Soliton wave solutions of ion-acoustic waves a cold plasma with negative ions

Abstract

This article investigates the computational complex wave solutions of the modified Korteweg–de Vries equation combined with an adverse order of the Korteweg–de Vries model. This model was derived in 2017, where the recursion and inverse recursion operators are employed to select the integrable merged MKdV with a negative MKdV model. This integrable property is tested utilizing the Painlevé property. Verosky gave the description and properties of the opposing order recursion operator. We handle this model by implementing eleven contemporary techniques. We obtain a novel formula of complex solitary wave solutions for this model. Complex solitary wave solutions describe wave propagation, and it is also considered more mathematically concise tools to explain more details about the physical properties of models. The main goals of our paper are a comparison between these methods and introducing a novel modified method. All solutions are checked for accuracy by putting them back into the model via two different software (Maple 17 and Mathematica 12).

Keywords

merged MKdV with negative MKdV equation recursion operator solitary wave solutions traveling solutions

AMS classification: 02.30.Jr; 02.30.Hq; 02.60.-x

Introduction

The recursion operator is regarded as a fundamental instrument in nonlinear wave theory.^1–3 This operator is based on the expansion of a power series across fields and the representation of momentum. Calculating the recursion operator using the Hamiltonian is one of the techniques. The recursion operator has many features, including the following:

1. It allows writing the families of equations integrable by a given spectral issue in a compact design.

2. It supports picking up the wholesome family of the equation.

3. It is accompanied by the Hamiltonian approach of integrable equations.

4. It provides the generating operator for the family of Hamiltonian structures.

For an example of using recursion operator:

This operator is used to the KdV equation to derive all of its families and determine the family’s integrability. It is written as follows

\frac{\partial u (x, t)}{\partial t} + \partial L^{n} u = 0,

(1.1)

where [L = ∂² + 2 u + 2 ∂⁻¹u∂]. Also, the integrability tested by using Hamiltonian structures that has the next formula

{[F, H]}_{n} = \int_{- \infty}^{\infty} d x \frac{δ F}{δ u (x)} \partial L^{n} \frac{δ H}{δ u},

(1.2)

where [n = 0, ±1, ± 2, …]. As previously stated, there are two types of recursion operators: weak recursion operators and robust recursion operators. This moniker refers to the vulnerable recursion operators that enable iteratively generating the infinite family of Hamiltonian equations. Simultaneously, the strong recursion operator generates an endless family of equations from a given equation.

Recursion operators are also used on the modified KdV equation, and it has the following formula⁴

φ (ν) = \partial_{x}^{2} + p_{3} ν^{2} + p_{3} ν_{x} \partial_{x}^{- 1} (. ν) .

(1.3)

The modified KdV equation describes the nonlinear wave propagation in a variety of polarity-symmetric physical formations.^5–7 Additionally, it defined nonlinear ion-acoustic waves as a collisionless plasma composed of adiabatic warm ions, a weakly relativistic electron beam, and non-isothermal electrons.^8,9 The negative order equation is denoted by ref.¹⁰

φ (ν_{t}) = - ν_{x} .

(1.4)

Now, combining the modified Korteweg–de Vries equation combined with an adverse order of the same models, this model has the nested formula¹¹

ν_{t} + p_{1} ν^{2} ν_{x} + ν_{x x x} + ν_{x x t} + p_{3} ν^{2} ν_{t} + p_{3} ν_{x} \partial^{- 1} (ν ν_{t}) = 0.

(1.5)

Differentiating equation (1.5) obtains

\begin{array}{l} ν_{x t} + p_{1} ν^{2} ν_{x x} + p_{2} ν ν_{x}^{2} + ν_{x x x x} + ν_{x x x t} + p_{3} ν^{2} ν_{x t} + p_{2} ν ν_{x} ν_{t} - \frac{ν_{x x}}{ν_{x}} \\ (ν_{t} + p_{1} ν^{2} ν_{x} + ν_{x x x} + ν_{x x t} + p_{3} ν^{2} ν_{t}) = 0. where, p_{2} = 2 p_{1} . \end{array}

(1.6)

Numerous approaches are generated by analyzing nonlinear partial differential equations with an integer order, a fractional order, a complementary order, or an adverse order. For a more detailed discussion of these techniques, see: Refs.^12–27.

This article combines the modified Korteweg–de Vries equation with an adverse order of the same models to generate novel traveling wave solutions using over eleven recent analytical techniques.^28–39

The remainder of this article is organized as follows. In Application, we applied some recent methods on the combining equation [Exp $(- ϕ (θ))$ -expansion method, an improved $(G^{'} / G)$ -expansion method, direct algebraic method, the generalized Kudryashov method, an extended tanh-function method, the simplest equation method, an extended Fan-expansion method, Riccati expansion method, an improved F-Expansion method, generalization Sinh-Gorden expansion method, and modified Khater method]. In Result and discussion, we represent our obtained solutions and show the convergence between our solutions and those obtained with different methods. In Conclusion, we demonstrate the conclusion of our paper.

Application

Using the next wave transformation $[ν (x, t) = Q (ϑ), ϑ = l_{2} t + l_{1} x]$ transforms the nonlinear partial differential equation form of equation (1.6) to the following nonlinear ordinary differential equation.⁴⁰

l_{1}^{2} (l_{1} + l_{2}) (l_{1}^{2} (Q^{(4)} Q^{'} - Q^{(3)} Q^{″}) + p_{2} Q {(Q^{'})}^{3}) = 0.

(2.1)

Evaluating the value of balance between the terms of equation (2.1) by using the next formula of balance rule

{\begin{cases} D [\frac{d^{ς} Q (ϑ)}{d ϑ^{ς}}] \Rightarrow N + ς, \\ D [Q^{ς} {(\frac{d^{ε} Q (ϑ)}{d ϑ^{ε}})}^{𝔖}] \Rightarrow N + ς + 𝔖 (N + ε), \end{cases}

(2.2)

and the following auxiliary equations of the abovementioned computational schemes, respectively

\begin{array}{l} [ϕ^{'} (ϑ) = σ e^{ϕ (ϑ)} + \frac{1}{e^{ϕ (ϑ)}} + ρ, \frac{G^{''} (ϑ)}{G (ϑ)} = - μ, f^{'} (ϑ) = l_{3} f {(ϑ)}^{2} + l_{2} f (ϑ) + l_{1}, Q^{'} (ϑ) = Q {(ϑ)}^{2} - Q (ϑ), ϕ^{'} (ϑ) = d + ϕ {(ϑ)}^{2}, f^{'} (ϑ) \\ = c_{2} f {(ϑ)}^{2} + c_{1} f (ϑ), ϕ^{'} {(ϑ)}^{2} = ζ_{3}^{2} ϕ {(ϑ)}^{4} + 2 ζ_{2} ζ_{3} ϕ {(ϑ)}^{3} + (ζ_{2}^{2} + 2 ζ_{1} ζ_{3}) ϕ {(ϑ)}^{2} + 2 ζ_{1} ζ_{2} ϕ (ϑ) + ζ_{1}^{2}, Q^{'} (ϑ) \\ = ς_{3} Q {(ϑ)}^{2} + ς_{2} Q (ϑ) + ς_{1}, ϕ^{'} (ϑ) = ϕ {(ϑ)}^{2} + ϱ, f^{'} (ξ) \to \frac{δ k^{f (ϑ)} + ϱ k^{- f (ϑ)} + χ}{ln (k)}] \end{array}

, get N = 1. Thus, the general solutions of equation (2.1) based on the considered computational techniques are given by

Q (ϑ) = {\begin{cases} \sum_{i = - N}^{N} a_{i} e^{- i ϕ (ϑ)} = a_{- 1} e^{ϕ (ϑ)} + a_{1} e^{- ϕ (ϑ)} + a_{0}, \\ \sum_{i = 1}^{N} (\frac{a_{i} G^{'} (ϑ)}{G (ϑ)} + b_{i} {(\frac{G^{'} ϑ)}{G (ϑ)})}^{i - 1} \sqrt{σ (\frac{{(\frac{G^{'} (ϑ)}{G (ϑ)})}^{2}}{μ} + 1)}) + a_{0} = \frac{a_{1} G^{'} (ϑ)}{G (ϑ)} + a_{0} + b_{1} \sqrt{σ (\frac{G^{'} {(ϑ)}^{2}}{μ G {(ϑ)}^{2}} + 1)}, \\ \sum_{i = - N}^{N} a_{i} f {(ϑ)}^{i} = \frac{a_{- 1}}{f (ϑ)} + a_{1} f (ϑ) + a_{0}, \\ \frac{\sum_{i = 0}^{m} a_{i} Q {(ϑ)}^{i}}{\sum_{j = 0}^{N} b_{j} Q {(ϑ)}^{j}} = \frac{a_{2} Q {(ϑ)}^{2} + a_{1} Q (ϑ) + a_{0}}{b_{1} Q (ϑ) + b_{0}}, \\ \sum_{i = - N}^{N} a_{i} ϕ {(ϑ)}^{i} = \frac{a_{- 1}}{ϕ (ϑ)} + a_{1} ϕ (ϑ) + a_{0}, \\ \sum_{i = 0}^{n} a_{i} f {(ϑ)}^{i} = a_{1} f (ϑ) + a_{0}, \\ \sum_{i = 0}^{N} a_{i} ϕ {(ϑ)}^{i} = a_{1} ϕ (ϑ) + a_{0}, \\ \sum_{i = 1}^{N} a_{i} Q {(ϑ)}^{i} + \sum_{i = 1}^{N} b_{i} Q {(ϑ)}^{- i} + a_{0} = a_{1} Q (ϑ) + a_{0} + \frac{b_{1}}{Q (ϑ)}, \\ \sum_{i = - N}^{N} a_{i} {(μ + ϕ (ϑ))}^{i} = \frac{a_{- 1}}{μ + ϕ (ϑ)} + a_{1} (μ + ϕ (ϑ)) + a_{0}, \\ \sum_{i = 1}^{N} \cosh^{i - 1} (W) [B_{i} \sinh (W) + A_{i} \cosh (W)] + A_{0} = A_{1} \cosh (W) + A_{0} + B_{1} \sinh (W), \\ \sum_{i = 1}^{N} a_{i} k^{i f (ϑ)} + \sum_{i = 1}^{N} b_{i} k^{- i f (ϑ)} + a_{0} = a_{1} k^{f (ϑ)} + a_{0} + b_{1} k^{- f (ϑ)} \end{cases}

(2.3)

where a_i, b_i are arbitrary constants to be determined later. Handling equation (2.1) with equation (2.3) along its auxiliary equations gives the value of the abovementioned parameters and explicit solutions of equation (1.5) as follows:

Exp (− ϕ(ϑ)) -expansion method

Employing Exp (− ϕ(ϑ)) -expansion method gives the following sets of the abovementioned parameters:

Set I

a_{- 1} \to a_{1} σ, a_{0} \to \frac{a_{1} ρ}{2}, l_{1} \to \frac{i a_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where (p_{2} < 0, i = \sqrt{- 1}) .

