Dispersive solitary wave solutions of new coupled Konno-Oono, Higgs field and Maccari equations and their applications

Mostafa M.A. Khater; Aly R. Seadawy; Dianchen Lu

doi:10.1016/j.jksus.2017.11.003

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Original article

30 (

); 417-423

doi:

10.1016/j.jksus.2017.11.003

Dispersive solitary wave solutions of new coupled Konno-Oono, Higgs field and Maccari equations and their applications

Mostafa M.A. Khater^{^a}, Aly R. Seadawy^{^b,⁎}, Dianchen Lu^{^a,⁎}

a Department of Mathematics, Faculty of Science, Jiangsu University, China

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

⁎Corresponding authors. Aly742001@yahoo.com (Aly R. Seadawy), dclu@ujs.edu.cn (Dianchen Lu)

Received: 2017-10-21, Accepted: 2017-11-23, Published: 2017-07-01

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

Peer review under responsibility of King Saud University.

Abstract

In this research we apply generalized Exp-Function method to obtain exact, solitary and new soliton wave solutions of new coupled Konno-Oono equation, Higgs field equation and Maccari equation via generalized Exp-Function method which are very substantial models in define a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space, introduces quantum field (or the Higgs field) to illustrate the generation mechanism of mass for gauge bosons and described the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics and others. generalized Exp-Function method is very sturdy, fabulous, felicitous and effective method to get exact, solitary and new soliton wave solution of nonlinear partial differential equations (PDEs.). We present a contrasting between the results of this modern method and another method and show that how the results that obtained by this method is much closed to cover many different methods in this field and not just that but also get a new solitary and soliton wave solutions which give a wide range of solutions that help all researchers who apply these models in our life.

Keywords

New coupled Konno-Oono equation

Higgs field equation

Maccari equation

Generalized Exp-Function method

Traveling wave solutions

Solitary wave solutions

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1 Introduction

From the beginning of the universe, there exist many of phenomenal phenomena in different fields in the life for example (Mathematical Physics, Biology, Chemistry, fluid mechanics, hydrodynamics, optics, and plasma physics and so on, but because of ignorance of the causes of these phenomena and do not know even how to occur or how to make use of them. Humanity has been lagging behind in scientific progress. This is the Dark Age continued until the emergence of partial differential equations (PDEs.) which can represent many of these phenomena. However all of these but still the problem persists as we cannot understand what is the physical meaning of these phenomena. In 1965 when Zabusky & Kruskal introduced the mean of soliton and showed how possible that every natural phenomena in different fields which help us to know a lot of information about the physical meaning of these phenomena. From this day, the scientific race began between all scientists and researchers to discover suitable methods to solve these phenomena to be able to apply in our life for example the $(\frac{G^{'}}{G})$ – expansion method, Novel $(\frac{G^{'}}{G})$ – expansion method, modified $(\frac{G^{'}}{G})$ – expansion method, the $(\frac{G^{'}}{G}, \frac{1}{G})$ – expansion method, the $e^{- ϕ (ξ)}$ – expansion method, extended $e^{- ϕ (ξ)}$ – expansion method, the extended tanh-function method, the Kudryashov, modified Kudryashov methods, The improved $tan (\frac{ϕ}{2})$ – expansion method, modified simple equation method and so on (Seadawy, 2014, 2017; Arshad et al., 2016; Liu et al., 2001; Naher et al., 2012; Seadawy et al., 2017a,b,c,d; Yang et al., 2016, 2017a,b; Lu et al., 2017a,b; Ebaid, 2007; Pava, 2007; El-Wakil and Abdou, 2007; Wazwaz, 2004; Khater, 2015; Gao et al., 2017; Lee and Sakthivel, 2013, 2014; Chun and Sakthivel, 2010; Zhou et al., 2003; Seadawy and Mostafa, 2017 ). Generalized Exp-Function method is considered the latest method in this area as it just discovered from just one year and also it contain the results of some methods so that, these methods can be considered as special case of Generalized Exp-Function method.

In this research, we treat with three important models in three different fields to get traveling wave solutions of these models by using Generalized Exp-Function method and for further information and steps of Generalized Exp-Function method; you can see Khater et al. (2017). These equations called new coupled Konno-Oono equation, Higgs field equation, and Maccari equation. In the following order, we give an introduction for each one of them. Firstly: new coupled Konno-Oono equation which defines a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space (Konno and Oono, 1994; Konno and Kakuhata, 1995; Souleymanou et al., 2012 ). In 1990, Konno and Oono (1994) presented more general version of coupled integrable dispersionless system defined as

(1.1)

\{\begin{matrix} q_{xt} - 2 α {qr}_{x} - 2 β {qs}_{x} + γ {(rs)}_{x} = 0, \\ r_{xt} - α {rr}_{x} - 2 β (2 {qq}_{x} + r_{x} s) - 2 γ q_{x} r = 0, \\ s_{xt} - 2 β {ss}_{x} + 2 α (2 {qq}_{x} + {rs}_{x}) - 2 γ {sq}_{x} = 0, \end{matrix}

where

(α, β, γ)

are arbitrary constants. This system appears geometrically as the parallel transport of each point of the curve along the direction of time where the connection is magnetic-valued (Khalique, 2012). When special values of some coefficients is taken in Eq. (1.1a), this system converted into new Konno-Oono equation system which is a coupled integrable dispersionless equations as following:

(1.2)

\{\begin{matrix} v_{t} + 2 {uu}_{x} = 0, \\ u_{xt} - 2 vu = 0, \end{matrix}

where

(u & v)

are functions in

(x, t)

. This new Konno-Oono equation system attract attention some scientist from all over the world. They have investigated this model in terms of new and different properties by using some mathematical approaches.

