010) first time introduced a nonlinear evolution equation with the help of theoretical and experimental work [Citation1]. The KS equation describe the phenomena of pressure waves in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. The KS equation given [Citation1], as (1) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (1) In Equation (Equation1(1) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (1) ), $u (x, t)$ is a function which denote to density, heat transfer and viscosity models, $γ, ε, κ, ν, δ,$ are real constant parameters. If $κ = ε = δ = ν = 0$ , then Equation (Equation1(1) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (1) ), change into Korteweg-de Vries equation [Citation2], as (2) $u_{t} + γ u u_{x} + u_{x x x} = 0.$ (2) If $κ = ε = δ = 0,$ then Equation (Equation1(1) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (1) ), reduced into Korteweg-de Vries Burgers equation [Citation3], as (3) $u_{t} + γ u u_{x} + u_{x x x} - ν u_{x x} = 0.$ (3) The Equation (Equation1(1) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (1) ), is the general form of KdV equation and KdVB equation. When $γ = ε = 1, ν = δ = 0,$ then Equation (Equation1(1) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (1) ), obtain [Citation4], as (4) $u_{t} + u u_{x} + u_{x x x} - (u u_{x x})_{x} - κ u_{x} u_{x x} = 0.$ (4) The author using the modification form of truncated expansion technique to investigate the solitary wave solutions of Equation (Equation4(4) $u_{t} + u u_{x} + u_{x x x} - (u u_{x x})_{x} - κ u_{x} u_{x x} = 0.$ (4) ), under the following condition $κ = - 3, κ = - 4,$ [Citation4]. Recently, many researchers found different types of solutions of KS equation by applying different techniques at different conditions of parameters, as in these references [Citation5–16].

In the last few decades, a lots of research have been done to determine the exact solutions of nonlinear evolution equations (NLEEs). The investigations of exact solutions of NLEEs have a great deal to know the structure, provide better information and its applications. Therefore, to calculate the exact and solitary solutions of nonlinear partial differential equations (NLPDEs), many researchers and mathematician introduced a lot of methods. Such as, Hirota bilinear method, Exp-function technique, Jacobian elliptic method, Trial equation method, extended simple equation method, exp $(- ψ (η))$ -expansion method, Backlund transformation, improved extended Fan-Subequation method, Darboux transformation, sinh-cosh method, sech-tanh method, extended direct algebraic method, extended auxiliary equation mapping method, improved F-expansion method [Citation17–29] and many researcher in applications of mathematical methods [Citation30–42].

In this current work, we investigated the exact travelling and solitary solutions of nonlinear KS equation by implementing the modified mathematical method [Citation43–50].

This research work is arranged as follows, we explain the introduction in Section 1. We describe the proposed technique in Section 2. We implemented the described technique on KS equation and found the exact travelling and solitary solutions in Section 3. This work end at conclusion in Section 4.

2. Description of proposed method

Here we describe the main future of the modified mathematical technique for finding the solutions of nonlinear partial differential equations (PDEs). We consider the general form of nonlinear PDEs as (5) $Q (u, u_{t}, u_{x}, u_{t t}, u_{x x}, u_{x t}, \dots . .) = 0.$ (5) Where Q denote to polynomial function of $u (x, t)$ and their derivatives. We explain the features of modified mathematical technique as

$S t e p 1.$ We consider the transformations of travelling wave as (6) $u (x, t) = U (ζ), ζ = (x - λ t) .$ (6) In Equation (Equation6(6) $u (x, t) = U (ζ), ζ = (x - λ t) .$ (6) ), λ is the wave frequency. We obtain the ODE of Equation (Equation5(5) $Q (u, u_{t}, u_{x}, u_{t t}, u_{x x}, u_{x t}, \dots . .) = 0.$ (5) ), as (7) $R (U, U^{'}, U^{''}, U^{'''}, \dots) = 0.$ (7) In Equation (Equation7(7) $R (U, U^{'}, U^{''}, U^{'''}, \dots) = 0.$ (7) ), R denote to polynomial function in $U (ζ)$ and their derivative.

$S t e p 2.$ We consider the trial solution of Equation (Equation7(7) $R (U, U^{'}, U^{''}, U^{'''}, \dots) = 0.$ (7) ), as (8) $\begin{aligned} U (ζ) & = \sum_{i = 0}^{N} a_{i} Ψ (ζ)^{i} + \sum_{i = - 1}^{- N} b_{- i} Ψ (ζ)^{i} \\ + \sum_{i = 2}^{N} c_{2} Ψ (ζ)^{i - 2} Ψ^{'} (ζ) + \sum_{i = 1}^{N} d_{i} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{i} . \end{aligned}$ (8) Here ( $a_{i}, b_{i}, c_{i}, d_{i})$ are constants which calculate later, the derivatives of $Ψ (ζ)$ satisfy the following auxiliary equation (9) $\begin{aligned} (Ψ^{'} (ζ))^{2} & = β_{1} Ψ^{2} (ζ) + β_{2} Ψ^{3} (ζ) + β_{3} Ψ^{4} (ζ); \\ Ψ^{''} (ζ) & = β_{1} Ψ (ζ) + \frac{3}{2} β_{2} Ψ^{2} (ζ) + 2 β_{3} Ψ^{3} (ζ) . \end{aligned}$ (9) In Equation (Equation9(9) $\begin{aligned} (Ψ^{'} (ζ))^{2} & = β_{1} Ψ^{2} (ζ) + β_{2} Ψ^{3} (ζ) + β_{3} Ψ^{4} (ζ); \\ Ψ^{''} (ζ) & = β_{1} Ψ (ζ) + \frac{3}{2} β_{2} Ψ^{2} (ζ) + 2 β_{3} Ψ^{3} (ζ) . \end{aligned}$ (9) ), $β_{i}^{,} s$ are real constants which are found later.