Set II

a_{0} \to \frac{a_{- 1} ρ}{2 σ}, a_{1} \to 0, l_{1} \to \frac{i a_{- 1} \sqrt{p_{2}}}{2 \sqrt{3} σ}, where (p_{2} < 0, i = \sqrt{- 1}) .

Set III

a_{- 1} \to 0, a_{0} \to \frac{a_{1} ρ}{2}, l_{1} \to \frac{i a_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where (p_{2} < 0, i = \sqrt{- 1}) .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For ρ² − 4σ > 0, σ ≠ 0, the solutions are given by

\begin{array}{l} ν_{I, 1} (x, t) = \frac{1}{2} a_{1} (- \frac{4 σ}{ρ + \sqrt{ρ^{2} - 4 σ} tanh (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))} \\ - \sqrt{ρ^{2} - 4 σ} tanh (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))), \end{array}

(2.4)

\begin{array}{l} ν_{I, 2} (x, t) = \frac{1}{2} a_{1} ((- \sqrt{ρ^{2} - 4 σ}) coth (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧)) \\ - \frac{4 σ}{ρ + \sqrt{ρ^{2} - 4 σ} coth (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))} \end{array}

(2.5)

ν_{II, 1} (x, t) = \frac{- 1}{2 σ} a_{- 1} \sqrt{ρ^{2} - 4 σ} \tanh (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧)),

(2.6)

ν_{II, 2} (x, t) = \frac{- 1}{2 σ} a_{- 1} \sqrt{ρ^{2} - 4 σ} coth (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧)),

(2.7)

ν_{III, 1} (x, t) = \frac{1}{2} a_{1} (ρ - \frac{4 σ}{ρ + \sqrt{ρ^{2} - 4 σ} \tanh (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))}),

(2.8)

ν_{III, 2} (x, t) = \frac{1}{2} a_{1} (ρ - \frac{4 σ}{ρ + \sqrt{ρ^{2} - 4 σ} coth (\frac{1}{2} \sqrt{ρ^{2} - 4 σ} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))}) .

(2.9)

For ρ² − 4σ > 0, σ = 0, the solutions are given by

ν_{I, 3} (x, t) = \frac{1}{2} a_{1} ρ coth (\frac{1}{12} ρ (i \sqrt{3} a_{1} \sqrt{p_{2}} x + 6 l_{2} t + 6 ℧)),

(2.10)

ν_{III, 3} (x, t) = \frac{1}{2} a_{1} ρ coth (\frac{1}{12} ρ (i \sqrt{3} a_{1} \sqrt{p_{2}} x + 6 l_{2} t + 6 ℧)) .

(2.11)

For ρ² − 4σ = 0, σ≐0, ρ≐0, the solutions are given by

ν_{I, 4} (x, t) = \frac{2 a_{1}}{ρ^{2}} (\frac{3 ρ^{3}}{i \sqrt{3} a_{1} \sqrt{p_{2}} ρ x + 6 l_{2} ρ t + 6 ρ ℧ + 12} - \frac{12 σ}{i \sqrt{3} a_{1} \sqrt{p_{2}} x + 6 l_{2} t + 6 ℧} - ρ σ),

(2.12)

ν_{II, 3} (x, t) = \frac{a_{- 1}}{2 ρ^{2}} (- \frac{8}{\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧} + \frac{ρ^{3}}{σ} - 4 ρ),

(2.13)

ν_{III, 4} (x, t) = \frac{6 a_{1} ρ}{i \sqrt{3} a_{1} \sqrt{p_{2}} ρ x + 6 l_{2} ρ t + 6 ρ ℧ + 12} .

(2.14)

For ρ² − 4σ = 0, σ = 0, ρ = 0, the solutions are given by

ν_{I, 5} (x, t) = \frac{a_{1}}{\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧},

(2.15)

ν_{III, 5} (x, t) = \frac{a_{1}}{\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧} .

(2.16)

For ρ² − 4σ < 0, σ ≠ 0, the solutions are given by

\begin{array}{l} ν_{I, 6} (x, t) = \frac{1}{2} a_{1} (\sqrt{4 σ - ρ^{2}} \tan (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧)) \\ - \frac{4 σ}{ρ - \sqrt{4 σ - ρ^{2}} \tan (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))}), \end{array}

(2.17)

\begin{array}{l} ν_{I, 7} (x, t) = \frac{1}{2} a_{1} (\sqrt{4 σ - ρ^{2}} \cot (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧)) \\ - \frac{4 σ}{ρ - \sqrt{4 σ - ρ^{2}} \cot (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))}), \end{array}

(2.18)

ν_{II, 4} (x, t) = \frac{a_{- 1} \sqrt{4 σ - ρ^{2}}}{2 σ} \tan (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧)),

(2.19)

ν_{II, 5} (x, t) = \frac{a_{- 1} \sqrt{4 σ - ρ^{2}}}{2 σ} \cot (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧)),

(2.20)

ν_{III, 6} (x, t) = \frac{1}{2} a_{1} (ρ - \frac{4 σ}{ρ - \sqrt{4 σ - ρ^{2}} \tan (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))}),

(2.21)

ν_{III, 7} (x, t) = \frac{1}{2} a_{1} (ρ - \frac{4 σ}{ρ - \sqrt{4 σ - ρ^{2}} \cot (\frac{1}{2} \sqrt{4 σ - ρ^{2}} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + l_{2} t + ℧))}) .

(2.22)

Improved (G^′/G) -expansion method

Employing improved (G^′/G) -expansion method gives the following sets of the abovementioned parameters:

Set I

a_{0} \to 0, a_{1} \to 0, p_{2} \to - \frac{12 μ l_{1}^{2}}{b_{1}^{2} σ} .

Set II

a_{0} \to 0, b_{1} \to 0, p_{2} \to - \frac{12 l_{1}^{2}}{a_{1}^{2}} .

Set III

a_{0} \to 0, b_{1} \to \frac{a_{1} \sqrt{μ}}{\sqrt{σ}}, l_{1} \to \frac{i a_{1} \sqrt{p_{2}}}{\sqrt{3}} where (p_{2} < 0, i = \sqrt{- 1})

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For μ > 0, the solutions are given by

ν_{I, 1} (x, t) = b_{1} \sqrt{\frac{(C_{1}^{2} + C_{2}^{2}) σ}{{(C_{1} \sin (\sqrt{μ} (l_{2} t + l_{1} x)) + C_{2} \cos (\sqrt{μ} (l_{2} t + l_{1} x)))}^{2}}},

(2.23)

ν_{II, 1} (x, t) = \frac{a_{1} \sqrt{μ} (C_{1} - C_{2} \tan (\sqrt{μ} (l_{2} t + l_{1} x)))}{C_{1} \tan (\sqrt{μ} (l_{2} t + l_{1} x)) + C_{2}},

(2.24)

ν_{III, 1} (x, t) = a_{1} \sqrt{μ} (\begin{array}{l} \frac{\sqrt{\frac{(C_{1}^{2} + C_{2}^{2}) σ}{{(C_{1} \sin (\sqrt{μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}})) + C_{2} \cos (\sqrt{μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}})))}^{2}}}}{\sqrt{σ}} \\ + \frac{- C_{2} + \frac{C_{1}^{2} + C_{2}^{2}}{C_{2} + C_{1} \tan (\sqrt{μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}}))}}{C_{1}}) . \end{array}

(2.25)

For μ < 0, the solutions are given by

ν_{I, 2} (x, t) = b_{1} \sqrt{\frac{(C_{1}^{2} - C_{2}^{2}) σ}{{(C_{1} \cos (\sqrt{μ} (l_{2} t + l_{1} x)) + C_{2} sinh (\sqrt{- μ} (l_{2} t + l_{1} x)))}^{2}}},

(2.26)

ν_{II, 2} (x, t) = \frac{a_{1} \sqrt{- μ} (C_{2} \cos (\sqrt{μ} (l_{2} t + l_{1} x)) + C_{1} sinh (\sqrt{- μ} (l_{2} t + l_{1} x)))}{C_{1} \cos (\sqrt{μ} (l_{2} t + l_{1} x)) + C_{2} sinh (\sqrt{- μ} (l_{2} t + l_{1} x))},

(2.27)

\begin{array}{l} ν_{III, 2} (x, t) = a_{1} \sqrt{μ} (\sqrt{\frac{(C_{1}^{2} - C_{2}^{2})}{{(C_{1} \cos (\sqrt{μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}})) + C_{2} sinh (\sqrt{- μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}})))}^{2}}} \\ + \frac{1}{\frac{C_{1} \sqrt{μ}}{C_{2} \sqrt{- μ}} + \tan (\sqrt{μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}}))} \\ - \frac{C_{1} \sin (\sqrt{μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}}))}{C_{1} \cos (\sqrt{μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}})) + C_{2} sinh (\sqrt{- μ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{\sqrt{3}}))}) . \end{array}

(2.28)

Direct algebraic method

Employing the direct algebraic method gives the following sets of the abovementioned parameters:

Set I

a_{0} \to \frac{a_{- 1} h_{2}}{2 h_{1}}, a_{1} \to \frac{a_{- 1} h_{3}}{h_{1}}, p_{2} \to - \frac{12 h_{1}^{2} l_{1}^{2}}{a_{- 1}^{2}} .

Set II

a_{0} \to \frac{a_{- 1} h_{2}}{2 h_{1}}, a_{1} \to 0, p_{2} \to - \frac{12 h_{1}^{2} l_{1}^{2}}{a_{- 1}^{2}} .

Set III

a_{- 1} \to 0, a_{0} \to \frac{a_{1} h_{2}}{2 h_{3}}, p_{2} \to - \frac{12 h_{3}^{2} l_{1}^{2}}{a_{1}^{2}} .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For h₂ = 0, h₁h₃ > 0, the solutions are given by

ν_{I, 1} = \frac{2 a_{- 1} \sqrt{h_{1} h_{3}}}{h_{1}} \csc (2 \sqrt{h_{1} h_{3}} (ζ + l_{2} t + l_{1} x)),

(2.29)

ν_{II, 1} = \frac{a_{- 1} \sqrt{h_{1} h_{3}}}{h_{1}} \cot (\sqrt{h_{1} h_{3}} (ζ + l_{2} t + l_{1} x)),

(2.30)

ν_{II, 2} = \frac{a_{- 1} \sqrt{h_{1} h_{3}}}{h_{1}} \tan (\sqrt{h_{1} h_{3}} (ζ + l_{2} t + l_{1} x)),

(2.31)

ν_{III, 1} = \frac{a_{1} \sqrt{h_{1} h_{3}}}{h_{3}} \tan (\sqrt{h_{1} h_{3}} (ζ + l_{2} t + l_{1} x)),

(2.32)

ν_{III, 2} = \frac{a_{1} \sqrt{h_{1} h_{3}}}{h_{3}} \cot (\sqrt{h_{1} h_{3}} (ζ + l_{2} t + l_{1} x)) .