Secondly: A complex couple Higgs field equation which introduces quantum field (or the Higgs field) to illustrate the generation mechanism of mass for gauge bosons (Hon and Fan, 2009; Hase and Satsuma, 1988). The general form of the complex couple Higgs field equation be in the following from

(1.3)

\{\begin{matrix} u_{tt} - u_{xx} - α_{1} u + β_{1} | u |^{2} u - 2 uv = 0, \\ v_{tt} + v_{xx} - β_{1} {(| u |^{2})}_{xx} = 0, \end{matrix}

where

β_{1} & α_{1}

are arbitrary constant.

Thirdly: The (2 + 1)-dimensional nonlinear complex coupled Maccari equations are the second complex coupled equations that will be discussed. The complex coupled Maccari equations is a nonlinear evolution equation described the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics and others (Rostamy et al., 2011; Zhao, 2008; Arshad et al., 2017; Jabbari et al., 2011; Bekir, 2009 ). Complex coupled Maccari equation derived from the Kadomtsev–Petviashvili equation (the best known two-dimensional generalizations of the KdV equation) and can be written in the form

(1.4)

\{\begin{matrix} {iu}_{t} + u_{xx} + uv = 0, \\ v_{t} + v_{y} + {(| u |^{2})}_{x} = 0 . \end{matrix}

This equation is called integrable (2 + 1)-dimensional nonlinear Maccari system (Maccari, 1996).

The remnant of this paper is systematized as follows: In Section 2, we apply generalized Exp-Function method to get the exact solutions of (NLPDEs.) pointed out above. In Section 3, we illustrate our solutions and what is the difference between our results and that obtained by using different methods and also what is the new in this paper which makes our paper is suitable for publication. In Section 4, conclusions are given.

2 Application

In this section, we apply generalized Exp-Function method for these three models (new coupled Konno-Oono equation, Higgs field equation and Maccari equation) and also we show the exact traveling wave solutions and solitary wave solutions of each one of these models.

2.1

2.1 New coupled Konno-Oono equation

By using traveling wave transformation $u (x, t) = u (ξ) & v (x, t) = v (ξ)$ where $ξ = k (x - ct)$ on Eq. (1.2), we obtain:

(2.1.1)

\{\begin{matrix} - {ckv}^{'} + 2 {kuu}^{'} = 0, \\ - {ck}^{2} u^{″} - 2 uv = 0 . \end{matrix}

Integrating first equation in the system (2.1.1) and submit the result into second equation in the same system, we obtain

(2.1.2)

ρ u^{″} + 2 u^{3} + 2 δ u = 0,

where

(δ & ρ = c^{2} k^{2})

are a constant of integration. Balance the highest order derivatives and nonlinear terms appearing in Eq. (2.1.2)

\Rightarrow (u^{″} & u^{3}) \Rightarrow (N = 1)

. So that, by using Generalized Exp-Function method, we get the exact solution of Eq. (2.1.2) be in the following form:

(2.1.3)

u (ξ) = a_{0} + a_{1} a^{f (ξ)} .

Substituting (2.1.3) and its derivative into Eq. (2.1.2) and collecting all term with the same power of $[a^{if (ξ)} where i = 0, 1, 2, 3]$ , we get the system of algebraic equations by solving it, we obtain: $ρ = \frac{- a_{1}^{2}}{σ^{2}}, δ = \frac{a_{1}^{2} (- β^{2} + 4 α σ)}{4 σ^{2}}, a_{0} = \frac{a_{1} β}{2 σ}, a_{1} = a_{1} .$

So that, the exact traveling wave solution will be in the form:

(2.1.4)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} a^{f (ξ)} .

Thus, the solitary traveling wave solutions:

When ( $β^{2} - α σ < 0 & σ \neq 0$ )

(2.1.5)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} + \frac{\sqrt{- (β^{2} - α σ)}}{σ} \tan (\frac{\sqrt{- (β^{2} - α σ)}}{2} ξ)],

(2.1.6)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} + \frac{\sqrt{- (β^{2} - α σ)}}{σ} \cot (\frac{\sqrt{- (β^{2} - α σ)}}{2} ξ)] .

When ( $β^{2} - α σ > 0 & σ \neq 0$ )

(2.1.7)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} - \frac{\sqrt{(β^{2} - α σ)}}{σ} \tanh (\frac{\sqrt{(β^{2} - α σ)}}{2} ξ)],

(2.1.8)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} - \frac{\sqrt{(β^{2} - α σ)}}{σ} \coth (\frac{\sqrt{(β^{2} - α σ)}}{2} ξ)] .

When ( $β^{2} - α^{2} > 0 & σ \neq 0 & σ = - α$ )

(2.1.9)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{β^{2} + α^{2}}}{α} \tanh (\frac{\sqrt{β^{2} + α^{2}}}{2} ξ)],

(2.1.10)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{β^{2} + α^{2}}}{α} \coth (\frac{\sqrt{β^{2} + α^{2}}}{2} ξ)] .