$S t e p 3.$ We balance the terms of nonlinear and derivative of higher order in Equation (Equation7(7) $R (U, U^{'}, U^{''}, U^{'''}, \dots) = 0.$ (7) ), determined N of Equation (Equation8(8) $\begin{aligned} U (ζ) & = \sum_{i = 0}^{N} a_{i} Ψ (ζ)^{i} + \sum_{i = - 1}^{- N} b_{- i} Ψ (ζ)^{i} \\ + \sum_{i = 2}^{N} c_{2} Ψ (ζ)^{i - 2} Ψ^{'} (ζ) + \sum_{i = 1}^{N} d_{i} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{i} . \end{aligned}$ (8) ).

$S t e p 4.$ Putting Equation (Equation8(8) $\begin{aligned} U (ζ) & = \sum_{i = 0}^{N} a_{i} Ψ (ζ)^{i} + \sum_{i = - 1}^{- N} b_{- i} Ψ (ζ)^{i} \\ + \sum_{i = 2}^{N} c_{2} Ψ (ζ)^{i - 2} Ψ^{'} (ζ) + \sum_{i = 1}^{N} d_{i} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{i} . \end{aligned}$ (8) ) in Equation (Equation9(9) $\begin{aligned} (Ψ^{'} (ζ))^{2} & = β_{1} Ψ^{2} (ζ) + β_{2} Ψ^{3} (ζ) + β_{3} Ψ^{4} (ζ); \\ Ψ^{''} (ζ) & = β_{1} Ψ (ζ) + \frac{3}{2} β_{2} Ψ^{2} (ζ) + 2 β_{3} Ψ^{3} (ζ) . \end{aligned}$ (9) ) and collecting every coefficients of $Ψ^{^{'} j} (ζ) Ψ^{i} (ζ) (i = 1, 2, 3, \dots N; j = 0, 1),$ then every coefficients make zero and get a system of equation, solve these system of equations using any computer software, the values of these parameters $(a_{i}, b_{i}, c_{i}, d_{i}),$ are found.

$S t e p 5.$ Substituting parameters values which are obtained and $Ψ (ζ)$ in Equation (Equation9(9) $\begin{aligned} (Ψ^{'} (ζ))^{2} & = β_{1} Ψ^{2} (ζ) + β_{2} Ψ^{3} (ζ) + β_{3} Ψ^{4} (ζ); \\ Ψ^{''} (ζ) & = β_{1} Ψ (ζ) + \frac{3}{2} β_{2} Ψ^{2} (ζ) + 2 β_{3} Ψ^{3} (ζ) . \end{aligned}$ (9) ), then we obtain the required solutions of Equation (Equation5(5) $Q (u, u_{t}, u_{x}, u_{t t}, u_{x x}, u_{x t}, \dots . .) = 0.$ (5) ).

3. KS equation

Here we apply the described technique to construct the solitary wave solutions for the KS equation. (10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) We apply transformation of the wave (11) $u (x, t) = U (ζ), ζ = (x - λ t) .$ (11) Substituting Equation (Equation11(11) $u (x, t) = U (ζ), ζ = (x - λ t) .$ (11) ) in Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) and integrating once w.r to ζ with zero integration constant, we obtain the following: (12) $\begin{aligned} - λ U + \frac{γ}{2} U^{2} + U^{''} - ε U U^{''} - \frac{κ}{2} (U^{'})^{2} - ν U^{'} - δ U U^{'} = 0. \end{aligned}$ (12) We balance the term of nonlinear and derivative of higher order in Equation (Equation12(12) $\begin{aligned} - λ U + \frac{γ}{2} U^{2} + U^{''} - ε U U^{''} - \frac{κ}{2} (U^{'})^{2} - ν U^{'} - δ U U^{'} = 0. \end{aligned}$ (12) ), we get $N = 2.$ Trial solution of Equation (Equation12(12) $\begin{aligned} - λ U + \frac{γ}{2} U^{2} + U^{''} - ε U U^{''} - \frac{κ}{2} (U^{'})^{2} - ν U^{'} - δ U U^{'} = 0. \end{aligned}$ (12) ), take as (13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) Substituting Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ) in Equation (Equation12(12) $\begin{aligned} - λ U + \frac{γ}{2} U^{2} + U^{''} - ε U U^{''} - \frac{κ}{2} (U^{'})^{2} - ν U^{'} - δ U U^{'} = 0. \end{aligned}$ (12) ) and collect every coefficients of $Ψ^{^{'} j} (ζ) Ψ^{i} (ζ) (i = 1, 2, 3, \dots N; j = 0, 1),$ compare every coefficients to zero. We obtain a system of equations. These system of equations solve by using the computer software Mathematica, the values of constants obtained are as follows:

Case-I (14) $\begin{aligned} a_{0} & = - \frac{\begin{array}{l} \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} \\ - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1) \end{array}}{2 β_{1} κ}, \\ a_{1} & = - β_{2} d_{2}, a_{2} = - β_{3} d_{2}, \\ b_{1} & = b_{1}, b_{2} = c_{2} = d_{1} = 0, d_{2} = d_{2}, \\ λ & = - \frac{(ε + κ) \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + 2 β_{1} ε}{2 κ}, \\ γ & = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (14)