(2.33)

For h₂ = 0, h₁h₃ < 0, the solutions are given by

ν_{I, 2} = - \frac{2 a_{- 1} \sqrt{- h_{1} h_{3}}}{h_{1}} csch (2 (\sqrt{- h_{1} h_{3}} (l_{2} t + l_{1} x) \mp \frac{log (ζ)}{2})),

(2.34)

ν_{II, 3} = - \frac{a_{- 1} \sqrt{- h_{1} h_{3}}}{h_{1}} \coth (\sqrt{- h_{1} h_{3}} (l_{2} t + l_{1} x) \mp \frac{log (ζ)}{2}),

(2.35)

ν_{II, 4} = - \frac{a_{- 1} \sqrt{- h_{1} h_{3}}}{h_{1}} \tanh (\sqrt{- h_{1} h_{3}} (l_{2} t + l_{1} x) \mp \frac{log (ζ)}{2}),

(2.36)

ν_{III, 3} = \frac{a_{1} \sqrt{- h_{1} h_{3}}}{h_{3}} \tanh (\sqrt{- h_{1} h_{3}} (l_{2} t + l_{1} x) \mp \frac{log (ζ)}{2}),

(2.37)

ν_{III, 4} = \frac{a_{1} \sqrt{- h_{1} h_{3}}}{h_{3}} \coth (\sqrt{- h_{1} h_{3}} (l_{2} t + l_{1} x) \mp \frac{log (ζ)}{2}) .

(2.38)

For h₁ = 0, h₂ > 0, the solutions are given by

ν_{III, 5} = - \frac{a_{1} h_{2}}{h_{3} (h_{3} e^{h_{2} (ζ + l_{2} t + l_{1} x)} - 1)} - \frac{a_{1} h_{2}}{2 h_{3}} .

(2.39)

For h₁ = 0, h₂ < 0, the solutions are given by

ν_{III, 6} = \frac{a_{1}}{h_{3} e^{h_{2} (ζ + l_{2} t + l_{1} x)} + 1} + \frac{a_{1} h_{2}}{2 h_{3}} - a_{1} .

(2.40)

For

4 h_{1} h_{3} > h_{2}^{2}

, the solutions are given by

\begin{array}{l} ν_{I, 3} = \frac{a_{- 1} \sqrt{4 h_{1} h_{3} - h_{2}^{2}}}{2 h_{1}} \tan (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x)) \\ - \frac{2 a_{- 1} h_{3}}{h_{2} - \sqrt{4 h_{1} h_{3} - h_{2}^{2}} \tan (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x))} \end{array}

(2.41)

\begin{array}{l} ν_{I, 4} = \frac{a_{- 1} \sqrt{4 h_{1} h_{3} - h_{2}^{2}}}{2 h_{1}} \cot (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x)) \\ - \frac{2 a_{- 1} h_{3}}{h_{2} - \sqrt{4 h_{1} h_{3} - h_{2}^{2}} \cot (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x))} \end{array}

(2.42)

ν_{II, 5} = \frac{a_{- 1} h_{2}}{2 h_{1}} - \frac{2 a_{- 1} h_{3}}{h_{2} - \sqrt{4 h_{1} h_{3} - h_{2}^{2}} \tan (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x))},

(2.43)

ν_{II, 6} = \frac{a_{- 1} h_{2}}{2 h_{1}} - \frac{2 a_{- 1} h_{3}}{h_{2} - \sqrt{4 h_{1} h_{3} - h_{2}^{2}} \cot (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x))},

(2.44)

ν_{III, 7} = \frac{a_{1} \sqrt{4 h_{1} h_{3} - h_{2}^{2}}}{2 h_{3}} \tan (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x)),

(2.45)

ν_{III, 8} = \frac{a_{1} \sqrt{4 h_{1} h_{3} - h_{2}^{2}}}{2 h_{3}} \cot (\frac{1}{2} \sqrt{4 h_{1} h_{3} - h_{2}^{2}} (ζ + l_{2} t + l_{1} x)) .

(2.46)

Generalized Kudryashov method

Employing the generalized Kudryashov method gives the following sets of the abovementioned parameters:

Set I

a_{0} \to - \frac{a_{2} b_{0}}{2 b_{1}}, a_{1} \to \frac{2 a_{2} b_{0} - a_{2} b_{1}}{2 b_{1}}, p_{2} \to - \frac{12 b_{1}^{2} l_{1}^{2}}{a_{2}^{2}} .

Set II

a_{0} \to \frac{a_{2} b_{0} (2 b_{0} + b_{1})}{2 b_{1}^{2}}, a_{1} \to \frac{2 a_{2} b_{0} - a_{2} b_{1}}{2 b_{1}}, p_{2} \to - \frac{12 b_{1}^{2} l_{1}^{2}}{a_{2}^{2}} .

Set III

a_{0} \to \frac{a_{2}}{2}, a_{1} \to - a_{2}, b_{1} \to - 2 b_{0}, p_{2} \to - \frac{48 b_{0}^{2} l_{1}^{2}}{a_{2}^{2}} .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas

ν_{I} (x, t) = \frac{a_{2}}{2 b_{1}} (\frac{2}{A e^{l_{2} t + l_{1} x} + 1} - 1),

(2.47)

ν_{II} (x, t) = \frac{a_{2}}{2 b_{1}^{2}} (\frac{2 b_{1}}{A e^{l_{2} t + l_{1} x} + 1} - \frac{2 b_{1} (b_{0} + b_{1})}{b_{0} (A e^{l_{2} t + l_{1} x} + 1) + b_{1}} + 2 b_{0} + b_{1}),

(2.48)

ν_{III} (x, t) = \frac{a_{2}}{2 b_{0} (1 - \frac{2}{A^{2} e^{2 l_{2} t + 2 l_{1} x} + 1})} .

(2.49)

Extended tanh-function method

Employing the extended tanh-function gives the following sets of the abovementioned parameters:

Set I

a_{- 1} \to 0, a_{0} \to 0, p_{2} \to - \frac{12 l_{1}^{2}}{a_{1}^{2}} .

Set II

a_{- 1} \to a_{1} d, a_{0} \to 0, p_{2} \to - \frac{12 l_{1}^{2}}{a_{1}^{2}} .

Set III

a_{0} \to 0, a_{1} \to 0, p_{2} \to - \frac{12 d^{2} l_{1}^{2}}{a_{- 1}^{2}}

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For d < 0, the solutions have been evaluated as following

ν_{I, 1} (x, t) = a_{1} \sqrt{d} \tan (\sqrt{d} (l_{2} t + l_{1} x)),

(2.50)

ν_{I, 2} (x, t) = a_{1} (- \sqrt{d}) \cot (\sqrt{d} (l_{2} t + l_{1} x)),

(2.51)

ν_{II, 1} (x, t) = 2 a_{1} \sqrt{d} \csc (2 \sqrt{d} (l_{2} t + l_{1} x)),

(2.52)

ν_{II, 2} (x, t) = - 2 a_{1} \sqrt{d} \csc (2 \sqrt{d} (l_{2} t + l_{1} x)),

(2.53)

ν_{III, 1} (x, t) = \frac{a_{- 1}}{\sqrt{d}} \cot (\sqrt{d} (l_{2} t + l_{1} x)),

(2.54)

ν_{III, 2} (x, t) = - \frac{a_{- 1}}{\sqrt{d}} \tan (\sqrt{d} (l_{2} t + l_{1} x)) .

(2.55)

For d > 0, the solutions have been evaluated as following

ν_{I, 3} (x, t) = a_{1} \sqrt{d} \tan (\sqrt{d} (l_{2} t + l_{1} x)),

(2.56)

ν_{I, 4} (x, t) = a_{1} (- \sqrt{d}) \cot (\sqrt{d} (l_{2} t + l_{1} x)),

(2.57)

ν_{II, 3} (x, t) = 2 a_{1} \sqrt{d} \csc (2 \sqrt{d} (l_{2} t + l_{1} x)),

(2.58)

ν_{II, 4} (x, t) = - 2 a_{1} \sqrt{d} \csc (2 \sqrt{d} (l_{2} t + l_{1} x)),

(2.59)

ν_{III, 3} (x, t) = \frac{a_{- 1}}{\sqrt{d}} \cot (\sqrt{d} (l_{2} t + l_{1} x)),

(2.60)

ν_{III, 4} (x, t) = - \frac{a_{- 1}}{\sqrt{d}} \tan (\sqrt{d} (l_{2} t + l_{1} x)) .

(2.61)

For d = 0, the solutions have been evaluated as following

ν_{I, 5} (x, t) = - \frac{a_{1}}{l_{2} t + l_{1} x},

(2.62)

ν_{II, 5} (x, t) = - \frac{a_{1} (d {(l_{2} t + l_{1} x)}^{2} + 1)}{l_{2} t + l_{1} x} .

(2.63)

Simplest equation method

Employing the simplest equation method gives the following sets of the abovementioned parameters:

Set I For c₂ = − 1

a_{0} \to - \frac{1}{2} a_{1} c_{1}, l_{1} \to \frac{i a_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where (p_{2} < 0, i = \sqrt{- 1}) .

Set I For c₁ = 1, c₂ = − 1

a_{0} \to - \frac{a_{1}}{2}, l_{1} \to \frac{i a_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where (p_{2} < 0, i = \sqrt{- 1}) .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas

ν_{I, 1} (x, t) = \frac{1}{2} a_{1} c_{1} \tanh (\frac{1}{2} c_{1} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + J + l_{2} t)),

(2.64)

ν_{I, 2} (x, t) = - \frac{1}{2} a_{1} c_{1} \tanh (\frac{1}{2} c_{1} (\frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}} + J + l_{2} t)),

(2.65)

ν_{II, 1} (x, t) = \frac{1}{2} a_{1} \tanh (\frac{1}{12} (6 l_{2} t + i \sqrt{3} a_{1} \sqrt{p_{2}} x)) .