When ( $β^{2} - α^{2} < 0 & σ \neq 0 & σ = - α$ )

(2.1.11)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{- (β^{2} + α^{2})}}{α} \tan (\frac{\sqrt{- (β^{2} + α^{2})}}{2} ξ)],

(2.1.12)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{- (β^{2} + α^{2})}}{α} \cot (\frac{\sqrt{- (β^{2} + α^{2})}}{2} ξ)] .

When ( $β^{2} - α^{2} < 0 & σ \neq 0 & σ = α$ )

(2.1.13)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{- (β^{2} - α^{2})}}{α} \tan (\frac{\sqrt{- (β^{2} - α^{2})}}{2} ξ)],

(2.1.14)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{- (β^{2} - α^{2})}}{α} \cot (\frac{\sqrt{- (β^{2} - α^{2})}}{2} ξ)] .

When ( $β^{2} - α^{2} > 0 & σ \neq 0 & σ = α$ )

(2.1.15)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{(β^{2} - α^{2})}}{α} \tanh (\frac{\sqrt{(β^{2} - α^{2})}}{2} ξ)],

(2.1.16)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{(β^{2} - α^{2})}}{α} \coth (\frac{\sqrt{(β^{2} - α^{2})}}{2} ξ)] .

When $(α σ < 0 & σ \neq 0 & β = 0)$

(2.1.17)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\sqrt{\frac{- α}{σ}} \tanh (\frac{\sqrt{- α σ}}{2} ξ)] .

(2.1.18)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\sqrt{\frac{- α}{σ}} \coth (\frac{\sqrt{- α σ}}{2} ξ)] .

When $(β = 0 & α = - σ)$

(2.1.19)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- (1 + e^{2 α ξ}) \pm \sqrt{2 (e^{4 α ξ} + 1)}}{e^{2 α ξ} - 1}],

(2.1.20)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- (1 + e^{2 α ξ}) \pm \sqrt{e^{4 α ξ} + 6 e^{2 α ξ} + 1}}{e^{2 α ξ}}] .

When $(α = σ = 0)$

(2.1.21)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- (1 + e^{2 β ξ}) \pm \sqrt{2 (e^{4 β ξ} + 1)}}{e^{2 β ξ} - 1}],

(2.1.22)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- (1 + e^{2 β ξ}) \pm \sqrt{e^{4 β ξ} + 6 e^{2 β ξ} + 1}}{e^{2 β ξ}}] .

When $(β^{2} = α σ)$

(2.1.23)

u (ξ) = \frac{a_{1} β}{2 σ} + \frac{- α a_{1} (β ξ + 2)}{β^{2} ξ} .

When $(β = k & α = 2 k & σ = 0)$

(2.1.24)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [e^{k ξ} - 1] .

When $(β = k & σ = 2 k & α = 0)$

(2.1.25)

u (ξ) = \frac{a_{1} β}{2 σ} + \frac{a_{1} e^{k ξ}}{1 - e^{k ξ}} .

When $(2 β = α + σ)$

(2.1.26)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{1 - α e^{0.5 (α - σ) ξ}}{1 - σ e^{0.5 (α - σ) ξ}}],

(2.1.27)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{α e^{0.5 (α - σ) ξ} + 1}{- σ e^{0.5 (α - σ) ξ} - 1}] .

When $(- 2 β = α + σ)$

(2.1.28)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{e^{0.5 (α - σ) ξ} + α}{e^{0.5 (α - σ) ξ} + σ}] .

When $(α = 0)$

(2.1.29)

u (ξ) = \frac{a_{1} β}{2 σ} + \frac{β a_{1} e^{β ξ}}{1 + 0.5 σ e^{β ξ}} .

When $(β = α = σ \neq 0)$

(2.1.30)

u (ξ) = \frac{a_{1} β}{2 σ} - \frac{a_{1} (α ξ + 2)}{α ξ} .

When $(β = σ = 0)$

(2.1.31)

u (ξ) = \frac{a_{1} β}{2 σ} + 0.5 α a_{1} ξ .

When $(β = α = 0)$

(2.1.32)

u (ξ) = \frac{a_{1} β}{2 σ} - \frac{2 a_{1}}{σ ξ} .

When $(β = 0 & α = σ)$

(2.1.33)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} \tan (\frac{α ξ + C}{2}) .

When $(σ = 0)$

(2.1.34)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [e^{β ξ} - \frac{α}{2 β}] .

Where $k, C$ are arbitrary constant.

2.2

2.2 Higgs field equation

Using the traveling wave transformation $[u (x, t) = e^{i θ} U (ξ), v (x, t) = V (ξ)]$ where $[ξ = x + ct & θ = px + rt]$ , on Eq. (1.3), we obtain:

(2.2.1)

\{\begin{matrix} (c^{2} - 1) U^{″} + (p^{2} - r^{2} - α_{1}) U + β_{1} U^{3} - 2 UV = 0, \\ (c^{2} + 1) U^{'} - 2 β_{1} {(U^{'})}^{2} - 2 β_{1} {UU}^{″} = 0 . \end{matrix}

Integrate the second equation of the system twice with zero constant of integration and submit the result in first equation in the system, we get

(2.2.2)

{AU}^{″} + BU + {CU}^{2} = 0,

where

[A = (c^{4} - 1) & B = (p^{2} - r^{2} - α_{1}) & C = β_{1} (c^{2} - 1)]