Substituting Equation (Equation14(14) $\begin{aligned} a_{0} & = - \frac{\begin{array}{l} \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} \\ - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1) \end{array}}{2 β_{1} κ}, \\ a_{1} & = - β_{2} d_{2}, a_{2} = - β_{3} d_{2}, \\ b_{1} & = b_{1}, b_{2} = c_{2} = d_{1} = 0, d_{2} = d_{2}, \\ λ & = - \frac{(ε + κ) \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + 2 β_{1} ε}{2 κ}, \\ γ & = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (14) ) in Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ), we get the solutions of Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) as (15) $\begin{aligned} u_{1} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (15) (16) $\begin{aligned} u_{2} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (16) (17) $\begin{aligned} u_{3} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (17) Case-II (18) $\begin{aligned} a_{0} & = \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ}, a_{1} = - β_{2} d_{2}, a_{2} = - β_{3} d_{2}, b_{1} = b_{1}, \\ b_{2} & = c_{2} = d_{1} = 0, d_{2} = d_{2}, λ = \frac{(ε + κ) \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - 2 β_{1} ε}{2 κ}, \\ γ & = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (18) Substituting Equation (Equation18(18) $\begin{aligned} a_{0} & = \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ}, a_{1} = - β_{2} d_{2}, a_{2} = - β_{3} d_{2}, b_{1} = b_{1}, \\ b_{2} & = c_{2} = d_{1} = 0, d_{2} = d_{2}, λ = \frac{(ε + κ) \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - 2 β_{1} ε}{2 κ}, \\ γ & = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (18) ) in Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ), we obtain the solutions of Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) as (19) $\begin{aligned} u_{4} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (19) (20) $\begin{aligned} u_{5} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ β_{1} d_{2} (- {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}) - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (20) (21) $\begin{aligned} u_{6} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (21)

Case-III (22) $\begin{aligned} a_{0} & = \frac{- A β_{2} (ε - 2 κ) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) - 4 β_{3} β_{1}^{3} d_{2} ε^{3} κ + β_{1}^{2} ε κ (4 β_{3} ε + β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}))}{β_{1} κ (β_{2}^{2} (4 ε κ^{2} - ε^{3}) + 4 β_{1} β_{3} ε^{3})}, \\ a_{1} & = - β_{2} d_{2}, a_{2} = - β_{3} d_{2}, b_{1} = \frac{2 (A - 2 β_{1} β_{2} κ (ε + κ))}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})}, b_{2} = c_{2} = d_{1} = 0, \\ d_{2} & = d_{2}, λ = \frac{- 2 A β_{2} (ε + κ) - 4 β_{3} β_{1}^{2} ε^{2} (2 ε + κ) + β_{2}^{2} β_{1} ε^{2} (2 ε + 5 κ)}{β_{2}^{2} (ε^{3} - 4 ε κ^{2}) - 4 β_{1} β_{3} ε^{3}}, γ = β_{1} (2 ε + κ), δ = ν = 0, \\ w h e r e \\ A & = \sqrt{β_{1}^{2} ε^{2} κ (β_{2}^{2} (2 ε + 5 κ) - 4 β_{1} β_{3} (2 ε + κ))} . \end{aligned}$ (22) Substituting Equation (Equation22(22) $\begin{aligned} a_{0} & = \frac{- A β_{2} (ε - 2 κ) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) - 4 β_{3} β_{1}^{3} d_{2} ε^{3} κ + β_{1}^{2} ε κ (4 β_{3} ε + β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}))}{β_{1} κ (β_{2}^{2} (4 ε κ^{2} - ε^{3}) + 4 β_{1} β_{3} ε^{3})}, \\ a_{1} & = - β_{2} d_{2}, a_{2} = - β_{3} d_{2}, b_{1} = \frac{2 (A - 2 β_{1} β_{2} κ (ε + κ))}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})}, b_{2} = c_{2} = d_{1} = 0, \\ d_{2} & = d_{2}, λ = \frac{- 2 A β_{2} (ε + κ) - 4 β_{3} β_{1}^{2} ε^{2} (2 ε + κ) + β_{2}^{2} β_{1} ε^{2} (2 ε + 5 κ)}{β_{2}^{2} (ε^{3} - 4 ε κ^{2}) - 4 β_{1} β_{3} ε^{3}}, γ = β_{1} (2 ε + κ), δ = ν = 0, \\ w h e r e \\ A & = \sqrt{β_{1}^{2} ε^{2} κ (β_{2}^{2} (2 ε + 5 κ) - 4 β_{1} β_{3} (2 ε + κ))} . \end{aligned}$ (22) ) in Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ), we get the solutions of Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) as (23) $\begin{aligned} u_{7} (x, t) & = \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} + β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{2 β_{1} (A - 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2} κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ - \frac{β_{3} β_{1}^{2} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (23) (24) $\begin{aligned} u_{8} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (A - 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (24) (25) $\begin{aligned} u_{9} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{2 (A - 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (β_{2}^{2} (4 κ^{2} - ε^{2}) + 4 β_{1} β_{3} ε^{2})} . \end{aligned}$ (25)

Case-IV (26) $\begin{aligned} a_{0} & = \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ}, a_{1} = - β_{2} d_{2}, b_{1} = b_{1}, a_{2} = b_{2} = c_{2} = d_{1} = 0, \\ d_{2} & = d_{2}, λ = \frac{(ε + κ) \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} - 2 β_{1} ε}{2 κ}, γ = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (26) Substituting Equation (Equation26(26) $\begin{aligned} a_{0} & = \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ}, a_{1} = - β_{2} d_{2}, b_{1} = b_{1}, a_{2} = b_{2} = c_{2} = d_{1} = 0, \\ d_{2} & = d_{2}, λ = \frac{(ε + κ) \sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} - 2 β_{1} ε}{2 κ}, γ = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (26) ) in Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ), the solutions of Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) are given as (27) $\begin{aligned} u_{10} (x, t) & = \frac{1}{4} β_{1} d_{2} ϵ (4 \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + \frac{ϵ c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}) \\ + \frac{β_{2} (\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} - 2 β_{1}) + b_{1} κ (β_{2}^{2} - 2 β_{1}^{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1))}{2 β_{1} β_{2} κ} . \end{aligned}$ (27) (28) $\begin{aligned} u_{11} (x, t) & = \frac{1}{2} (\frac{2 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{β_{1} κ} - b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2}) . \end{aligned}$ (28) (29) $\begin{aligned} u_{12} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} + β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) . \end{aligned}$ (29)