(2.66)

Extended fan-expansion method

Employing the extended Fan-expansion method gives the following sets of the abovementioned parameters

a_{0} \to \frac{a_{1} ζ_{2}}{2 ζ_{3}}, p_{2} \to - \frac{12 ζ_{3}^{2} l_{1}^{2}}{a_{1}^{2}} .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For $ζ_{2}^{2} - 4 ζ_{3} ζ_{1} > 0, ζ_{2} ζ_{3} \neq 0, ζ_{3} ζ_{1} \neq 0$ , the solutions have been evaluated as following

ν_{1} (x, t) = - \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{2 ζ_{3}} tanh (\frac{1}{2} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)),

(2.67)

ν_{2} (x, t) = - \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{2 ζ_{3}} coth (\frac{1}{2} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)),

(2.68)

\begin{matrix} ν_{3} (x, t) = - \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{2 ζ_{3}} (tanh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) \pm i sech (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x))), \end{matrix}

(2.69)

ν_{4} (x, t) = - \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{2 ζ_{3}} (coth (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) \pm csch (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x))),

(2.70)

\begin{array}{l} ν_{5} (x, t) = - \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{4 ζ_{3}} tanh (\frac{1}{4} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{4} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) + 1), \end{array}

(2.71)

ν_{6} (x, t) = \frac{a_{1} (\sqrt{(ζ_{2}^{2} - 4 ζ_{1} ζ_{3}) (A^{2} + B^{2})} - A \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} cosh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)))}{2 ζ_{3} (A sinh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) + B)},

(2.72)

ν_{7} (x, t) = - \frac{a_{1} (\sqrt{(ζ_{2}^{2} - 4 ζ_{1} ζ_{3}) (B^{2} - A^{2})} + A \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} sinh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)))}{2 ζ_{3} (A cosh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) + B)},

(2.73)

ν_{8} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} - \frac{4 ζ_{1}}{ζ_{2} - \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} tanh (\frac{1}{2} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x))}),

(2.74)

ν_{9} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} - \frac{4 ζ_{1}}{ζ_{2} - \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} coth (\frac{1}{2} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x))}),

(2.75)

\begin{array}{l} ν_{10} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} + (4 ζ_{1} cosh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x))) / (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} sinh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) \\ - (ζ_{2} cosh (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) \pm i \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}))), \end{array}

(2.76)

\begin{array}{l} ν_{11} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} + (4 ζ_{1} / (csch (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} c o s h (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) \\ \pm \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}) - ζ_{2}))), \end{array}

(2.77)

\begin{array}{l} ν_{12} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} - 4 ζ_{1} / ((ζ_{2} - \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} cosh (2 \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) csch (\frac{1}{2} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} \times (l_{2} t + l_{1} x)))), \end{array}

(2.78)

where A and B are two non-zero constants satisfying B² − A² > 0.

For $ζ_{2}^{2} - 4 ζ_{3} ζ_{1} < 0, ζ_{2} ζ_{3} \neq 0, ζ_{3} ζ_{1} \neq 0$ , the solutions have been evaluated as following

ν_{13} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} - \frac{4 ζ_{1}}{ζ_{2} - \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} tanh (\frac{1}{2} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x))}),

(2.79)

ν_{14} (x, t) = - \frac{a_{1} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \cot (\frac{1}{2} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x))}{2 ζ_{3}},

(2.80)

\begin{matrix} ν_{15} (x, t) = \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{2 ζ_{3}} (\tan (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) \pm \sec (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x))), \end{matrix}

(2.81)

ν_{16} (x, t) = - \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{2 ζ_{3}} (\cot (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) \pm csc (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x))),

(2.82)

\begin{matrix} ν_{17} (x, t) = - \frac{a_{1} \sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}}}{4 ζ_{3}} \tan (\frac{1}{4} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) (\cot^{2} (\frac{1}{4} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) - 1), \end{matrix}

(2.83)

ν_{18} (x, t) = \frac{a_{1} (\sqrt{(ζ_{2}^{2} - 4 ζ_{1} ζ_{3}) (- (A - B) (A + B))} - A \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \cos (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)))}{2 ζ_{3} (A \sin (\sqrt{ζ_{2}^{2} - 4 ζ_{1} ζ_{3}} (l_{2} t + l_{1} x)) + B)},

(2.84)

ν_{19} (x, t) = \frac{a_{1} (A \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \sin (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) - \sqrt{(ζ_{2}^{2} - 4 ζ_{1} ζ_{3}) (B^{2} - A^{2})})}{2 ζ_{3} (A \cos (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) + B)},

(2.85)

ν_{20} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} - \frac{4 ζ_{1}}{ζ_{2} + \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \tan (\frac{1}{2} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x))}),

(2.86)

\begin{array}{l} ν_{21} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} + 4 ζ_{1} sinh (\frac{1}{2} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) / (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \cos (\frac{1}{2} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) \\ - ζ_{2} \sin (\frac{1}{2} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)))), \end{array}

(2.87)

\begin{array}{l} ν_{22} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} - 4 ζ_{1} \cos (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) / (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \sin (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) \\ + (ζ_{2} \cos (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) \pm \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}}))) \end{array}

(2.88)

\begin{array}{l} ν_{23} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} + 4 ζ_{1} / (csc (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \cos (\sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) \\ \pm \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}}) - ζ_{2})), \end{array}

(2.89)

\begin{array}{l} ν_{24} (x, t) = \frac{1}{2} a_{1} (\frac{ζ_{2}}{ζ_{3}} - 4 ζ_{1} / (ζ_{2} - \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \cos (2 \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} (l_{2} t + l_{1} x)) csc (\frac{1}{2} \sqrt{4 ζ_{1} ζ_{3} - ζ_{2}^{2}} \\ \times (l_{2} t + l_{1} x)))), \end{array}

(2.90)

where A and B are two non-zero constants satisfying B² − A² < 0.

Generalized Riccati expansion method

Employing the generalized Riccati expansion method gives the following sets of the abovementioned parameters:

Set I

a_{0} \to \frac{a_{1} ς_{2}}{2 ς_{3}}, b_{1} \to \frac{a_{1} ς_{1}}{ς_{3}}, p_{2} \to - \frac{12 l_{1}^{2} ς_{3}^{2}}{a_{1}^{2}} .

Set II

a_{0} \to \frac{b_{1} ς_{2}}{2 ς_{1}}, a_{1} \to 0, p_{2} \to - \frac{12 l_{1}^{2} ς_{1}^{2}}{b_{1}^{2}} .

Set III

a_{0} \to \frac{a_{1} ς_{2}}{2 ς_{3}}, b_{1} \to 0, p_{2} \to - \frac{12 l_{1}^{2} ς_{3}^{2}}{a_{1}^{2}} .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For $℧ = ς_{2}^{2} - 4 ς_{1} ς_{3} > 0, ς_{2} ς_{3} ≐ 0, ς_{3} ς_{1} ≐ 0$ , the solutions have been evaluated as following

\begin{matrix} ν_{I, 1} (x, t) = \frac{1}{2} a_{1} (- \frac{4 ς_{1}}{\sqrt{℧} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}} - \frac{\sqrt{℧} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3}}), \end{matrix}

(2.91)

\begin{matrix} ν_{I, 2} (x, t) = \frac{1}{2} a_{1} (- \frac{\sqrt{℧} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3}} - \frac{4 ς_{1}}{\sqrt{℧} coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}}), \end{matrix}

(2.92)

\begin{array}{l} ν_{I, 3} (x, t) = \frac{1}{2} a_{1} (- \frac{\sqrt{℧}}{ς_{3}} (\tanh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i sech (\sqrt{℧} (l_{2} t + l_{1} x))) \\ - \frac{4 ς_{1}}{ς_{2} + \sqrt{℧} (tanh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i sech (\sqrt{℧} (l_{2} t + l_{1} x)))}) \end{array}

(2.93)

\begin{array}{l} ν_{I, 4} (x, t) = \frac{1}{2} a_{1} (- \frac{\sqrt{℧}}{ς_{3}} (\coth (\sqrt{℧} (l_{2} t + l_{1} x)) \pm csch (\sqrt{℧} (l_{2} t + l_{1} x))) \\ - \frac{4 ς_{1}}{\sqrt{℧} (coth (\sqrt{℧} (l_{2} t + l_{1} x)) \pm csch (\sqrt{℧} (l_{2} t + l_{1} x))) + ς_{2}}) \end{array}

(2.94)

\begin{array}{l} ν_{I, 5} (x, t) = \frac{1}{4} a_{1} (- \frac{\sqrt{℧}}{ς_{3}} \tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + 1) \\ - \frac{16 ς_{1}}{\sqrt{℧} (tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + coth (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x))) + 2 ς_{2}}) \end{array}

(2.95)

\begin{array}{l} ν_{I, 6} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{℧ (A^{2} + B^{2})} - A \sqrt{℧} c o s h (\sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3} (A sinh (\sqrt{℧} (l_{2} t + l_{1} x)) + B)} \\ - \frac{4 ς_{1}}{\frac{A \sqrt{℧} cosh (\sqrt{℧} (l_{2} t + l_{1} x)) - \sqrt{℧ (A^{2} + B^{2})}}{A sinh (\sqrt{℧} (l_{2} t + l_{1} x)) + B} + ς_{2}}) \end{array}

(2.96)

\begin{matrix} ν_{I, 7} (x, t) = \frac{1}{2} a_{1} (- \frac{\sqrt{℧ (B^{2} - A^{2})} + A \sqrt{℧} s i n h (\sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3} (A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B)} \\ - \frac{4 ς_{1}}{\frac{\sqrt{℧ (B^{2} - A^{2})} + A \sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x))}{A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B} + ς_{2}}), \end{matrix}

(2.97)

\begin{matrix} ν_{I, 8} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{℧} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3}} - \frac{4 ς_{1}}{ς_{2} - \sqrt{℧} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}), \end{matrix}

(2.98)

\begin{matrix} ν_{I, 9} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{℧} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3}} + \frac{4 ς_{1}}{\sqrt{℧} coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2}}), \end{matrix}

(2.99)

\begin{array}{l} ν_{I, 10} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1} c o s h (\sqrt{℧} (l_{2} t + l_{1} x))}{\sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x)) - (ς_{2} cosh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i \sqrt{℧})} \\ + \frac{1}{ς_{3}} sech (\sqrt{℧} (l_{2} t + l_{1} x)) (- (ς_{2} c o s h (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i \sqrt{℧})) \\ + \sqrt{℧} tanh (\sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}), \end{array}

(2.100)

\begin{array}{l} ν_{I, 11} (x, t) = a_{1} (\frac{csch (\sqrt{℧} (l_{2} t + l_{1} x)) (\sqrt{℧} c o s h (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧})}{2 ς_{3}} \\ + \frac{2 ς_{1}}{csch (\sqrt{℧} (l_{2} t + l_{1} x)) (\sqrt{℧} c o s h (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧}) - ς_{2}} \end{array}

