Accepted Manuscript

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

Original article

Journal of

King Saud University

-Science

King Saud University

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

PII: DOI:

Reference:

S1018-3647(17)31092-3 https://doi.org/10.1016/j.jksus.2017.11.003 JKSUS 553

To appear in:

Journal of King Saud University - Science

Received Date: 21 October 2017

Accepted Date: 23 November 2017

Please cite this article as: M.M.A. Khater, A.R. Seadawy, D. Lu, Dispersive solitary wave solutions of New Coupled Konno-Oono, Higgs field and Maccari equations and their applications, Journal of King Saud University -Science (2017), doi: https://doi.org/10.1016/j.jksus.2017.11.003

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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

1 1 O H4 1 H4

Mostafa M.A. Khater1, Aly R. Seadawy12 and Dianchen Lu' 1 Department of Mathematics, Faculty of Science, Jiangsu University, China. Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia.

Corresponding Authors:(Aly Seadway, Dianchen Lu) E-mail:Aly742001@yahoo.com, dclu@ujs.edu.cn

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.

' K d J*

Keywords:New coupled Konno-Oono equation; Higgs field equation; Maccari equation; Generalized Exp-Function method; Traveling wave solutions; Solitary wave solutions.

PACS/topics:02.30.Ik, 02.30.Jr, 05.45.Yv

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 naturalphenomena 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 -expansion method, Novel -expansion method, modified -expansion meth°^ the ^j>~j -

expansion method, the e-^® -expansion method, extended e-^® -expansion method, the extended tanh-function method, the Kudryashov, modified Kudryashov methods, The improved

tan (^-expansion method, modified simple equation method and so on. [1]- [24]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 [25]. These equations called new coupled Konno-Oono equation, Higgs field equation, and Maccari equation. In the following order, we give anintroduction 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 [26]-[29]. In 1990, Konno et al. [26] presented more general version of coupled integrable dispersionless system defined as

qxt-2aqrx-2pqsx+ y (r s)x = 0, rxt- arrx-2p (2qqx+ rxs)-2yqxr = 0, (1.1)

ysxt-2fissx + 2a(2qqx+rsx)-2ysqx = 0,

where (a, fí, y) 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[30]. 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:

(Vt + 2uux = 0,

\ux t — 2 v u = 0, (

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 [31, 32]. The general form of the complex couple Higgs field equation be in the following from

^ (1-3)

(ut t — uxx — aiu + Pi\u\2 u — 2 u v = 0,

f vtt + v x x — Pi (|u|2)xx = 0, where Pi&aiare 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 [33]-[37]. 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

ri ut + uxx + uv = 0,

(vt + vy + (M2)x = 0. (L4)

This equation is called integrable (2 + 1)-dimensional nonlinear Maccari system [38]. 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 givei

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. New coupled Konno-Oono equation:

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

Of—c k v' + 2 k uu' = 0, (2 11)

I —ck2u'' — 2 uv = 0. ( ' ' )

Integrating first equation in the system (2.1.1) and submit the result into second equation in the

same system, we obtain

pu'' + 2u3 + 25u = 0J (2.1.2)

where(5 & p = c2k2) are a constant of integration. Balance the highest order derivatives and nonlinear terms appearing in Eq. (2.1.2)^ (u''&u3) ^ (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:

utf) = a0 + a1af(2.1.3) Substituting (2.1.3) and its derivative into Eq. (2.1.2) and collecting all term with the same power of[a* f^where i = 0,1,2,3\,we get the system of algebraic equations by solving it, we obtain:

-ai _ —_

P G2 g2

a\(-p2 + 4œg)

4 G2 2 G

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

Thus, the solitary traveling wave solutions: When (P2 - œg < 0 &g * 0)

_aiP _ aQ — , ai — ai.

m aiP _L_

u(0 _-—+ai 2 G

aiP _L_

u(0 _-—+ai 2 G

-P j-{p2 - ŒG) --\--tan

When (P2 - œg > 0 &g * 0)

-p + y-w - œg) jj^y-(p2 - âô) ^

'V-(P2 - œg) V 2

m aiP _L_

u(0 _ -— + ai 2 G

2iP + n 17 + ai

œg) (V(P2 - ŒG)

tanh I-E

-P V(P2 - œg) , (v(p2-œg)