Case-V (30) $\begin{aligned} a_{0} & = - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)}, a_{1} = - β_{2} d_{2}, \\ b_{1} & = \frac{2 (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ))}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)}, a_{2} = b_{2} = c_{2} = d_{1} = 0, d_{2} = d_{2}, \\ λ & = \frac{β_{1} β_{2} ε^{2} (2 ε + 5 κ) + 2 (ε + κ) \sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)}}{β_{2} ε (ε^{2} - 4 κ^{2})}, γ = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (30) Substituting Equation (Equation30(30) $\begin{aligned} a_{0} & = - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)}, a_{1} = - β_{2} d_{2}, \\ b_{1} & = \frac{2 (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ))}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)}, a_{2} = b_{2} = c_{2} = d_{1} = 0, d_{2} = d_{2}, \\ λ & = \frac{β_{1} β_{2} ε^{2} (2 ε + 5 κ) + 2 (ε + κ) \sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)}}{β_{2} ε (ε^{2} - 4 κ^{2})}, γ = β_{1} (2 ε + κ), δ = ν = 0. \end{aligned}$ (30) ) in Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ), we obtain the solutions of Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) as (31) $\begin{aligned} u_{13} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{2 β_{1} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{3} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (31) (32) $\begin{aligned} u_{14} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} \\ + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} . \end{aligned}$ (32) (33) $\begin{aligned} u_{15} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} (η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] \\ {+ ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) β_{2} d_{2} (- (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)) \\ + \frac{2 (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (33)

Case-VI (34) $\begin{aligned} a_{0} & = a_{0}, a_{1} = - β_{2} - d_{2}, a_{2} = - β_{3} d_{2}, b_{1} = b_{2} = c_{2} = d_{1} = 0, \\ d_{2} & = d_{2}, λ = \frac{1}{2} γ (a_{0} + β_{1} d_{2}) . \end{aligned}$ (34) Substituting Equation (Equation34(34) $\begin{aligned} a_{0} & = a_{0}, a_{1} = - β_{2} - d_{2}, a_{2} = - β_{3} d_{2}, b_{1} = b_{2} = c_{2} = d_{1} = 0, \\ d_{2} & = d_{2}, λ = \frac{1}{2} γ (a_{0} + β_{1} d_{2}) . \end{aligned}$ (34) ) in Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ), the solutions of Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) are given as (35) $\begin{aligned} u_{16} (x, t) & = a_{0} + \frac{β_{1} d_{2} ϵ^{2} {c s c h}^{4} (\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0}))}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} + β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} . \end{aligned}$ (35) (36) $\begin{aligned} u_{17} (x, t) & = a_{0} + \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) . \end{aligned}$ (36) (37) $\begin{aligned} u_{18} (x, t) & = a_{0} + β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} . \end{aligned}$ (37) Case-VII (38) $\begin{aligned} a_{0} & = \frac{1}{4} (- 4 β_{1} d_{2} - \frac{3}{ε + κ} + \frac{2}{ε} + \frac{2}{κ}), a_{1} = - β_{2} d_{2}, a_{2} = - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2})}{4 β_{1} ε^{2}} \\ b_{1} & = \frac{β_{1} (2 ε + κ)}{2 β_{2} κ (ε + κ)}, b_{2} = c_{2} = d_{1} = 0, d_{2} = d_{2}, λ = \frac{1}{2} β_{1} (\frac{κ}{ε} + 2), \\ γ & = β_{1} (2 ε + κ), δ = ν = 0, β_{3} = \frac{β_{2}^{2} (ε^{2} - 4 κ^{2})}{4 β_{1} ε^{2}} . \end{aligned}$ (38) Substituting Equation (Equation38(38) $\begin{aligned} a_{0} & = \frac{1}{4} (- 4 β_{1} d_{2} - \frac{3}{ε + κ} + \frac{2}{ε} + \frac{2}{κ}), a_{1} = - β_{2} d_{2}, a_{2} = - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2})}{4 β_{1} ε^{2}} \\ b_{1} & = \frac{β_{1} (2 ε + κ)}{2 β_{2} κ (ε + κ)}, b_{2} = c_{2} = d_{1} = 0, d_{2} = d_{2}, λ = \frac{1}{2} β_{1} (\frac{κ}{ε} + 2), \\ γ & = β_{1} (2 ε + κ), δ = ν = 0, β_{3} = \frac{β_{2}^{2} (ε^{2} - 4 κ^{2})}{4 β_{1} ε^{2}} . \end{aligned}$ (38) ) in Equation (Equation13(13) $\begin{aligned} U (ζ) & = a_{0} + a_{1} Ψ (ζ) + a_{2} Ψ (ζ)^{2} + \frac{b_{1}}{Ψ (ζ)} + \frac{b_{2}}{Ψ (ζ)^{2}} \\ + c_{2} Ψ^{'} (ζ) + d_{1} \frac{Ψ^{'} (ζ)}{Ψ (ζ)} + d_{2} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{2} . \end{aligned}$ (13) ), the solutions of Equation (Equation10(10) $\begin{aligned} u_{t} + γ u u_{x} + u_{x x x} - ε (u u_{x x})_{x} - κ u_{x} u_{x x} \\ - ν u_{x x} - δ (u u_{x})_{x} = 0. \end{aligned}$ (10) ) are obtained as (39) $\begin{aligned} u_{19} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + 4 β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{β_{1} d_{2} (ε^{2} - 4 κ^{2}) {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{ε^{2}} \\ + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} - \frac{2 β_{1}^{2} (2 ε + κ) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{2} κ (ε + κ)}) . \end{aligned}$ (39) (40) $\begin{aligned} u_{20} (x, t) & = \frac{1}{16} (\frac{16 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + 4 (- 4 β_{1} d_{2} - \frac{3}{ε + κ} + \frac{2}{ε} + \frac{2}{κ}) - \frac{4 β_{1} \sqrt{\frac{β_{1}}{β_{3}}} (2 ε + κ) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2} κ (ε + κ)} \\ + 8 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{3} ε^{2}}) . \end{aligned}$ (40)