(2.101)

\begin{matrix} ν_{I, 12} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{℧} \coth (\sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3}} + \frac{4 ς_{1}}{\sqrt{℧} coth (\sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2}}), \end{matrix}

(2.102)

\begin{matrix} ν_{II, 1} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\sqrt{℧} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}}), \end{matrix}

(2.103)

\begin{matrix} ν_{II, 2} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\sqrt{℧} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}}), \end{matrix}

(2.104)

\begin{matrix} ν_{II, 3} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{ς_{2} + \sqrt{℧} (\tanh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i sech (\sqrt{℧} (l_{2} t + l_{1} x)))}), \end{matrix}

(2.105)

\begin{matrix} ν_{II, 4} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\sqrt{℧} (\coth (\sqrt{℧} (l_{2} t + l_{1} x)) \pm csch (\sqrt{℧} (l_{2} t + l_{1} x))) + ς_{2}}), \end{matrix}

(2.106)

\begin{matrix} ν_{II, 5} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{8 ς_{3}}{\sqrt{℧} (\tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + coth (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x))) + 2 ς_{2}}), \end{matrix}

(2.107)

\begin{matrix} ν_{II, 6} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\frac{A \sqrt{℧} \cosh (\sqrt{℧} (l_{2} t + l_{1} x)) - \sqrt{℧ (A^{2} + B^{2})}}{A sinh (\sqrt{℧} (l_{2} t + l_{1} x)) + B} + ς_{2}}), \end{matrix}

(2.108)

\begin{matrix} ν_{II, 7} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\frac{\sqrt{℧ (B^{2} - A^{2})} + A \sqrt{℧} \sinh (\sqrt{℧} (l_{2} t + l_{1} x))}{A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B} + ς_{2}}), \end{matrix}

(2.109)

\begin{matrix} ν_{II, 8} (x, t) = \frac{b_{1} \sqrt{℧}}{2 ς_{1}} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.110)

\begin{matrix} ν_{II, 9} (x, t) = \frac{b_{1} \sqrt{℧}}{2 ς_{1}} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.111)

ν_{II, 10} (x, t) = \frac{b_{1}}{2 ς_{1}} (sech (\sqrt{℧} (l_{2} t + l_{1} x)) (- (ς_{2} c o s h (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i \sqrt{℧})) + \sqrt{℧} tanh (\sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}),

(2.112)

\begin{matrix} ν_{II, 11} (x, t) = \frac{b_{1}}{2 ς_{1}} csch (\sqrt{℧} (l_{2} t + l_{1} x)) (\sqrt{℧} c o s h (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧}), \end{matrix}

(2.113)

\begin{matrix} ν_{II, 12} (x, t) = \frac{b_{1} \sqrt{℧}}{4 ς_{1}} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + 1), \end{matrix}

(2.114)

\begin{matrix} ν_{III, 1} (x, t) = - \frac{a_{1} \sqrt{℧}}{2 ς_{3}} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.115)

\begin{matrix} ν_{III, 2} (x, t) = - \frac{a_{1} \sqrt{℧}}{2 ς_{3}} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.116)

\begin{matrix} ν_{III, 3} (x, t) = - \frac{a_{1} \sqrt{℧}}{2 ς_{3}} (\tanh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i sech (\sqrt{℧} (l_{2} t + l_{1} x))), \end{matrix}

(2.117)

\begin{matrix} ν_{III, 4} (x, t) = - \frac{a_{1} \sqrt{℧}}{2 ς_{3}} (\coth (\sqrt{℧} (l_{2} t + l_{1} x)) \pm csch (\sqrt{℧} (l_{2} t + l_{1} x))), \end{matrix}

(2.118)

\begin{matrix} ν_{III, 5} (x, t) = - \frac{a_{1} \sqrt{℧}}{4 ς_{3}} \tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + 1), \end{matrix}

(2.119)

\begin{matrix} ν_{III, 6} (x, t) = \frac{a_{1} (\sqrt{℧ (A^{2} + B^{2})} - A \sqrt{℧} \cosh (\sqrt{℧} (l_{2} t + l_{1} x)))}{2 ς_{3} (A sinh (\sqrt{℧} (l_{2} t + l_{1} x)) + B)}, \end{matrix}

(2.120)

\begin{matrix} ν_{III, 7} (x, t) = - \frac{a_{1} (\sqrt{℧ (B^{2} - A^{2})} + A \sqrt{℧} \sinh (\sqrt{℧} (l_{2} t + l_{1} x)))}{2 ς_{3} (A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B)}, \end{matrix}

(2.121)

\begin{matrix} ν_{III, 8} (x, t) = \frac{1}{2} a_{1} (\frac{ς_{2}}{ς_{3}} - \frac{4 ς_{1}}{ς_{2} - \sqrt{℧} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}), \end{matrix}

(2.122)

\begin{matrix} ν_{III, 9} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1}}{\sqrt{℧} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2}} + \frac{ς_{2}}{ς_{3}}), \end{matrix}

(2.123)

\begin{matrix} ν_{III, 10} (x, t) = \frac{1}{2} a_{1} (\frac{ς_{2}}{ς_{3}} + \frac{4 ς_{1} \cosh (\sqrt{℧} (l_{2} t + l_{1} x))}{\sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x)) - (ς_{2} cosh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm i \sqrt{℧})}), \end{matrix}

(2.124)

\begin{matrix} ν_{III, 11} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1}}{csch (\sqrt{℧} (l_{2} t + l_{1} x)) (\sqrt{℧} \cosh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧}) - ς_{2}} + \frac{ς_{2}}{ς_{3}}), \end{matrix}

(2.125)

\begin{matrix} ν_{III, 12} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1}}{\sqrt{℧} \cosh (\sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2}} + \frac{ς_{2}}{ς_{3}}), \end{matrix}

(2.126)

where A and B are two non-zero real constants satisfying B² − A² > 0.

For $℧ = ς_{2}^{2} - 4 ς_{1} ς_{3} < 0, ς_{2} ς_{3} ≐ 0, ς_{3} ς_{1} ≐ 0$ , the solutions have been evaluated as following

\begin{matrix} ν_{I, 13} (x, t) = \frac{1}{2} a_{1} (- \frac{4 ς_{1}}{\sqrt{℧} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}} - \frac{\sqrt{℧}}{ς_{3}} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))), \end{matrix}

(2.127)

\begin{matrix} ν_{I, 14} (x, t) = \frac{1}{2} a_{1} (- \frac{\sqrt{℧}}{ς_{3}} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) - \frac{4 ς_{1}}{\sqrt{℧} coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}}), \end{matrix}

(2.128)

\begin{array}{l} ν_{I, 15} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{- ℧}}{ς_{3}} (\tan (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm sech (\sqrt{℧} (l_{2} t + l_{1} x))) \\ + \frac{4 ς_{1}}{\sqrt{- ℧} (\tan (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm sech (\sqrt{℧} (l_{2} t + l_{1} x))) - ς_{2}}) \end{array}

(2.129)

\begin{array}{l} ν_{I, 16} (x, t) = \frac{1}{2} a_{1} (- \frac{\sqrt{- ℧}}{ς_{3}} (\cot (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm csc (\sqrt{- ℧} (l_{2} t + l_{1} x))) \\ - \frac{4 ς_{1}}{\sqrt{- ℧} (\cot (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm csc (\sqrt{- ℧} (l_{2} t + l_{1} x))) + ς_{2}} \end{array}

(2.130)

\begin{array}{l} ν_{I, 17} (x, t) = \frac{1}{4} a_{1} (- \frac{16 ς_{1}}{\sqrt{℧} tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + 1) + 2 ς_{2}} \\ - \frac{\sqrt{℧}}{ς_{3}} tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + 1)), \end{array}

(2.131)

\begin{array}{l} ν_{I, 18} (x, t) = \frac{a_{1}}{2} (\frac{1}{A ς_{3}} csc (\sqrt{- ℧} (l_{2} t + l_{1} x)) (\sqrt{A^{2} - B^{2} - ℧} - A + \sqrt{- ℧} cosh (\sqrt{℧} (l_{2} t + l_{1} x))) \\ - \frac{4 A ς_{1}}{(A - \sqrt{A^{2} - B^{2} - ℧}) csc (\sqrt{- ℧} (l_{2} t + l_{1} x)) + A ς_{2} + (- \sqrt{℧}) coth (\sqrt{℧} (l_{2} t + l_{1} x))}) \end{array}

(2.132)

\begin{array}{l} ν_{I, 19} (x, t) = \frac{1}{2} a_{1} (\frac{A \sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x)) - \sqrt{℧ (B^{2} - A^{2})}}{ς_{3} (A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B)} \\ - \frac{4 ς_{1}}{\frac{\sqrt{℧ (B^{2} - A^{2})} - A \sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x))}{A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B} + ς_{2}}) \end{array}

(2.133)

\begin{matrix} ν_{I, 20} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{℧}}{ς_{3}} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) - \frac{4 ς_{1}}{ς_{2} - \sqrt{℧} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}), \end{matrix}

(2.134)

\begin{matrix} ν_{I, 21} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{℧}}{ς_{3}} coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + \frac{4 ς_{1}}{\sqrt{℧} coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2}}), \end{matrix}

(2.135)

\begin{array}{l} ν_{I, 22} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1} cosh (\sqrt{℧} (l_{2} t + l_{1} x))}{\sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x)) - (ς_{2} cosh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{- ℧})} \\ + \frac{1}{ς_{3}} (sech (\sqrt{℧} (l_{2} t + l_{1} x)) (- (ς_{2} cosh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{- ℧})) \\ + \sqrt{℧} tanh (\sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2})), \end{array}

(2.136)

\begin{array}{l} ν_{I, 23} (x, t) = a_{1} (\frac{1}{2 ς_{3}} csc (\sqrt{- ℧} (l_{2} t + l_{1} x)) (\sqrt{- ℧} \cos (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧}) \\ + \frac{2 ς_{1}}{csc (\sqrt{- ℧} (l_{2} t + l_{1} x)) (\sqrt{- ℧} \cos (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧}) - ς_{2}}), \end{array}

(2.137)

\begin{matrix} ν_{I, 24} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{℧} coth (\sqrt{℧} (l_{2} t + l_{1} x))}{ς_{3}} + \frac{4 ς_{1}}{\sqrt{℧} coth (\sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2}}), \end{matrix}

(2.138)

\begin{matrix} ν_{II, 13} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\sqrt{℧} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}}), \end{matrix}

(2.139)

\begin{matrix} ν_{II, 14} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\sqrt{℧} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}}), \end{matrix}