When (P2- a2 >0&g*0&g_ -a)

(2.1.4)

(2.1.5)

(2.1.6)

(2.1.7)

(2.1.8)

m aiP _L_

h0U(E) _-— + ai 2 G

P VP2 + a2 VP2 + a2

—\--tanh I-E

ft\ aiP _L_

u(E) _-— + ai

P VP2 + a2 , VP2 + a2

- +-joth I---E

When (P2- a2 <0&g*0&g_ -a)

(2.1.9)

u(f) = -— + ai 2 o

u(f) = -— + ai 2 o

P V—(P2 + a2) /V—(P2 + a2) c

- +-tan I---f

a a \ 2

P V—(P2 + a2) /V—(P2 + a2) c

- +-cot I---f

When (P2 — a2 < 0 & o * 0 & o = a)

u(f) = -— + ai 2 o

u(f) = -— + ai 2 o

P V—(P2 —a2) /V—P' — a

---I--tan I-

P V—(P2 —a2)

---I--cot

When (P2 — a2 > 0 & o * 0 & o = a)

u(f) = -—+ ai 2 o

u(f) = -—+ ai 2 o

—P+VP

(2.1.12)

V(P2 — a2)

tanh I---f

P V(P2 —a2) , /V(P2 — a2) c

- +-coth I---f

When (ao < 0 & o * 0 & P = 0)

(2.1.13)

(2.1.14)

(2.1.15)

(2.1.16)

i (P = <

Ou(f) = -— + ai 2o

I—a /V—a o N i_tanh^__f

u(f) = -—+ ai 2 o

I—a /V—a o N

vcoth(—f

When (P = 0 &a = —o)

(2.1.17)

(2.1.18)

u(f) = -— + ai 2 o

— (1 + e2± V2 (e4^ + 1)

i2a( _ 1

u(f) =-— + ai 2 o

— (1 + e2 ± Ve4a^ + 6e2a^ + 1

When (a = o = 0)

u(f) = -— + ai 2 o

— (1 + e2^)±V2(e4^ + 1)

u(f) = -— + ai 2 o

(1 + e2^)±Ve4^ + 6e2^ + 1l

e2^ ^VT

(2.1.21)

When (P2 = a o)

aiP —aai(Pf +

u(f) = To+—

When (P = k &a = 2 k&o = 0)

u(f)=0i-P+ai[ek^ —1]. 2 o

When (P = k &o = 2 k&a = 0)

When (2 P = a + o)

u(f) = -

2 o 1 —

1— a e0.5 (a-aO£-

1 — o e0.5 (a-aO£

u(f) = ^7+ ai

e0.5 (a-aO£ + 1

When (—2 P = a + o)

u(f) = -—+ ai 2 o

—o e0.5 (a-aO£ — 1 e0.5(a-a)^ + a

When (a = 0)

e0.5 (a-o"H + o

(2.1.22)

(2.1.23)

(2.1.24)

(2.1.25)

(2.1.26)

(2.1.27)

aiP Paie^ u(f) = -— +

When (P = a = o * 0)

When (P = o = 0)

When (P = a = 0)

When (P = 0 &a = o)

u(f) =

2 o 1 + 0.5 o e^' ai P ai(a f + 2)

u(f) = —--+ 0.5 a ai f.

u(f) =

ai P 2ai

2 o of' ai P , . /a f + C\

ai p /

u(f) =--+ ai tan (-

v 2o i V 2

When (o = 0)

ai P r o <r a 2 o L 2 p

(2.1.30)

(2.1.31)

(2.1.32)

(2.1.33)

(2.1.34)

Where k, C are arbitrary constant.