(41) $\begin{aligned} u_{21} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + (4 ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ + \frac{2 β_{1} (2 ε + κ) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2} κ (ε + κ)} - 4 (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ β_{2} d_{2} - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{1} ε^{2}}) . \end{aligned}$ (41)

4. Results and discussion

Many researchers determined different types of solutions of KS equation by applying different techniques. In this recent work, we have found new and more general exact travelling and solitary wave solutions, the important thing in this study is the trial solution of Equation (Equation8(8) $\begin{aligned} U (ζ) & = \sum_{i = 0}^{N} a_{i} Ψ (ζ)^{i} + \sum_{i = - 1}^{- N} b_{- i} Ψ (ζ)^{i} \\ + \sum_{i = 2}^{N} c_{2} Ψ (ζ)^{i - 2} Ψ^{'} (ζ) + \sum_{i = 1}^{N} d_{i} {(\frac{Ψ^{'} (ζ)}{Ψ (ζ)})}^{i} . \end{aligned}$ (8) ) uses the range of four parameters that have different structures. The values of constant parameters $a_{i}, b_{i}, c_{i}, d_{i},$ are collected by applying the Mathematica, then Equation (Equation9(9) $\begin{aligned} (Ψ^{'} (ζ))^{2} & = β_{1} Ψ^{2} (ζ) + β_{2} Ψ^{3} (ζ) + β_{3} Ψ^{4} (ζ); \\ Ψ^{''} (ζ) & = β_{1} Ψ (ζ) + \frac{3}{2} β_{2} Ψ^{2} (ζ) + 2 β_{3} Ψ^{3} (ζ) . \end{aligned}$ (9) ) has different types of solutions which are hyperbolic, rational and trigonometric functions. As a result, are obtained new families of exact travelling and solitary wave solutions with the help of this powerful technique. Now we discuss the differences and similarities of our new obtained solutions with already that have been found in the past literature by using different techniques.

In previous literature, many authors have been determined various types of solutions of KS equation such as elliptic, trigonometric, hyperbolic, rational functions including dark, bright, kink, anti-kink, periodic wave solutions with the help of modified truncated expansion method, dynamical system, bifurcation technique, backlund transformation, lie symmetry analysis, F-expansion method, improved subequation method, the $(\frac{G^{'}}{G})$ -expansion method [Citation5–16]. But our obtained solutions have different structures in the shape of dark, bright, singular, combined, kink wave solitons, periodic solitary wave, traveling wave (see Figures ).

Figure 1. (a) Dark soliton solution for Equation (Equation15(15) $\begin{aligned} u_{1} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (15) ) and (b) bright soliton solution for Equation (Equation16(16) $\begin{aligned} u_{2} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (16) ).

Figure 1. (a) Dark soliton solution for Equation (Equation15(15) u1(x,t)=β1d2ϵcoth12β1x−λt+ζ0+1+β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12−b1β1ϵcoth12β1x−λt+ζ0+1β2−β12β3d2ϵcoth12β1x−λt+ζ0+12β22−b12β22κ2+4β1β1−b12β3κ2−b1β2κ+2β1β1d2κ+12β1κ.(15) ) and (b) bright soliton solution for Equation (Equation16(16) u2(x,t)=144β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−2b1β1β3ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−β1d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12+2β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−2β22b12κ2−4β1β3b12κ2+4β12−b1β2κ+2β1β1d2κ+1β1κ.(16) ).

Figure 2. (a) Bright soliton solution for Equation (Equation17(17) $\begin{aligned} u_{3} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (17) ), and (b) dark soliton solution for Equation (Equation19(19) $\begin{aligned} u_{4} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (19) ).

Figure 2. (a) Bright soliton solution for Equation (Equation17(17) u3(x,t)=b1−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+ϵ2ηp2+1coshβ1x−λt+ζ0+1−psinhβ1x−λt+ζ02β1d2ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12−b12β22κ2+4β1β1−b12β3κ2−b1β2κ+2β1β1d2κ+12β1κ.(17) ), and (b) dark soliton solution for Equation (Equation19(19) u4(x,t)=β1d2ϵcoth12β1x−λt+ζ0+1+β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12−b1β1ϵcoth12β1x−λt+ζ0+1β2−β12β3d2ϵcoth12β1x−λt+ζ0+12β22+b12β22κ2+4β1β1−b12β3κ2+b1β2κ−2β1β1d2κ+12β1κ.(19) ).

Figure 3. (a) Combined dark–bright soliton solutions for Equation (Equation20(20) $\begin{aligned} u_{5} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ β_{1} d_{2} (- {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}) - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (20) ) and (b) bright soliton solutions for Equation (Equation21(21) $\begin{aligned} u_{6} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (21) ).

Figure 3. (a) Combined dark–bright soliton solutions for Equation (Equation20(20) u5x,t=144β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02β1d2−ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12−2b1β1β3ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1+2β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1+2β22b12κ2−4β1β3b12κ2+4β12+b1β2κ−2β1β1d2κ+1β1κ.(20) ) and (b) bright soliton solutions for Equation (Equation21(21) u6(x,t)=b1−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+ϵ2ηp2+1coshβ1x−λt+ζ0+1−psinhβ1x−λt+ζ02β1d2ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12+b12β22κ2+4β1β1−b12β3κ2+b1β2κ−2β1β1d2κ+12β1κ.(21) ).

Figure 4. (a) Dark soliton solution for Equation (Equation23(23) $\begin{aligned} u_{7} (x, t) & = \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} + β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{2 β_{1} (A - 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2} κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ - \frac{β_{3} β_{1}^{2} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (23) ) and (b) Kink wave soliton solution for Equation (Equation24(24) $\begin{aligned} u_{8} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (A - 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (24) ).