(2.140)

\begin{matrix} ν_{II, 15} (x, t) = \frac{1}{2} b_{1} (\frac{4 ς_{3}}{\sqrt{- ℧} (\tan (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm sech (\sqrt{℧} (l_{2} t + l_{1} x))) - ς_{2}} + \frac{ς_{2}}{ς_{1}}), \end{matrix}

(2.141)

\begin{matrix} ν_{II, 16} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\sqrt{- ℧} (\cot (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm csc (\sqrt{- ℧} (l_{2} t + l_{1} x))) + ς_{2}}), \end{matrix}

(2.142)

\begin{matrix} ν_{II, 17} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{8 ς_{3}}{\sqrt{℧} \tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + 1) + 2 ς_{2}}), \end{matrix}

(2.143)

\begin{matrix} ν_{II, 18} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - 4 A ς_{3} / ((A - \sqrt{A^{2} - B^{2} - ℧}) c s c (\sqrt{- ℧} (l_{2} t + l_{1} x)) + A ς_{2} + (- \sqrt{℧}) \\ \times coth (\sqrt{℧} (l_{2} t + l_{1} x)))), \end{matrix}

(2.144)

\begin{matrix} ν_{II, 19} (x, t) = \frac{1}{2} b_{1} (\frac{ς_{2}}{ς_{1}} - \frac{4 ς_{3}}{\frac{\sqrt{℧ (B^{2} - A^{2})} - A \sqrt{℧} \sinh (\sqrt{℧} (l_{2} t + l_{1} x))}{A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B} + ς_{2}}), \end{matrix}

(2.145)

\begin{matrix} ν_{II, 20} (x, t) = \frac{b_{1} \sqrt{℧}}{2 ς_{1}} \tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.146)

\begin{matrix} ν_{II, 21} (x, t) = \frac{b_{1} \sqrt{℧}}{2 ς_{1}} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.147)

\begin{array}{l} ν_{II, 22} (x, t) = \frac{b_{1}}{2 ς_{1}} (sech (\sqrt{℧} (l_{2} t + l_{1} x)) (- (ς_{2} \cosh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{- ℧})) \\ + \sqrt{℧} tanh (\sqrt{℧} (l_{2} t + l_{1} x)) + ς_{2}), \end{array}

(2.148)

\begin{matrix} ν_{I I, 23} (x, t) = \frac{b_{1}}{2 ς_{1}} c s c (\sqrt{- ℧} (l_{2} t + l_{1} x)) (\sqrt{- ℧} \cos (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧}), \end{matrix}

(2.149)

\begin{matrix} ν_{II, 24} (x, t) = \frac{b_{1} \sqrt{℧}}{4 ς_{1}} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) + 1), \end{matrix}

(2.150)

\begin{matrix} ν_{III, 13} (x, t) = - \frac{a_{1} \sqrt{℧}}{2 ς_{3}} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.151)

\begin{matrix} ν_{III, 14} (x, t) = - \frac{a_{1} \sqrt{℧}}{2 ς_{3}} coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)), \end{matrix}

(2.152)

\begin{matrix} ν_{III, 15} (x, t) = \frac{a_{1} \sqrt{- ℧}}{2 ς_{3}} (tan (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm sech (\sqrt{℧} (l_{2} t + l_{1} x))), \end{matrix}

(2.153)

\begin{matrix} ν_{III, 16} (x, t) = - \frac{a_{1} \sqrt{- ℧}}{2 ς_{3}} (cot (\sqrt{- ℧} (l_{2} t + l_{1} x)) \pm csc (\sqrt{- ℧} (l_{2} t + l_{1} x))), \end{matrix}

(2.154)

\begin{matrix} ν_{III, 17} (x, t) = - \frac{a_{1} \sqrt{℧}}{4 ς_{3}} tanh (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) ({coth}^{2} (\frac{1}{4} \sqrt{℧} (l_{2} t + l_{1} x)) + 1), \end{matrix}

(2.155)

\begin{matrix} ν_{III, 18} (x, t) = \frac{a_{1}}{2 A ς_{3}} csc (\sqrt{- ℧} (l_{2} t + l_{1} x)) (\sqrt{A^{2} - B^{2} - ℧} - A + \sqrt{- ℧} cosh (\sqrt{℧} (l_{2} t + l_{1} x))), \end{matrix}

(2.156)

\begin{matrix} ν_{III, 19} (x, t) = \frac{a_{1} (A \sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x)) - \sqrt{℧ (B^{2} - A^{2})})}{2 ς_{3} (A cosh (\sqrt{℧} (l_{2} t + l_{1} x)) + B)}, \end{matrix}

(2.157)

\begin{matrix} ν_{III, 20} (x, t) = \frac{1}{2} a_{1} (\frac{ς_{2}}{ς_{3}} - \frac{4 ς_{1}}{ς_{2} - \sqrt{℧} tanh (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x))}), \end{matrix}

(2.158)

\begin{matrix} ν_{III, 21} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1}}{\sqrt{℧} \coth (\frac{1}{2} \sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2}} + \frac{ς_{2}}{ς_{3}}), \end{matrix}

(2.159)

\begin{matrix} ν_{III, 22} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1} cosh (\sqrt{℧} (l_{2} t + l_{1} x))}{\sqrt{℧} sinh (\sqrt{℧} (l_{2} t + l_{1} x)) - (ς_{2} cosh (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{- ℧})} + \frac{ς_{2}}{ς_{3}}), \end{matrix}

(2.160)

\begin{matrix} ν_{I I I, 23} (x, t) = \frac{1}{2} a_{1} (\frac{4 ς_{1}}{\csc (\sqrt{- ℧} (l_{2} t + l_{1} x)) (\sqrt{- ℧} \cos (\sqrt{℧} (l_{2} t + l_{1} x)) \pm \sqrt{℧}) - ς_{2}} + \frac{ς_{2}}{ς_{3}}), \end{matrix}

(2.161)

\begin{matrix} ν_{III, 24} (x, t) = \frac{a_{1} (ς_{2} \sqrt{- ℧} \cosh (\sqrt{℧} (l_{2} t + l_{1} x)) - (ς_{2}^{2} - 4 ς_{1} ς_{3}) \sin (\sqrt{- ℧} (l_{2} t + l_{1} x)))}{2 ς_{3} (\sqrt{- ℧} cosh (\sqrt{℧} (l_{2} t + l_{1} x)) - ς_{2} \sin (\sqrt{- ℧} (l_{2} t + l_{1} x)))}, \end{matrix}

(2.162)

where A and B are two non-zero real constants satisfying B² − A² < 0.

For $\begin{matrix} ς_{1} = 0, ς_{2} ς_{3} = 0 \end{matrix}$ , the solutions have been evaluated as following

\begin{matrix} ν_{I, 25} (x, t) = \frac{a_{1}}{2 ς_{2}} (\frac{ς_{2}^{2} (\frac{2}{i e^{ς_{2} (l_{2} t + l_{1} x)} + 1} - 1)}{ς_{3}} - \frac{2 ς_{1} (i + e^{ς_{2} (- (l_{2} t + l_{1} x))})}{i}), \end{matrix}

(2.163)

\begin{matrix} ν_{I, 26} (x, t) = \frac{a_{1} ς_{1}}{ς_{2}} (i (- e^{ς_{2} (- (l_{2} t + l_{1} x))}) - 1) + \frac{a_{1} ς_{2}}{2 ς_{3}} (\frac{2 i}{i + e^{ς_{2} (l_{2} t + l_{1} x)}} - 1), \end{matrix}

(2.164)

\begin{matrix} ν_{II, 25} (x, t) = \frac{b_{1} ς_{3}}{ς_{2}} (- \frac{e^{ς_{2} (- (l_{2} t + l_{1} x))}}{i} - 1) + \frac{b_{1} ς_{2}}{2 ς_{1}}, \end{matrix}

(2.165)

\begin{matrix} ν_{II, 26} (x, t) = \frac{b_{1} ς_{3}}{ς_{2}} (i (- e^{ς_{2} (- (l_{2} t + l_{1} x))}) - 1) + \frac{b_{1} ς_{2}}{2 ς_{1}}, \end{matrix}

(2.166)

\begin{matrix} ν_{III, 25} (x, t) = \frac{a_{1} ς_{2}}{2 ς_{3}} (\frac{2}{i e^{ς_{2} (l_{2} t + l_{1} x)} + 1} - 1), \end{matrix}

(2.167)

\begin{matrix} ν_{III, 26} (x, t) = \frac{a_{1} ς_{2}}{2 ς_{3}} (\frac{2 i}{i + e^{ς_{2} (l_{2} t + l_{1} x)}} - 1) . \end{matrix}

(2.168)

For

\begin{matrix} ς_{1} = ς_{2} = 0, ς_{3} = 0 \end{matrix}

, the solutions have been evaluated as following

\begin{matrix} ν_{I, 27} (x, t) = \frac{1}{2} a_{1} (\frac{ς_{2} - 2 ς_{1} (ς_{3} (l_{2} t + l_{1} x) + ϑ)}{ς_{3}} - \frac{2}{ς_{3} (l_{2} t + l_{1} x) + ϑ}), \end{matrix}

(2.169)

\begin{matrix} ν_{II, 27} (x, t) = \frac{b_{1}}{2 ς_{1}} (ς_{2} - 2 ς_{1} (ς_{3} (l_{2} t + l_{1} x) + ϑ)), \end{matrix}

(2.170)

\begin{matrix} ν_{III, 27} (x, t) = \frac{1}{2} a_{1} (\frac{ς_{2}}{ς_{3}} - \frac{2}{ς_{3} (l_{2} t + l_{1} x) + ϑ}) . \end{matrix}

(2.171)

Improved F-expansion method

Employing the improved F-expansion method gives the following sets of the abovementioned parameters:

Set I

a_{- 1} \to 0, a_{0} \to a_{1} (- μ), l_{1} \to \frac{i a_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where (p_{2} < 0, i = \sqrt{- 1}) .

Set II

a_{- 1} \to a_{1} (μ^{2} + ϱ), a_{0} \to a_{1} (- μ), l_{1} \to \frac{i a_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where (p_{2} < 0, i = \sqrt{- 1}) .