. Balance the highest order derivatives and nonlinear terms appearing in Eq. (2.2.2)

\Rightarrow (U^{″} & U^{3}) \Rightarrow (N = 1)

. So that, by using Generalized Exp-Function method, we get the exact solution of Eq. (2.2.2). That solutionis the same solution of Eq. (2.1.2) and is in the form (2.1.3). Substituting (2.1.3) and its derivative into Eq. (2.2.2) and collecting all term with the same power of

[a^{if (ξ)} where i = 0, 1, 2, 3]

, we get the system of algebraic equations by solving it, we obtain:

B = \frac{- 2 A σ (- a_{0}^{2} σ + α a_{1}^{2})}{a_{1}^{2}}, C = \frac{- 2 A σ^{2}}{a_{1}^{2}}, β = \frac{2 σ a_{0}}{a_{1}}, a_{0} = a_{0}, a_{1} = a_{1} .

So that, the exact traveling wave solution will be in the form:

(2.2.3)

u (ξ) = a_{0} + a_{1} a^{f (ξ)} .

Thus, the solitary traveling wave solutions:

When ( $β^{2} - α σ < 0 & σ \neq 0$ )

(2.2.4)

u (ξ) = a_{0} + a_{1} [\frac{- β}{σ} + \frac{\sqrt{- (β^{2} - α σ)}}{σ} \tan (\frac{\sqrt{- (β^{2} - α σ)}}{2} ξ)],

(2.2.5)

u (ξ) = a_{0} + a_{1} [\frac{- β}{σ} + \frac{\sqrt{- (β^{2} - α σ)}}{σ} \cot (\frac{\sqrt{- (β^{2} - α σ)}}{2} ξ)] .

When ( $β^{2} - α σ > 0 & σ \neq 0$ )

(2.2.5)

u (ξ) = a_{0} + a_{1} [\frac{- β}{σ} - \frac{\sqrt{(β^{2} - α σ)}}{σ} \tanh (\frac{\sqrt{(β^{2} - α σ)}}{2} ξ)],

(2.2.6)

u (ξ) = a_{0} + a_{1} [\frac{- β}{σ} - \frac{\sqrt{(β^{2} - α σ)}}{σ} \coth (\frac{\sqrt{(β^{2} - α σ)}}{2} ξ)] .

When ( $β^{2} - α^{2} > 0 & σ \neq 0 & σ = - α$ )

(2.2.7)

u (ξ) = a_{0} + a_{1} [\frac{β}{α} + \frac{\sqrt{β^{2} + α^{2}}}{α} \tanh (\frac{\sqrt{β^{2} + α^{2}}}{2} ξ)],

(2.2.8)

u (ξ) = a_{0} + a_{1} [\frac{β}{α} + \frac{\sqrt{β^{2} + α^{2}}}{α} \coth (\frac{\sqrt{β^{2} + α^{2}}}{2} ξ)] .

When ( $β^{2} - α^{2} < 0 & σ \neq 0 & σ = - α$ )

(2.2.9)

u (ξ) = a_{0} + a_{1} [\frac{β}{α} + \frac{\sqrt{- (β^{2} + α^{2})}}{α} \tan (\frac{\sqrt{- (β^{2} + α^{2})}}{2} ξ)],

(2.2.10)

u (ξ) = a_{0} + a_{1} [\frac{β}{α} + \frac{\sqrt{- (β^{2} + α^{2})}}{α} \cot (\frac{\sqrt{- (β^{2} + α^{2})}}{2} ξ)] .

When ( $β^{2} - α^{2} < 0 & σ \neq 0 & σ = α$ )

(2.2.11)

u (ξ) = a_{0} + a_{1} [- \frac{β}{α} + \frac{\sqrt{- (β^{2} - α^{2})}}{α} \tan (\frac{\sqrt{- (β^{2} - α^{2})}}{2} ξ)],

(2.2.12)

u (ξ) = a_{0} + a_{1} [- \frac{β}{α} + \frac{\sqrt{- (β^{2} - α^{2})}}{α} \cot (\frac{\sqrt{- (β^{2} - α^{2})}}{2} ξ)] .

When ( $β^{2} - α^{2} > 0 & σ \neq 0 & σ = α$ )

(2.2.13)

u (ξ) = a_{0} + a_{1} [- \frac{β}{α} + \frac{\sqrt{(β^{2} - α^{2})}}{α} \tanh (\frac{\sqrt{(β^{2} - α^{2})}}{2} ξ)],

(2.2.14)

u (ξ) = a_{0} + a_{1} [- \frac{β}{α} + \frac{\sqrt{(β^{2} - α^{2})}}{α} \coth (\frac{\sqrt{(β^{2} - α^{2})}}{2} ξ)] .

When $(α σ < 0 & σ \neq 0 & β = 0)$

(2.2.15)

u (ξ) = a_{0} + a_{1} [\sqrt{\frac{- α}{σ}} \tanh (\frac{\sqrt{- α σ}}{2} ξ)],

(2.2.16)

u (ξ) = a_{0} + a_{1} [\sqrt{\frac{- α}{σ}} \coth (\frac{\sqrt{- α σ}}{2} ξ)] .