2.2. Higgs field equation:

Using the traveling wave transformation [u(x, t) = eie U(f), v(x, t) = 7(f)] where[f = x + ct&0 = px + rt], on Eq. (1.3), we obtain:

f(c2 — 1)U'' + (p2 — r2 — ajU + Pi^3 — 2 U 7 = 0,

I (c2 + 1)U'' — 2 P^U')2 — 2 Pi U U'' = 0. 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

(2.2.1)

4 U'' + B U + C U2 = 0, (2.2.2)

where [4 = (c4 — 1)&B = (p2 — r2 — ai)&C = Pi(c2 — 1)] . Balance the highest order derivatives and nonlinear terms appearing in Eq. (2.2.2)^ (U''& U3) ^ (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[aif(^where i = 0,1,2,3\, we get the system of algebraic equations by solving it, we obtain:

—2 Ag (-ag g + a al) —2 Ag2 2 g aQ

■, C — 2 i ft — i aQ — aQ, ax — a^.

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

u(E) _ aQ + Thus, the solitary traveling wave solutions: When (P2 - œg <0&g *0)

u(E) _ aQ + ai

P V-(P2 - œg) (V-P^-aG

u(E) _ aQ + ai

-l+VE^^jot,

When (P2 - œg > 0 &g * 0)

u(E) _ aQ + ai

u(E) _ aQ + ai

-P V(P2 - œg)

---tanh

-P V(P2 - œg)

fV(P2 - œg)

aothl^^E

When (P2- a2 >0&g*0&g_ -a)

(2.2.3)

(2.2.4)

(2.2.5)

(2.2.5)

(2.2.6)

u(0 = CLq+CLI

u(E) _ aQ + ai

P VP2 + a2 r /VF +

- + aa

—\--aoth

VP2 + a

When (P2- a2 <0&g*0&g_ -a)

(2.2.7)

(2.2.8)

u(E) _ aQ + ai

P V-(P2 + a2) NE<PPT±a2)

- +-tan I---E

(2.2.9)

u(f) = a0 + ai When (P2 — a2 < 0 & o * 0 & o = a)

P V—(P2 + a2) /V—(P2 + a2) c

- +-cot I---f

u(f) = a0 + ai

P , V—(P2 — a2^ (V—(P2 — a2) ,„

—a—tan(—2—fii-

— + aa

u(f) = a0 + ai When (P2 — a2 > 0 & o * 0 & o = a)

P , V—(P2 — a2) /V—P^ — a

u(f) = a0 + ai

(2.2.11)

u(f) = a0 + ai When (ao < 0 & o * 0 & P = 0)

P + V(P2 —a2) coth IV(P2 —a2)

—- + aa

a / V—a o u(f) = a0 + ai I I— tanh I —-— f

u(f) = a0 + ai When (P = 0 &a = —o)

a / V—a o coth I —-— f

u(f) = a0 + ai

(1 + e2 «£) ± V2 (e4^ + 1)

u(f) = a0 + ai When (a = o = 0)

— (1 + e2± Ve4a^ + 6e2a^ + 1

(2.2.12)

(2.2.13)

(2.2.14)

(2.2.15)

(2.2.16)

(2.2.17)

u(f) = a0 + ai

—(1 + e2^)±V2(e4^ + 1)

• —1

(2.2.19)

u(f) = a0 + ai

(1 + e2^T) ± Ve4^ + 6e2^ + 1]

When (P2 = a o)

u(f) = a0

a ai (P f + 2)

When (P = k &a = 2 k&o = 0)

When (P = k &o = 2 k&a = 0)

When (2 P = a + o)

u(f) = a0 + ai[ek^ —1]. 0)

aiek^ u(f) = a0 -kT

u(f) = a0 + ai I-

(2.2.20)

0.5 (a-a-)f

u(f) = a0 + ai

— oe0.5(a-^)T ae°5(a-ff)T + 1

When (—2 P = a + o) When (a = 0) When (P = a = o * 0) When (P = o = 0)

u(f) = a0 + ai

— o e0.5 (a-aOT — 1

e0.5(a-a)T + a

e0.5 (a-aOT + o

u(f) = a0 +

u(f) = a0

1 + 0.5 o ai(a f + 2)

(2.2.21)

(2.2.22)

(2.2.23)

(2.2.24)

(2.2.25)

(2.2.26)

(2.2.27)

(2.2.28)

u(E) — aQ + 0.5 a1a (2.2.29)

When (ft — a —0)

u($) — aQ-

When (ft — 0&a — g)

When (g — 0)

u(0 — aQ----. (2.2.30)

utf) — aQ + a1tan (^^r^J- (2.2.31)

u(%) — aQ + a1 Where k, C are arbitrary constant.