Set III

a_{0} \to - \frac{a_{- 1} μ}{μ^{2} + ϱ}, a_{1} \to 0, l_{1} \to \frac{a_{- 1} \sqrt{p_{2}}}{2 \sqrt{3} \sqrt{- μ^{4} - 2 μ^{2} ϱ - ϱ^{2}}}, where (p_{2} < 0, i = \sqrt{- 1}) .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For ϱ < 0, the solutions have been evaluated as following

ν_{I, 1} (x, t) = a_{1} \sqrt{ϱ} tan (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}})),

(2.172)

ν_{I, 2} (x, t) = a_{1} (- \sqrt{ϱ}) cot (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}})),

(2.173)

ν_{II, 1} (x, t) = a_{1} (\frac{μ^{2} + ϱ}{μ + \sqrt{ϱ} tan (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))} + \sqrt{ϱ} \tan (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))),

(2.174)

ν_{II, 2} (x, t) = a_{1} (\frac{μ^{2} + ϱ}{μ - \sqrt{ϱ} cot (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))} - \sqrt{ϱ} \cot (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))),

(2.175)

ν_{III, 1} (x, t) = a_{- 1} (\frac{1}{\sqrt{ϱ} tan (\sqrt{ϱ} (\frac{a_{- 1} \sqrt{p_{2}} x}{2 \sqrt{3} \sqrt{- μ^{4} - 2 μ^{2} ϱ - ϱ^{2}}} + l_{2} t)) + μ} - \frac{μ}{μ^{2} + ϱ}),

(2.176)

ν_{III, 2} (x, t) = a_{- 1} (\frac{1}{μ - \sqrt{ϱ} cot (\sqrt{ϱ} (\frac{a_{- 1} \sqrt{p_{2}} x}{2 \sqrt{3} \sqrt{- μ^{4} - 2 μ^{2} ϱ - ϱ^{2}}} + l_{2} t))} - \frac{μ}{μ^{2} + ϱ}) .

(2.177)

For ϱ > 0, the solutions have been evaluated as following

ν_{I, 3} (x, t) = a_{1} \sqrt{ϱ} \tan (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}})),

(2.178)

ν_{I, 4} (x, t) = a_{1} (- \sqrt{ϱ}) cot (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}})),

(2.179)

ν_{II, 3} (x, t) = a_{1} (\frac{μ^{2} + ϱ}{μ + \sqrt{ϱ} \tan (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))} + \sqrt{ϱ} \tan (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))),

(2.180)

ν_{II, 4} (x, t) = a_{1} (\frac{μ^{2} + ϱ}{μ - \sqrt{ϱ} cot (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))} - \sqrt{ϱ} \cot (\sqrt{ϱ} (l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}))),

(2.181)

ν_{III, 3} (x, t) = a_{- 1} (\frac{1}{\sqrt{ϱ} tan (\sqrt{ϱ} (\frac{a_{- 1} \sqrt{p_{2}} x}{2 \sqrt{3} \sqrt{- μ^{4} - 2 μ^{2} ϱ - ϱ^{2}}} + l_{2} t)) + μ} - \frac{μ}{μ^{2} + ϱ}),

(2.182)

ν_{III, 4} (x, t) = a_{- 1} (\frac{1}{μ - \sqrt{ϱ} cot (\sqrt{ϱ} (\frac{a_{- 1} \sqrt{p_{2}} x}{2 \sqrt{3} \sqrt{- μ^{4} - 2 μ^{2} ϱ - ϱ^{2}}} + l_{2} t))} - \frac{μ}{μ^{2} + ϱ}) .

(2.183)

For ϱ = 0, the solutions have been evaluated as following

ν_{I, 5} (x, t) = - \frac{a_{1}}{l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}},

(2.184)

ν_{II, 5} (x, t) = a_{1} (\frac{μ^{2} + ϱ}{μ - \frac{1}{l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}}} - \frac{1}{l_{2} t + \frac{i a_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}}),

(2.185)

ν_{III, 5} (x, t) = a_{- 1} (\frac{1}{μ - \frac{1}{\frac{a_{- 1} \sqrt{p_{2}} x}{2 \sqrt{3} \sqrt{- μ^{4} - 2 μ^{2} ϱ - ϱ^{2}}} + l_{2} t}} - \frac{1}{μ}) .

(2.186)

Generalization sinh-Gorden expansion method

Employing the generalization sinh-Gorden expansion method gives the following sets of the abovementioned parameters:

Case one When $W^{'} = sinh (W)$ , we get the following sets of solutions

Set I

A_{0} \to 0, A_{1} \to 0, l_{1} \to \frac{i B_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where p_{2} < 0, i = \sqrt{- 1} .

Set II

A_{0} \to 0, B_{1} \to 0, l_{1} \to \frac{i A_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where p_{2} < 0, i = \sqrt{- 1} .

Set III

A_{0} \to 0, B_{1} \to - A_{1}, l_{1} \to \frac{i A_{1} \sqrt{p_{2}}}{\sqrt{3}}, where p_{2} < 0, i = \sqrt{- 1} .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas

ν_{I, 1} (x, t) = i B_{1} sech (l_{2} t + \frac{i B_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}),

(2.187)

ν_{I, 2} (x, t) = i B_{1} csch (l_{2} t + \frac{i B_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}),

(2.188)

ν_{II, 1} (x, t) = - A_{1} tanh (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}),

(2.189)

ν_{II, 2} (x, t) = A_{1} (- coth (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{2 \sqrt{3}})),

(2.190)

ν_{III, 1} (x, t) = A_{1} (- tanh (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{\sqrt{3}}) - i sech (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{\sqrt{3}})),

(2.191)

ν_{III, 2} (x, t) = A_{1} (- \coth (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{\sqrt{3}})) - i A_{1} csch (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{\sqrt{3}}) .

(2.192)

Case one When

W^{'} = cosh (W)

, we get the following sets of solutions

Set A

A_{0} \to 0, A_{1} \to 0, l_{1} \to \frac{i B_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where p_{2} < 0, i = \sqrt{- 1} .

Set B

A_{0} \to 0, B_{1} \to 0, l_{1} \to \frac{i A_{1} \sqrt{p_{2}}}{2 \sqrt{3}}, where p_{2} < 0, i = \sqrt{- 1} .

Set C

A_{0} \to 0, B_{1} \to - A_{1}, l_{1} \to \frac{i A_{1} \sqrt{p_{2}}}{\sqrt{3}}, where p_{2} < 0, i = \sqrt{- 1} .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas

ν_{A, 1} (x, t) = B_{1} tan (l_{2} t + \frac{i B_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}),

(2.193)

ν_{A, 2} (x, t) = B_{1} (- cot (l_{2} t + \frac{i B_{1} \sqrt{p_{2}} x}{2 \sqrt{3}})),

(2.194)

ν_{B, 1} (x, t) = A_{1} \sec (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}),

(2.195)

ν_{B, 2} (x, t) = A_{1} csc (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{2 \sqrt{3}}),

(2.196)

ν_{C, 1} (x, t) = A_{1} (\sec (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{\sqrt{3}}) - \tan (l_{2} t + \frac{i A_{1} \sqrt{p_{2}} x}{\sqrt{3}})),

(2.197)

ν_{C, 2} (x, t) = A_{1} cot (\frac{1}{6} (3 l_{2} t + i \sqrt{3} A_{1} \sqrt{p_{2}} x)) .

(2.198)

Modified Khater method

Employing the modified Khater method gives the following sets of the abovementioned parameters:

Set I

a_{0} \to \frac{a_{1} χ}{2 δ}, b_{1} \to \frac{a_{1} ϱ}{δ}, p_{2} \to - \frac{12 δ^{2} l_{1}^{2}}{a_{1}^{2}} .

Set II

a_{0} \to \frac{a_{1} χ}{2 δ}, b_{1} \to 0, p_{2} \to - \frac{12 δ^{2} l_{1}^{2}}{a_{1}^{2}} .

Set III

a_{0} \to \frac{b_{1} χ}{2 ϱ}, a_{1} \to 0, p_{2} \to - \frac{12 l_{1}^{2} ϱ^{2}}{b_{1}^{2}} .

Thus, the solitary wave solutions of equation (1.5) can be formulated in the following formulas:

For χ² − 4δϱ < 0, δ ≠ 0, the solutions are given by

\begin{array}{l} ν_{I, 1} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{4 δ ϱ - χ^{2}}}{δ} \tan (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x)) \\ - \frac{4 ϱ}{χ - \sqrt{4 δ ϱ - χ^{2}} \tan (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x))}) \end{array}

(2.199)

\begin{array}{l} ν_{I, 2} (x, t) = \frac{1}{2} a_{1} (\frac{\sqrt{4 δ ϱ - χ^{2}}}{δ} \cot (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x)) \\ - \frac{4 ϱ}{χ - \sqrt{4 δ ϱ - χ^{2}} \cot (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x))}), \end{array}

(2.200)

ν_{II, 1} (x, t) = \frac{a_{1} \sqrt{4 δ ϱ - χ^{2}}}{2 δ} tan (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x)),

(2.201)

ν_{II, 2} (x, t) = \frac{a_{1} \sqrt{4 δ ϱ - χ^{2}}}{2 δ} \cot (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x)),

(2.202)

ν_{III, 1} (x, t) = \frac{1}{2} b_{1} (\frac{χ}{ϱ} - \frac{4 δ}{χ - \sqrt{4 δ ϱ - χ^{2}} tan (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x))}),

(2.203)

ν_{III, 2} (x, t) = \frac{1}{2} b_{1} (\frac{χ}{ϱ} - \frac{4 δ}{χ - \sqrt{4 δ ϱ - χ^{2}} \cot (\frac{1}{2} \sqrt{4 δ ϱ - χ^{2}} (l_{2} t + l_{1} x))}) .

(2.204)

For χ² − 4δϱ > 0, δ ≠ 0, the solutions are given by

\begin{array}{l} ν_{I, 3} (x, t) = \frac{1}{2} a_{1} (- \frac{4 ϱ}{\sqrt{χ^{2} - 4 δ ϱ} tanh (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x)) + χ} \\ - \frac{\sqrt{χ^{2} - 4 δ ϱ}}{δ} tanh (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x))), \end{array}

(2.205)

\begin{array}{l} ν_{I, 4} (x, t) = \frac{1}{2} a_{1} (- \frac{4 ϱ}{\sqrt{χ^{2} - 4 δ ϱ} coth (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x)) + χ} \\ - \frac{\sqrt{χ^{2} - 4 δ ϱ}}{δ} coth (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x))), \end{array}

(2.206)

ν_{II, 3} (x, t) = - \frac{a_{1} \sqrt{χ^{2} - 4 δ ϱ}}{2 δ} tanh (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x)),

(2.207)

ν_{II, 4} (x, t) = - \frac{a_{1} \sqrt{χ^{2} - 4 δ ϱ}}{2 δ} coth (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x)),

(2.208)

ν_{III, 3} (x, t) = \frac{1}{2} b_{1} (\frac{χ}{ϱ} - \frac{4 δ}{\sqrt{χ^{2} - 4 δ ϱ} tanh (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x)) + χ}),

(2.209)

ν_{III, 4} (x, t) = \frac{1}{2} b_{1} (\frac{χ}{ϱ} - \frac{4 δ}{\sqrt{χ^{2} - 4 δ ϱ} coth (\frac{1}{2} \sqrt{χ^{2} - 4 δ ϱ} (l_{2} t + l_{1} x)) + χ}) .