When $(β = 0 & α = - σ)$

(2.2.17)

u (ξ) = a_{0} + a_{1} [\frac{- (1 + e^{2 α ξ}) \pm \sqrt{2 (e^{4 α ξ} + 1)}}{e^{2 α ξ} - 1}],

(2.2.18)

u (ξ) = a_{0} + a_{1} [\frac{- (1 + e^{2 α ξ}) \pm \sqrt{e^{4 α ξ} + 6 e^{2 α ξ} + 1}}{e^{2 α ξ}}] .

When $(α = σ = 0)$

(2.2.19)

u (ξ) = a_{0} + a_{1} [\frac{- (1 + e^{2 β ξ}) \pm \sqrt{2 (e^{4 β ξ} + 1)}}{e^{2 β ξ} - 1}],

(2.2.20)

u (ξ) = a_{0} + a_{1} [\frac{- (1 + e^{2 β ξ}) \pm \sqrt{e^{4 β ξ} + 6 e^{2 β ξ} + 1}}{e^{2 β ξ}}] .

When $(β^{2} = α σ)$

(2.2.21)

u (ξ) = a_{0} - \frac{α a_{1} (β ξ + 2)}{β^{2} ξ} .

When $(β = k & α = 2 k & σ = 0)$

(2.2.22)

u (ξ) = a_{0} + a_{1} [e^{k ξ} - 1] .

When $(β = k & σ = 2 k & α = 0)$

(2.2.23)

u (ξ) = a_{0} + \frac{a_{1} e^{k ξ}}{1 - e^{k ξ}} .

When $(2 β = α + σ)$

(2.2.24)

u (ξ) = a_{0} + a_{1} [\frac{1 - α e^{0.5 (α - σ) ξ}}{1 - σ e^{0.5 (α - σ) ξ}}],

(2.2.25)

u (ξ) = a_{0} + a_{1} [\frac{α e^{0.5 (α - σ) ξ} + 1}{- σ e^{0.5 (α - σ) ξ} - 1}] .

When $(- 2 β = α + σ)$

(2.2.26)

u (ξ) = a_{0} + a_{1} [\frac{e^{0.5 (α - σ) ξ} + α}{e^{0.5 (α - σ) ξ} + σ}] .

When $(α = 0)$

(2.2.27)

u (ξ) = a_{0} + \frac{β a_{1} e^{β ξ}}{1 + 0.5 σ e^{β ξ}} .

When $(β = α = σ \neq 0)$

(2.2.28)

u (ξ) = a_{0} - \frac{a_{1} (α ξ + 2)}{α ξ} .

When $(β = σ = 0)$

(2.2.29)

u (ξ) = a_{0} + 0.5 a_{1} α ξ .

When $(β = α = 0)$

(2.2.30)

u (ξ) = a_{0} - \frac{2 a_{1}}{σ ξ} .

When $(β = 0 & α = σ)$

(2.2.31)

u (ξ) = a_{0} + a_{1} \tan (\frac{α ξ + C}{2}) .

When $(σ = 0)$

(2.2.32)

u (ξ) = a_{0} + a_{1} [e^{β ξ} - \frac{α}{2 β}] .

Where $k, C$ are arbitrary constant.

2.3

2.3 Maccari equation

Using the traveling wave transformation $[u (x, t) = e^{i θ} U (ξ) & v (x, y, t) = V (ξ)]$ where $[ξ = x + y + ct & θ = pxqy + rt]$ on Eq. (1.4), we obtain:

(2.3.1)

\{\begin{matrix} c (p^{2} + r) U - U^{″} - UV = 0, \\ (c + 1) V^{'} + 2 {UV}^{'} = 0 . \end{matrix}

Integrate second equation of the system (2.3.1) with zero constant of integration and submit the result into thefirst equation of the same system, we obtain:

(2.3.2)

lU - {MU}^{″} + U^{3} = 0,

where

[l = (1 - 2 p) (p^{2} + r) & M = (1 - 2 p)]

. Balance the highest order derivatives and nonlinear terms appearing in Eq. (2.3.2)

\Rightarrow (U^{″} & U^{3}) \Rightarrow (N = 1)

. So that, by using Generalized Exp-Function method, we get the exact solution of Eq. (2.2.2). That solutionis the same solution of Eq. (2.1.2) and is in the form (2.1.3). Substituting (2.1.3) and its derivative into Eq. (2.3.2) and collecting all term with the same power of

[a^{if (ξ)} where i = 0, 1, 2, 3]

, we get the system of algebraic equations by solving it, we obtain:

l = \frac{a_{1}^{2} (- β^{2} + 4 α σ)}{4 σ^{2}}, M = \frac{a_{1}^{2}}{2 σ^{2}}, a_{0} = \frac{a_{1} β}{2 σ}, a_{1} = a_{1} .

So that, the exact traveling wave solution will be in the form:

(2.3.3)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} a^{f (ξ)} .

Thus, the solitary traveling wave solutions:

When ( $β^{2} - α σ < 0 & σ \neq 0$ )

(2.3.4)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} + \frac{\sqrt{- (β^{2} - α σ)}}{σ} \tan (\frac{\sqrt{- (β^{2} - α σ)}}{2} ξ)],

(2.3.5)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} + \frac{\sqrt{- (β^{2} - α σ)}}{σ} \cot (\frac{\sqrt{- (β^{2} - α σ)}}{2} ξ)] .