^—Yft} (2-2-32)

2.3. Maccari equation:

Using the traveling wave transformation [u(x, t) — eieU(Q &v(x,y, t) — V(<f)\ where — x + y + ct&6 — p x q y + r t\on Eq. (1.4), we obtain:

«p2+r)U-U" — UV = 0,

I (c + 1)V' + 2UV' — 0. v ^

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:

IU — MU" + U3 — 0, (2.3.2)

where [I — (1 — 2 p)(p2 + r)&M — (1 — 2 p)\. Balance the highest order derivatives and nonlinear terms appearing in Eq. (2.3.2) ^ (U"& U3) ^ (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* f(^where i — 0,1,2,3\, we get the system of algebraic equations by solving it, we obtain:

a2(—ft2 + 4aG) al a1 ft

I —--—2-lM — o 21 aQ — ^—

4 g2 2 g2 2 G

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

Thus, the solitary traveling wave solutions: When (P2 - «g < 0 & g * 0)

u(f) = -—+ Ol

u(0 = -—+ Ol

-P | V-(P2 - «G^ ^v-(P2-«G) rii +--ton I --- f I I,

When (P2 - «g > 0 & g * 0)

u(0 = -— + Ol

U(0 = --+ Ol

(2.3.4)

-P - V(P2 - «g) tanh |V(P2 - «g)

-P V(P2-«G) cjth|V(P2-«G) f

When (P2-«2>0&g*0&g = -«)

(2.3.5)

(2.3.6)

(2.3.7)

m OlP+ p+VPï+1^, JJE+EE.*

2 G « « \ 2

When (P2-«2<0&g*0&g = -«)

u(f) = "2a" + Ol

P y-(P2 + «2) /V-(P2 + «2) c

- +-tOn I---f

« « 2

(2.3.8)

(2.3.8)

(2.3.9)

u(f) = -— + Ol

P V-(P2 + «2) /V-(P2 + «2) c

- +- cot I --- f

« « 2

When (P2 - «2 < 0 & g * 0 & g = «)

(2.3.10)

u($) —-— + ai 2 G

ft j—(ft2 — a2) (J—(ft2 — a2)

— - +-tan I---$

u($) —-— + ai

>2 „2

ft f V—(ft2 — a2) J J—(ft2—a2) ru

a+—a—cot(—2—$"

When (ft2 — a2 > 0 &g ^ 0 &g — a)

u($) —-—+a1

u($) —-—+a1

ft | V(ft2 — a2) ^ ,

---\--tanh .

When (aG < 0 &g ^ 0 & ft — 0)

(2.3.12)

l + V(ftT—ar) c0th(V(^ — aI) $

i^—a g n coth I —-— $

When (ft — 0&a — —g)

u($) — a1

u($}—a1 [R 0

u($) — a \—(1 + e2^)±V2(e*aS + 1)

,2a£ — 1

{l + e2a^)± VeAa$ + 6e2a$ + 1

When (a — g — 0)

(2.3.13)

(2.3.14)

(2.3.15)

(2.3.16)

(2.3.17)

(2.3.18)

u($) —-— + a1

(1 + e2^)±V2(e*PZ + 1)

>2PZ — 1

u($) —-— + a1

— (1 + e2^)± Ve4PS + 6e2PS + 1] v J~ (2.3.20)

When (ft2 — a g)

2 g ft2$

When (ft — k&a —2 k&G — 0)

When (ft — k&G — 2 k&a — 0)

a1ft aa1(ft$ + 2) u($) —-----. (2.3.21)

u($)—^+a1[ek^ — 1\. (2.3.22)

a1 ft a1ek1

uv—ff+f—eti- (2'3'23)

When (2 ft — a + f)

1— aeQ.s(a-<r)1]

u($) —-—+a1

1 — G eQS (a-*)l

When (—2 ft — a + f)

a1ft \ a e°-s + 1

rm, a1 ft \eQ's(a-a)1+ a u($) ——--+ a1

2 g ^eQ~s(a-a)1 + g

When (a — 0)