(2.210)

For δϱ > 0, ϱ ≠ 0, δ ≠ 0, χ = 0, the solutions are given by

ν_{I, 5} (x, t) = \frac{2 a_{1} ϱ}{\sqrt{δ ϱ}} csc (2 \sqrt{δ ϱ} (l_{2} t + l_{1} x)),

(2.211)

ν_{I, 6} (x, t) = - \frac{2 a_{1} ϱ}{\sqrt{δ ϱ}} csc (2 \sqrt{δ ϱ} (l_{2} t + l_{1} x)),

(2.212)

ν_{II, 5} (x, t) = \frac{a_{1} ϱ}{\sqrt{δ ϱ}} tan (\sqrt{δ ϱ} (l_{2} t + l_{1} x)),

(2.213)

ν_{II, 6} (x, t) = - \frac{a_{1} ϱ}{\sqrt{δ ϱ}} cot (\sqrt{δ ϱ} (l_{2} t + l_{1} x)),

(2.214)

ν_{III, 5} (x, t) = \frac{b_{1} δ}{\sqrt{δ ϱ}} cot (\sqrt{δ ϱ} (l_{2} t + l_{1} x)),

(2.215)

ν_{III, 6} (x, t) = - \frac{b_{1} δ}{\sqrt{δ ϱ}} tan (\sqrt{δ ϱ} (l_{2} t + l_{1} x)) .

(2.216)

For δϱ < 0, ϱ ≠ 0, δ ≠ 0, χ = 0, the solutions are given by

ν_{I, 7} (x, t) = \frac{2 a_{1} \sqrt{ϱ}}{\sqrt{δ}} csc (2 \sqrt{δ} \sqrt{ϱ} (l_{2} t + l_{1} x)),

(2.217)

ν_{I, 8} (x, t) = \frac{2 a_{1} ϱ}{\sqrt{- δ ϱ}} csch (2 \sqrt{- δ ϱ} (l_{2} t + l_{1} x)),

(2.218)

ν_{II, 7} (x, t) = \frac{a_{1} ϱ}{\sqrt{- δ ϱ}} tanh (\sqrt{- δ ϱ} (l_{2} t + l_{1} x)),

(2.219)

ν_{II, 8} (x, t) = \frac{a_{1} ϱ}{\sqrt{- δ ϱ}} coth (\sqrt{- δ ϱ} (l_{2} t + l_{1} x)),

(2.220)

ν_{III, 7} (x, t) = \frac{b_{1} \sqrt{δ}}{\sqrt{ϱ}} \cot (\sqrt{δ} \sqrt{ϱ} (l_{2} t + l_{1} x)),

(2.221)

ν_{III, 8} (x, t) = \frac{b_{1} \sqrt{- δ ϱ}}{ϱ} tanh (\sqrt{- δ ϱ} (l_{2} t + l_{1} x)) .

(2.222)

For χ = 0, ϱ = − δ, the solutions are given by

ν_{I, 9} (x, t) = 2 a_{1} csch (2 ϱ (l_{2} t + l_{1} x)),

(2.223)

ν_{II, 9} (x, t) = a_{1} coth (ϱ (l_{2} t + l_{1} x)),

(2.224)

ν_{III, 9} (x, t) = b_{1} tanh (ϱ (l_{2} t + l_{1} x)) .

(2.225)

For

χ = \frac{ϱ}{2} = κ, δ = 0

, the solutions are given by

ν_{III, 10} (x, t) = \frac{1}{4} b_{1} (\frac{4}{e^{κ (l_{2} t + l_{1} x)} - 2} + 1) .

(2.226)

For χ = δ = κ, ϱ = 0, the solutions are given by

ν_{I, 10} (x, t) = - \frac{1}{2} a_{1} coth (\frac{1}{2} κ (l_{2} t + l_{1} x)),

(2.227)

ν_{II, 10} (x, t) = - \frac{1}{2} a_{1} coth (\frac{1}{2} κ (l_{2} t + l_{1} x)) .

(2.228)

For χ = δ = 0, ϱ ≠ 0, the solutions are given by

ν_{III, 11} (x, t) = \frac{b_{1}}{l_{2} t ϱ + l_{1} x ϱ} .

(2.229)

For χ = ϱ = 0, δ ≠ 0, the solutions are given by

ν_{I, 11} (x, t) = - \frac{a_{1}}{δ l_{2} t + δ l_{1} x},

(2.230)

ν_{II, 11} (x, t) = - \frac{a_{1}}{δ l_{2} t + δ l_{1} x} .

(2.231)

For χ = 0, ϱ = δ, the solutions are given by

ν_{I, 12} (x, t) = 2 a_{1} csc (2 (C + l_{2} t ϱ + l_{1} x ϱ)),

(2.232)

ν_{II, 12} (x, t) = a_{1} tan (C + l_{2} t ϱ + l_{1} x ϱ),

(2.233)

ν_{III, 12} (x, t) = b_{1} cot (C + l_{2} t ϱ + l_{1} x ϱ) .

(2.234)

For δ = 0, χ ≠ 0, ϱ ≠ 0, the solutions are given by

ν_{III, 13} (x, t) = \frac{1}{2} b_{1} χ (\frac{1}{ϱ} - \frac{2}{ϱ - χ e^{χ (l_{2} t + l_{1} x)}}) .

(2.235)

For χ² − 4δϱ = 0, the solutions are given by

ν_{I, 13} (x, t) = \frac{a_{1}}{χ^{2}} (\frac{χ^{3}}{δ (l_{2} t χ + l_{1} χ x + 2)} - \frac{4 ϱ}{l_{2} t + l_{1} x} - 2 χ ϱ),

(2.236)

ν_{II, 13} (x, t) = \frac{a_{1}}{2 χ^{2}} (\frac{χ^{3}}{δ} - \frac{8 ϱ}{l_{2} t + l_{1} x} - 4 χ ϱ),

(2.237)

ν_{III, 14} (x, t) = \frac{b_{1} χ}{l_{2} t χ ϱ + l_{1} χ x ϱ + 2 ϱ} .

(2.238)

Result and discussion

The merging MKdV with a negative MKdV equation was used in this paper to create unique explicit traveling wave solutions using eleven current approaches. Numerous special formulas for exponential, trigonometric, and hyperbolic solutions have been employed to represent the investigated model’s wave solutions. These solutions define the physical qualities more straightforwardly and across a considerably broader range, which enables us to apply our obtained responses in the models’ application. Now, and in this portion of our article, we will examine our solutions and their relationships and the techniques themselves to make the primary notion of these approaches evident. From an essential examination of our obtained answers, we can observe that a large number of ways provide solutions. These solutions have two parts; the first part is closed to other solutions that got by another manner, for example, [Equations (2.4), (2.6), (2.8), (2.26), (2.28), (2.34), (2.36), (2.47), (2.48), (2.50), (2.64), (2.67), (2.71), (2.72), (2.91), (2.103), (2.115), (2.172), (2.173), (2.187), (2.189), (2.205), (2.207), (2.210)], and so on. However, the second part is new formulas of solutions which are the rest of the solutions. These solutions have been represented in 3D, 2D, contour, spherical, and polar plots to show more physical and dynamical behavior of the investigated model ( Figures 1–24) by using some special values of the above shown parameters.

Figure 1.

Breather solitary wave of equation (2.4) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 2.

Kink solitary wave of equation (2.6) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 3.

Periodic solitary wave of equation (2.8) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 4.

Singular solitary wave of equation (2.26) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 5.

Periodic solitary wave of equation (2.28) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 6.

Singular solitary wave of equation (2.34) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 7.

Kink solitary wave of equation (2.36) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 8.

Kink solitary wave of equation (2.47) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 9.

Singular solitary wave of equation (2.48) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 10.

Solitary wave of equation (2.50) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 11.

Kink wave of equation (2.64) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 12.

Kink wave of equation (2.67) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 13.

Singular wave of equation (2.71) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 14.

Kink wave of equation (2.72) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 15.

Breather wave of equation (2.91) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 16.

Kink wave of equation (2.103) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 17.

Solitary wave of equation (2.115) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 18.

Kink wave of equation (2.172) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 19.

Cone solitary wave of equation (2.173) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 20.

Breather wave of equation (2.187) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 21.

Kink solitary wave of equation (2.189) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 22.

Breather solitary wave of equation (2.205) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 23.

Kink solitary wave of equation (2.207) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Figure 24.

Periodic solitary wave of equation (2.210) in (a) 3D, (b) 2D, (c) contour, (d) spherical, and (d) polar plots.

Convergence and divergence of solutions may be traced back to the similarity of techniques in the auxiliary equation, which is often a remarkable instance of the Riccati equation.

Conclusion

We utilized eleven techniques to the Merged MKdV with a negative MKdV equation in this research. We obtained several innovative formulas for this model’s sophisticated solitary wave solutions. Complex solitary wave solutions are used to explain the propagation of waves. Additionally, it is considered a more mathematically succinct tool for elaborating on the physical features of models. A representation of complicated solutions simplifies the handling of wave superpositions significantly. The correctness of all solutions found has been verified using the Mathematica software.

Footnotes

Acknowledgments

We greatly thank Taif University for providing fund for this work through Taif University Researchers Supporting Project number (TURSP-2020/52), Taif University, Taif, Saudi Arabia.

Authors’ contribution

Dexu Zhao and Mostafa Khater have revised the conceptualization, data curation, and methodology. Dianchen Lu and Mostafa Khater have revised data curation, investigation, and software. Samir Salama and Piyaphong Yongphe have revised the physical meaning of the obtained solutions and raised the given graphs resolutions. All authors have read and agreed to the published version of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: We greatly thank Taif University for providing fund for this work through Taif University Researchers Supporting Project number (TURSP-2020/52), Taif University, Taif, Saudi Arabia.

Availability of data and material

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The used code of this study is available from the corresponding author upon reasonable request.

ORCID iD

Mostafa MA Khater

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Soliton wave solutions of ion-acoustic waves a cold plasma with negative ions

Abstract

Keywords

Introduction

Application

Exp (− ϕ(ϑ)) -expansion method

Improved (G′/G) -expansion method

Direct algebraic method

Generalized Kudryashov method

Extended tanh-function method

Simplest equation method

Extended fan-expansion method

Generalized Riccati expansion method

Improved F-expansion method

Generalization sinh-Gorden expansion method

Modified Khater method

Result and discussion

Conclusion

Footnotes

Acknowledgments

Authors’ contribution

Declaration of conflicting interests

Funding

Availability of data and material

Code availability

ORCID iD

References

Improved (G^′/G) -expansion method