When ( $β^{2} - α σ > 0 & σ \neq 0$ )

(2.3.6)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} - \frac{\sqrt{(β^{2} - α σ)}}{σ} \tanh (\frac{\sqrt{(β^{2} - α σ)}}{2} ξ)],

(2.3.7)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- β}{σ} - \frac{\sqrt{(β^{2} - α σ)}}{σ} \coth (\frac{\sqrt{(β^{2} - α σ)}}{2} ξ)] .

When ( $β^{2} - α^{2} > 0 & σ \neq 0 & σ = - α$ )

(2.3.8)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{β^{2} + α^{2}}}{α} \tanh (\frac{\sqrt{β^{2} + α^{2}}}{2} ξ)],

(2.3.8)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{β^{2} + α^{2}}}{α} \coth (\frac{\sqrt{β^{2} + α^{2}}}{2} ξ)] .

When ( $β^{2} - α^{2} < 0 & σ \neq 0 & σ = - α$ )

(2.3.9)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{- (β^{2} + α^{2})}}{α} \tan (\frac{\sqrt{- (β^{2} + α^{2})}}{2} ξ)],

(2.3.10)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{β}{α} + \frac{\sqrt{- (β^{2} + α^{2})}}{α} \cot (\frac{\sqrt{- (β^{2} + α^{2})}}{2} ξ)] .

When ( $β^{2} - α^{2} < 0 & σ \neq 0 & σ = α$ )

(2.3.11)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{- (β^{2} - α^{2})}}{α} \tan (\frac{\sqrt{- (β^{2} - α^{2})}}{2} ξ])],

(2.3.12)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{- (β^{2} - α^{2})}}{α} \cot (\frac{\sqrt{- (β^{2} - α^{2})}}{2} ξ)] .

When ( $β^{2} - α^{2} > 0 & σ \neq 0 & σ = α$ )

(2.3.13)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{(β^{2} - α^{2})}}{α} \tanh (\frac{\sqrt{(β^{2} - α^{2})}}{2} ξ)],

(2.3.14)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [- \frac{β}{α} + \frac{\sqrt{(β^{2} - α^{2})}}{α} \coth (\frac{\sqrt{(β^{2} - α^{2})}}{2} ξ)] .

When $(α σ < 0 & σ \neq 0 & β = 0)$

(2.3.15)

u (ξ) = a_{1} [\sqrt{\frac{- α}{σ}} \tanh (\frac{\sqrt{- α σ}}{2} ξ)],

(2.3.16)

u (ξ) = a_{1} [\sqrt{\frac{- α}{σ}} \coth (\frac{\sqrt{- α σ}}{2} ξ)] .

When $(β = 0 & α = - σ)$

(2.3.17)

u (ξ) = a_{1} [\frac{- (1 + e^{2 α ξ}) \pm \sqrt{2 (e^{4 α ξ} + 1)}}{e^{2 α ξ} - 1}],

(2.3.18)

u (ξ) = a_{1} [\frac{- (1 + e^{2 α ξ}) \pm \sqrt{e^{4 α ξ} + 6 e^{2 α ξ} + 1}}{e^{2 α ξ}}] .

When $(α = σ = 0)$

(2.3.19)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- (1 + e^{2 β ξ}) \pm \sqrt{2 (e^{4 β ξ} + 1)}}{e^{2 β ξ} - 1}],

(2.3.20)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{- (1 + e^{2 β ξ}) \pm \sqrt{e^{4 β ξ} + 6 e^{2 β ξ} + 1}}{e^{2 β ξ}}] .

When $(β^{2} = α σ)$

(2.3.21)

u (ξ) = \frac{a_{1} β}{2 σ} - \frac{α a_{1} (β ξ + 2)}{β^{2} ξ} .

When $(β = k & α = 2 k & σ = 0)$

(2.3.22)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [e^{k ξ} - 1] .

When $(β = k & σ = 2 k & α = 0)$

(2.3.23)

u (ξ) = \frac{a_{1} β}{2 σ} + \frac{a_{1} e^{k ξ}}{1 - e^{k ξ}} .

When $(2 β = α + σ)$

(2.3.24)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{1 - α e^{0.5 (α - σ) ξ}}{1 - σ e^{0.5 (α - σ) ξ}}],

(2.3.25)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{α e^{0.5 (α - σ) ξ} + 1}{- σ e^{0.5 (α - σ) ξ} - 1}] .

When $(- 2 β = α + σ)$

(2.3.26)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [\frac{e^{0.5 (α - σ) ξ} + α}{e^{0.5 (α - σ) ξ} + σ}] .

When $(α = 0)$

(2.3.27)

u (ξ) = \frac{a_{1} β}{2 σ} + \frac{a_{1} β e^{β ξ}}{1 + 0.5 σ e^{β ξ}} .

When $(β = α = σ \neq 0)$

(2.3.28)

u (ξ) = \frac{a_{1} β}{2 σ} - \frac{a_{1} (α ξ + 2)}{α ξ} .

When $(β = σ = 0)$

(2.3.29)

u (ξ) = 0.5 α a_{1} ξ .

When $(β = α = 0)$

(2.3.30)

u (ξ) = - \frac{2 a_{1}}{σ ξ} .

When $(β = 0 & α = σ)$

(2.3.31)

u (ξ) = a_{1} \tan (\frac{α ξ + C}{2}) .

When $(σ = 0)$

(2.3.32)

u (ξ) = \frac{a_{1} β}{2 σ} + a_{1} [e^{β ξ} - \frac{α}{2 β}] .