Figure 4. (a) Dark soliton solution for Equation (Equation23(23) u7(x,t)=β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12+β1d2ϵcoth12β1x−λt+ζ0+1−2β1A−2β1β2κ(ε+κ)ϵcoth12β1x−λt+ζ0+1β2κβ22ε2−4κ2−4β1β3ε2−β3β12d2ϵcoth12β1x−λt+ζ0+12β22+−Aβ2(ε−2κ)−4β3β12ε2κβ1d2ε−1+β22β1εκ(ε−2κ)β1d2(ε+2κ)+1β1εκ4β1β3ε2−β22ε2−4κ2.(23) ) and (b) Kink wave soliton solution for Equation (Equation24(24) u8(x,t)=β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−14β1d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12+12β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−β1β3A−2β1β2κ(ε+κ)ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1κβ22ε2−4κ2−4β1β3ε2+−Aβ2(ε−2κ)−4β3β12ε2κβ1d2ε−1+β22β1εκ(ε−2κ)β1d2(ε+2κ)+1β1εκ4β1β3ε2−β22ε2−4κ2.(24) ).

Figure 5. (a,b) Solitary wave solutions for Equation (Equation25(25) $\begin{aligned} u_{9} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{2 (A - 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (β_{2}^{2} (4 κ^{2} - ε^{2}) + 4 β_{1} β_{3} ε^{2})} . \end{aligned}$ (25) ) and Equation (Equation27(27) $\begin{aligned} u_{10} (x, t) & = \frac{1}{4} β_{1} d_{2} ϵ (4 \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + \frac{ϵ c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}) \\ + \frac{β_{2} (\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} - 2 β_{1}) + b_{1} κ (β_{2}^{2} - 2 β_{1}^{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1))}{2 β_{1} β_{2} κ} . \end{aligned}$ (27) ).

Figure 5. (a,b) Solitary wave solutions for Equation (Equation25(25) u9(x,t)=β1d2ϵ2ηp2+1coshβ1x−λt+ζ0−psinhβ1x−λt+ζ0+12ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12+2A−2β1β2κ(ε+κ)−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1κβ22ε2−4κ2−4β1β3ε2+−Aβ2(ε−2κ)−4β3β12ε2κβ1d2ε−1+β22β1εκ(ε−2κ)β1d2(ε+2κ)+1β1εκβ224κ2−ε2+4β1β3ε2.(25) ) and Equation (Equation27(27) u10(x,t)=14β1d2ϵ4coth12β1x−λt+ζ0+ϵcsch12β1x−λt+ζ04ϵcoth12β1x−λt+ζ0+12+β2b12β22κ2+4β12−2β1+b1κβ22−2β12ϵcoth12β1x−λt+ζ0+12β1β2κ.(27) ).

Figure 6. (a,b) Solitary traveling wave solutions for Equation (Equation28(28) $\begin{aligned} u_{11} (x, t) & = \frac{1}{2} (\frac{2 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{β_{1} κ} - b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2}) . \end{aligned}$ (28) ) and Equation (Equation29(29) $\begin{aligned} u_{12} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} + β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) . \end{aligned}$ (29) ).

Figure 6. (a,b) Solitary traveling wave solutions for Equation (Equation28(28) u11(x,t)=122β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02+b12β22κ2+4β12+b1β2κ−2β1β1d2κ+1β1κ−b1β1β3ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1+ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1β2β1β3d2.(28) ) and Equation (Equation29(29) u12(x,t)=b1−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+ϵ2ηp2+1coshβ1x−λt+ζ0+1−psinhβ1x−λt+ζ02β1d2ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−b12β22κ2+4β12+b1β2κ−2β1β1d2κ+12β1κ+β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1.(29) ).

Figure 7. (a) Solitary wave solution for Equation (Equation31(31) $\begin{aligned} u_{13} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{2 β_{1} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{3} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (31) ) and (b) solitary traveling wave solution for Equation (Equation32(32) $\begin{aligned} u_{14} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} \\ + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} . \end{aligned}$ (32) ).

Figure 7. (a) Solitary wave solution for Equation (Equation31(31) u13(x,t)=β1d2ϵcoth12β1x−λt+ζ0+1+β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12−2β1β12β22ε2κ(2ε+5κ)+2β1β2κ(ε+κ)ϵcoth12β1x−λt+ζ0+1β234κ3−ε2κ−β12β22ε2κ(2ε+5κ)+β1β2εκβ1d2(ε+2κ)+1β1β2εκ(ε+2κ).(31) ) and (b) solitary traveling wave solution for Equation (Equation32(32) u14(x,t)=β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β12β22ε2κ(2ε+5κ)+β1β2εκβ1d2(ε+2κ)+1β1β2εκ(ε+2κ)+12β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−β1β3β12β22ε2κ(2ε+5κ)+2β1β2κ(ε+κ)ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1β224κ3−ε2κ.(32) ).

Figure 8. (a) Solitary traveling wave solution for Equation (Equation33(33) $\begin{aligned} u_{15} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} (η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] \\ {+ ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) β_{2} d_{2} (- (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)) \\ + \frac{2 (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (33) ) and (b) bright soliton solution for Equation (Equation36(36) $\begin{aligned} u_{17} (x, t) & = a_{0} + \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) . \end{aligned}$ (36) ).

Figure 8. (a) Solitary traveling wave solution for Equation (Equation33(33) u15(x,t)=β1d2ϵ2ηp2+1coshβ1x−λt+ζ0−psinhβ1x−λt+ζ0+12ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02)β2d2−−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+2β12β22ε2κ(2ε+5κ)+2β1β2κ(ε+κ)−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1β224κ3−ε2κ−β12β22ε2κ(2ε+5κ)+β1β2εκβ1d2(ε+2κ)+1β1β2εκ(ε+2κ).(33) ) and (b) bright soliton solution for Equation (Equation36(36) u17(x,t)=a0+β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−14β1d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12+12β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1.(36) ).