Where $k, C$ are arbitrary constant.

3 Discuss the results

In this section, we will discuss our results and make a comparison between our results and that obtained by using a different method in the following steps:

3.1

3.1 Firstly: new coupled Konno-Oono equation (Choi et al., 2016):

You can see when you make a comparison between our results and that obtained by GülnurYel, Haci Mehmet Baskonus, HasanBulutwho used the sine-Gordon expansion method that our results are completely new about that obtained in this research. So that, our results will provide researchers with a wide range of the possibilities and capacities to use this wonderful model in the life.

3.2

3.2 Secondly: Higgs field equation (Yel et al., 2017)

You can see when you make a comparison between our results and that obtained by M.A. Abdelkawy, A.H. Bhrawy, E. Zerrad and A. Biswas who used tanh method that our solutions (2.2.15) and (2.2.16) are equivalent with them solutions (17) and (18) when the parameters take this values $[a_{0} = 0, C = (1 - c^{2}) β_{1} σ, β = 0]$ , our solution (2.2.30) is equivalent with them solution (19) when the parameters take this value $[a_{0} = 0, β = 0, a_{1} = \pm i \sqrt{\frac{(1 + c^{2})}{2 β_{1}}}]$ and our solution (2.2.31) is equivalent with them solution (20) when the parameters take this value $[a_{0} = 0, β = 0, a_{1} = \pm i \sqrt{\frac{2 (1 - c^{2}) (r^{2} (1 - c^{2}) + α_{1})}{2 β_{1} (c^{2} - 1)}}]$ . So that, it very clear that our method (Generalized Exp-Function method) covered all solutions that given by using tanh method.

3.3

3.3 Thirdly: Maccari equation (Yel et al., 2017)

You can see when you make a comparison between our results and that obtained by M.A. Abdelkawy, A.H. Bhrawy, E. Zerrad and A. Biswas who used tanh method that our solutions (2.3.15) and (2.3.16) are equivalent with them solutions (46) when the parameters take this values $[a_{0} = 0, β = 0, a_{1} = \pm i \sqrt{1 + (p^{2} + r) (1 - 2 p)}]$ , our solution (2.3.30) is equivalent with them solution (47) when the parameters take this value $[a_{0} = 0, β = 0, a_{1} = - \frac{σ \sqrt{2 - 4 p}}{2}]$ and our solution (2.3.31) is equivalent with them solution (48) when the parameters take this value $[a_{0} = 0, β = 0, a_{1} = \pm i \sqrt{1 + (p^{2} + r) (1 - 2 p)}]$ . So that, it very clear that our method (Generalized Exp-Function method) covered all solutions that given by using tanh method.

4 Conclusion

In this paper, we succeed in applying Generalized Exp-Function method and obtaining traveling wave solution and new solitary traveling wave solutions for each of the following models (new coupled Konno-Oono equation, Higgs field equation and Maccari equation). We believe that our results of these models will be useful for young researchers who are going to study the exact solutions of nonlinear partial differential equations (NLPDEs.). We also hope that our results will be interesting for some referees. According to above discussion and all solutions that obtained by using Generalized Exp-Function method, we can notice that: Generalized Exp-Function method is very simple, direct, effective and powerful method to apply it for many nonlinear evolution equations.

References

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Introduction

Application

New coupled Konno-Oono equation
Higgs field equation
Maccari equation

Discuss the results

Firstly: new coupled Konno-Oono equation (Choi et al., 2016):
Secondly: Higgs field equation (Yel et al., 2017)
Thirdly: Maccari equation (Yel et al., 2017)

Conclusion

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[1] Seadawy Aly R., . Travelling wave solutions of a weakly nonlinear two-dimensional higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves. Eur. Phys. J. Plus. 2017;132:29. 1:13
[Google Scholar]

[2] Arshad M., Seadawy Aly, Lu Dianchen, Wang Jun, . Travelling wave solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations. Results Phys.. 2016;6:1136-1145.
[Google Scholar]

[3] Liu Shikuo, . Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A. 2001;289(1):69-74.
[Google Scholar]

[4] Naher Hasibun, Aini Abdullah Farah, Ali Akbar M., . New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method. J. Appl. Math. 2012:2012.
[Google Scholar]

[5] Seadawy Aly R., Lu Dianchen, Khater Mostafa, . Solitary wave solutions for the generalized Zakharov-kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation. J. Ocean Eng. Sci.. 2017;2:137-142.
[Google Scholar]

[6] Yang, Xiao-Jun, Tenreiro Machado, J.A., Baleanu, Dumitru, Cattani, Carlo, 2016. On exact traveling-wave solutions for local fractional Korteweg-de Vries equation. Chaos: Interdiscip. J. Nonlinear Sci. 26(8), 084312

[7] Yang Xiao-Jun, Gao Feng, Srivastava H.M., . Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Comput. Math. Appl.. 2017;73(2):203-210.
[Google Scholar]

[8] Lu Dianchen, Seadawy Aly, Arshad M., Wang Jun, . New solitary wave solutions of (3+1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications. Results Phys.. 2017;7:899-909.
[Google Scholar]

[9] Ebaid A., . Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method. Phys. Lett. A. 2007;365(3):213-219.
[Google Scholar]

[10] Pava Jaime Angulo, . Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg–de Vries equations. J. Differ. Equ.. 2007;235(1):1-30.
[Google Scholar]

[11] El-Wakil S.A., Abdou M.A., . New exact travelling wave solutions using modified extended tanh-function method. Chaos Solitons Fractals. 2007;31(4):840-852.
[Google Scholar]

[12] Wazwaz Abdul-Majid, . The tanh method for traveling wave solutions of nonlinear equations. Appl. Math. Comput.. 2004;3:713-723.
[Google Scholar]

[13] Khater, Mostafa, M.A., 2015. Extended Exp (-(ξ))-Expansion Method for Solving the Generalized Hirota-Satsuma Coupled KdV System.