Figure 9. (a) Solitary wave solutions for Equation (Equation37(37) $\begin{aligned} u_{18} (x, t) & = a_{0} + β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} . \end{aligned}$ (37) ) and (b) bright soliton solution for Equation (Equation39(39) $\begin{aligned} u_{19} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + 4 β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{β_{1} d_{2} (ε^{2} - 4 κ^{2}) {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{ε^{2}} \\ + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} - \frac{2 β_{1}^{2} (2 ε + κ) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{2} κ (ε + κ)}) . \end{aligned}$ (39) ).

Figure 9. (a) Solitary wave solutions for Equation (Equation37(37) u18(x,t)=a0+β1d2ϵ2ηp2+1coshβ1x−λt+ζ0−psinhβ1x−λt+ζ0+12ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12.(37) ) and (b) bright soliton solution for Equation (Equation39(39) u19(x,t)=142ε+2κ−3ε+κ−4β1d2+4β1d2ϵcoth12β1x−λt+ζ0+1−β1d2ε2−4κ2ϵcoth12β1x−λt+ζ0+12ε2+β1d2ϵ2csch12β1x−λt+ζ04ϵcoth12β1x−λt+ζ0+12−2β12(2ε+κ)ϵcoth12β1x−λt+ζ0+1β22κ(ε+κ).(39) ).

Figure 10. (a,b) Solitary wave solutions for Equation (Equation40(40) $\begin{aligned} u_{20} (x, t) & = \frac{1}{16} (\frac{16 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + 4 (- 4 β_{1} d_{2} - \frac{3}{ε + κ} + \frac{2}{ε} + \frac{2}{κ}) - \frac{4 β_{1} \sqrt{\frac{β_{1}}{β_{3}}} (2 ε + κ) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2} κ (ε + κ)} \\ + 8 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{3} ε^{2}}) . \end{aligned}$ (40) ) and Equation (Equation41(41) $\begin{aligned} u_{21} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + (4 ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ + \frac{2 β_{1} (2 ε + κ) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2} κ (ε + κ)} - 4 (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ β_{2} d_{2} - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{1} ε^{2}}) . \end{aligned}$ (41) ).

Figure 10. (a,b) Solitary wave solutions for Equation (Equation40(40) u20(x,t)=11616β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02+4−4β1d2−3ε+κ+2ε+2κ−4β1β1β3(2ε+κ)ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1β2κ(ε+κ)+8β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−β22d2ε2−4κ2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12β3ε2.(40) ) and Equation (Equation41(41) u21(x,t)=142ε+2κ−3ε+κ−4β1d2+4ϵ2ηp2+1coshβ1x−λt+ζ0+1−psinhβ1x−λt+ζ02β1d2ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02+2β1(2ε+κ)−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1β2κ(ε+κ)−4−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1β2d2−β22d2ε2−4κ2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12β1ε2.(41) ).

In Figure (a), dark soliton solution for Equation (Equation15(15) $\begin{aligned} u_{1} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (15) ) and (b) bright soliton solution for Equation (Equation16(16) $\begin{aligned} u_{2} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (16) ), with these parameters values $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 8, η = 2, ζ_{0} = 0.3, λ = 4, κ = 0.4, b_{1} = 1.5, d_{2} = 2$ . In Figure , 3D (a) bright soliton solution for Equation (Equation17(17) $\begin{aligned} u_{3} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (17) ), and (b) dark soliton solution for Equation (Equation19(19) $\begin{aligned} u_{4} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (19) ), with these parameters values $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = 8, ζ_{0} = 0.3, λ = 4, κ = 0.4, b_{1} = 1.5, d_{2} = 2, p = 2$ and $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = 8, ζ_{0} = 0.3, λ = 4, κ = 1.4, b_{1} = 1.5, d_{2} = 2$ , respectively. In Figure (a), combined dark–bright soliton solutions for Equation (Equation20(20) $\begin{aligned} u_{5} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ β_{1} d_{2} (- {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}) - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (20) ) and (b) bright soliton solutions for Equation (Equation21(21) $\begin{aligned} u_{6} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (21) ), with these parameters values $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 8, η = 2, ζ_{0} = 0.3, λ = 4, κ = 1.4, b_{1} = 1.5, d_{2} = 2$ and $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 10, η = 2, ζ_{0} = 0.3, λ = 4, κ = 0.4, b_{1} = 1.5, d_{2} = 2, p = 0.5$ , respectively. In Figure (a), dark soliton solution for Equation (Equation23(23) $\begin{aligned} u_{7} (x, t) & = \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} + β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{2 β_{1} (A - 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2} κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ - \frac{β_{3} β_{1}^{2} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (23) ) and (b) Kink wave soliton solution for Equation (Equation24(24) $\begin{aligned} u_{8} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (A - 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (24) ), with these parameters values $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 8, η = 2, ζ_{0} = 0.3, λ = 4, ε = 0.5, κ = 1.4, d_{2} = 2, A = 1$ and $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 8, η = 2, ζ_{0} = 0.3, λ = 4, ε = 2.3 κ = 1.4, d_{2} = 2, A = 2$ , respectively. In Figure (a,b), solitary wave solutions for Equation (Equation25(25) $\begin{aligned} u_{9} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{2 (A - 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (β_{2}^{2} (4 κ^{2} - ε^{2}) + 4 β_{1} β_{3} ε^{2})} . \end{aligned}$ (25) ) and Equation (Equation27(27) $\begin{aligned} u_{10} (x, t) & = \frac{1}{4} β_{1} d_{2} ϵ (4 \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + \frac{ϵ c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}) \\ + \frac{β_{2} (\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} - 2 β_{1}) + b_{1} κ (β_{2}^{2} - 2 β_{1}^{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1))}{2 β_{1} β_{2} κ} . \end{aligned}$ (27) ), with these parameters values $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = 8, ζ_{0} = 0.3, λ = 4, ε = 2.5, κ = 1.4, d_{2} = 2, p = 2, A = 4$ and $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = 8, ζ_{0} = 0.3, λ = 4, κ = 1.4, b_{1} = 1, d_{2} = 2$ , respectively. In Figure (a,b), solitary traveling wave solutions for Equation (Equation28(28) $\begin{aligned} u_{11} (x, t) & = \frac{1}{2} (\frac{2 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{β_{1} κ} - b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2}) . \end{aligned}$ (28) ) and Equation (Equation29(29) $\begin{aligned} u_{12} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} + β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) . \end{aligned}$ (29) ), with these parameters values $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, κ = 1.4, b_{1} = 2, d_{2} = 2$ and $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, κ = 1.4, b_{1} = 2, d_{2} = 2, p = 2$ , respectively. In Figure (a), solitary wave solution for Equation (Equation31(31) $\begin{aligned} u_{13} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{2 β_{1} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{3} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (31) ) and (b) Solitary traveling wave solution for Equation (Equation32(32) $\begin{aligned} u_{14} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} \\ + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} . \end{aligned}$ (32) ), with these parameters values $β_{1} = 2, β_{2} = 4, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, ε = 2.5, κ = 1.4, d_{2} = 2$ and $β_{1} = 2, β_{2} = 4, ϵ = 18, η = - 8, ζ_{0} = 0.3, ε = 2.5, λ = 4, κ = 1.4, d_{2} = 2$ , respectively. In Figure (a), solitary traveling wave solution for Equation (Equation33(33) $\begin{aligned} u_{15} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} (η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] \\ {+ ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) β_{2} d_{2} (- (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)) \\ + \frac{2 (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (33) ) and (b) bright soliton solution for Equation (Equation36(36) $\begin{aligned} u_{17} (x, t) & = a_{0} + \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) . \end{aligned}$ (36) ), with these parameters values $β_{1} = 2, β_{2} = 4, ϵ = 18, η = - 6, ζ_{0} = 0.3, λ = 4, ε = 2.5, κ = 1.4, d_{2} = 2, p = 2$ and $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, a_{0} = 2, d_{2} = 1.5$ , respectively. In Figure (a), solitary wave solutions for Equation (Equation37(37) $\begin{aligned} u_{18} (x, t) & = a_{0} + β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} . \end{aligned}$ (37) ) and (b) bright soliton solution for Equation (Equation39(39) $\begin{aligned} u_{19} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + 4 β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{β_{1} d_{2} (ε^{2} - 4 κ^{2}) {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{ε^{2}} \\ + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} - \frac{2 β_{1}^{2} (2 ε + κ) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{2} κ (ε + κ)}) . \end{aligned}$ (39) ), with these parameters values $β_{1} = 2, β_{2} = 4, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, a_{0} = 2, d_{2} = 1.5, p = 0.5$ and $β_{1} = 2, β_{2} = 4, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, ε = 1.5, κ = 1.4, d_{2} = 2$ , respectively. In Figure (a,b), solitary wave solutions for Equation (Equation40(40) $\begin{aligned} u_{20} (x, t) & = \frac{1}{16} (\frac{16 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + 4 (- 4 β_{1} d_{2} - \frac{3}{ε + κ} + \frac{2}{ε} + \frac{2}{κ}) - \frac{4 β_{1} \sqrt{\frac{β_{1}}{β_{3}}} (2 ε + κ) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2} κ (ε + κ)} \\ + 8 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{3} ε^{2}}) . \end{aligned}$ (40) ) and Equation (Equation41(41) $\begin{aligned} u_{21} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + (4 ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ + \frac{2 β_{1} (2 ε + κ) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2} κ (ε + κ)} - 4 (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ β_{2} d_{2} - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{1} ε^{2}}) . \end{aligned}$ (41) ), with these parameters values $β_{1} = 2, β_{2} = 4, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, ε = 1.5, κ = 2.4, d_{2} = 2$ and $β_{1} = 2, β_{2} = 4, β_{3} = 2, ϵ = 18, η = - 8, ζ_{0} = 0.3, λ = 4, ε = 1.5, κ = 2.4, p = 1, d_{2} = 2$ , respectively.

We can conclude that from the above comparison and detailed discussion, our obtained solutions are new and more generally which have not formulated before by other techniques. It is proved that our modified technique is fruitful, reliable, straight forward and effective to investigate other nonlinear evolution equations.

5. Conclusion

We successfully constructed some new exact travelling and solitary wave solutions of nonlinear KS equation by applying the modified mathematical technique. Our solutions are different and new from other researcher found by using the different techniques before this work. Our new exact solutions obtained in the shape of dark solitons, bright solitons, travelling wave, solitary wave and periodic wave. These new solutions are more useful in the study of quantum plasma, optical fibres, dynamics of solitons, dynamics of fluid, problems of biomedical, mathematical physics, engineering and many other branches. The physical structure of new solutions shows the effectiveness and power of this technique. This research work completed by using the Mathematica. We can also apply this technique on other nonlinear evolution equations involves in optical fibre, Geo physics, mathematical physics, plasma physics, fluid dynamics, hydrodynamics, mechanics, mathematical biology, field of engineering and many other applied sciences.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Aly R. Seadawy http://orcid.org/0000-0002-7412-4773

Mujahid Iqbal http://orcid.org/0000-0003-1234-4321

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Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity

Abstract

1. Introduction

2. Description of proposed method

3. KS equation

4. Results and discussion

5. Conclusion

Disclosure statement

References

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Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity

Abstract

1. Introduction

2. Description of proposed method

3. KS equation

4. Results and discussion

5. Conclusion

Disclosure statement

ORCID

References

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