[14] Yang, Xiao-Jun, Tenreiro Machado, J.A., Baleanu, Dumitru, 2017. Exact Traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(4), 1740006 (7 pages).

[15] Gao, Feng, Yang, Xiao-Jun, Zhang, Yu-Feng, 2017. Exact traveling wave solutions for a new non-linear heat transfer equation. Therm. Sci. 21(4).

[16] Seadawy Aly R., Lu Dianchen, Khater Mostafa M.A., . Bifurcations of traveling wave solutions for Dodd–Bullough–Mikhailov equation and coupled Higgs equation and their applications. Chin. J. Phys.. 2017;55(4):1310-1318.
[Google Scholar]

[17] Lee Jonu, Sakthivel Rathinasamy, . Exact travelling wave solutions of a variety of Boussinesq-like equations. Chin. J. Phys.. 2014;52:939-957.
[Google Scholar]

[18] Chun Changbum, Sakthivel Rathinasamy, . Homotopy perturbation technique for solving two-point boundary value problems – comparison with other methods. Comput. Phys. Commun.. 2010;181:1021-1024.
[Google Scholar]

[19] Lee Jonu, Sakthivel Rathinasamy, . Direct approach for solving nonlinear evolution and two-point boundary value problems. Pramana J. Phys.. 2013;81:893-909.
[Google Scholar]

[20] Seadawy, Aly R., Lu, Dianchen, Khater, Mostafa M.A., 2017c. Bifurcations of solitary wave solutions for the three dimensional Zakharov-Kuznetsov-Burgers equation and Boussinesq equation with dual dispersion. Optik-Int. J. Light Electron Optics.

[21] Lu Dianchen, Seadawy Aly R., Khater MostafaM.A., . Bifurcations of new multi soliton solutions of the van der Waals normal form for fluidized granular matter via six different methods. Results Phys.. 2017;7:2028-2035.
[Google Scholar]

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

[23] Zhou Yubin, Wang Mingliang, Wang Yueming, . Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys. Lett. A. 2003;308(1):31-36.
[Google Scholar]

[24] Seadawy Aly R., Lu Dianchen, Khater Mostafa M.A., . New wave solutions for the fractional-order biological population model, time fractional burgers, Drinfel’d–Sokolov–Wilson and system of shallow water wave equations and their applications. Eur. J. Comput. Mech. 2017
[Google Scholar]

[25] Khater Mostafa M.A., Seadawy Aly R., Lu Dianchen, . Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and Time-fractional Cahn-Allen equation. Results Phys.. 2017;7:2325-2333.
[Google Scholar]

[26] Konno Kimiaki, Oono Hitoshi, . New coupled integrable dispersionless equations. J. Phys. Soc. Jpn.. 1994;63(2):377-378.
[Google Scholar]

[27] Konno Kimiaki, Kakuhata Hiroshi, . Interaction among growing, decaying and stationary solitons for coupled integrabledispersionless equations. J. Phys. Soc. Jpn.. 1995;64(8):2707-2709.
[Google Scholar]

[28] Souleymanou Abbagari, . Traveling wave-guide channels of a new coupled integrabledispersionless system. Commun. Theor. Phys.. 2012;57(1):10.
[Google Scholar]

[29] Khalique Chaudry Masood, . Exact solutions and conservation laws of a coupled integrabledispersionless system. Filomat. 2012;26(5):957-964.
[Google Scholar]

[30] Hon Y.C., Fan E.G., . A series of exact solutions for coupled Higgs field equation and coupled Schrödinger-Boussinesq equation. Nonlinear Anal.: Theory Methods Appl.. 2009;71(7):3501-3508.
[Google Scholar]

[31] Hase Yoko, Satsuma Junkichi, . An N-soliton solution for the nonlinear Schrödinger equation coupled to the Boussinesq equation. J. Phys. Soc. Jpn.. 1988;57(3):679-682.
[Google Scholar]

[32] Rostamy Davood, . The first integral method for solving Maccari’s system. Appl. Math.. 2011;2(02):258.
[Google Scholar]

[33] Zhao Hong, . Applications of the generalized algebraic method to special-type nonlinear equations. Chaos Solitons Fractals. 2008;36(2):359-369.
[Google Scholar]

[34] Arshad M., Seadawy Aly, Lu Dianchen, . Elliptic function and Solitary Wave solutions of the higher-order nonlinear Schrodinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability. Eur. Phys. J. Plus. 2017;132:371.
[Google Scholar]

[35] Jabbari A., Kheiri H., Bekir A., . Exact solutions of the coupled Higgs equation and the Maccari system using He’s semi-inverse method and-expansion method. Comput. Math. Appl.. 2011;62(5):2177-2186.
[Google Scholar]

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