^ a1ft + a1ft 1

When (ft — a — g ± 0)

i (ft— a —

(2.3.24)

(2.3.25)

(2.3.26)

u($) —m+ __(2.3.27)

u($) 2g+ 1 + 0.5g ep1' ( )

u($)—0±l—a1(a$;2\ (2.3.28)

2 g a $

When (ft — g — 0)

u($) — 0.5 aa1$. (2.3.29)

When (ft = a = 0)

When (0 = 0 &a = a)

When (a = 0)

u(<f) = a1tan ^

u(0 = -—+a1 2 a

af + C

Where k, C are arbitrary constant.

3. Discuss the results:

In this section, we will discuss our results and make a com] obtained by using adifferent method in the following s

(2.3.31)

(2.3.32)

between our results and that

Firstly: new coupled Konno-Oono equation[39]:

You can see when you make a comparison between our results and that obtained by GulnurYel, 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.

Secondly: Higgs field equation [40]:

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 [a0 = 0, C = (1 — c2)0± <r,fi = 0], our solution (2.2.30) is equivalent with them

a0 = 0,0 = 0, a1 = ±i

(1+c2)

and our

solution (19) when the parameters take this value solution (2.2.31) is equivalent with them solution (20) when the parameters take this value

a0 = 0,0 = 0, a1 = ±i

2 (l-c2)(r2(l-c2) + «i)

So that, it very clear that our method

2 Pi (c2-1)

(Generalized Exp-Function method) covered all solutions that given by using tanh method.

Thirdly: Maccari equation[40]:

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[a0 = 0, ft = 0, at = ±i jl + (p2 + r)(1 — 2 p) ], our solution (2.3.30) is

equivalent with them solution (47) when the parameters take this value la0 = 0, ft = 0, at =

j 12—4 pi

--—and our solution (2.3.31) is equivalent with them solution (48) when the parameters

take this value [a0 = 0,ft = 0, a1 = ±i + (p2 + 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

1 ] Aly R. Seadawy, Travelling wave solutions of a weakly nonlinear two-dimensional higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves, The European Physical Journal Plus 132 (2017) 29: 1:13.

2] M. Arshad, Aly Seadawy, Dianchen Lu and Jun Wang, Travelling wave solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations, Results in Physics 6 (2016) 1136-1145.

3] Liu, Shikuo, et al. "Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations." Physics Letters A 289.1 (2001): 69-74. ^^

4] Naher, Hasibun, Farah Aini Abdullah, and M. Ali Akbar. "New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method." Journal of Applied Mathematics 2012 (2012).

5] Aly R. Seadawy, Dianchen Lu and Mostafa Khater, Solitary wave solutions for the generalized Zakharov-kuznetsov- Benjamin-Bona-Mahony nonlinear evolution equation, Journal of Ocean Engineering and Science 2 (2017) 137-142.

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

7] Xiao-Jun Yang, Feng Gao, H.M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Computers & Mathematics with Applications,73(2), (2017) 203-210.

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

9] Ebaid, A. "Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method." Physics Letters A 365.3 (2007): 213-219.

10] Pava, Jaime Angulo. "Nonlinear stability of periodic traveling wave solutions to the Schrodinger and the modified Korteweg-de Vries equations." Journal of Differential Equations 235.1 (2007): 1-30.

11] El-Wakil, S. A., and M. A. Abdou. "New exact travelling wave solutions using modified extended tanh-function method." Chaos, Solitons& Fractals 31.4 (2007): 840-852.

12] Wazwaz, Abdul-Majid. "The tanh method for traveling wave solutions of nonlinear equations." Applied Mathematics and Computation154.3 (2004): 713-723.

13] Khater, Mostafa MA. "Extended Exp (-(^-Expansion Method for Solving the Generalized Hirota-Satsuma Coupled KdV System." (2015).

14] XIAO-JUN YANG, J. A. TENREIRO MACHADO, DUMITRU BALEANU, EXACT TRAVELING-WAVE SOLUTION FOR LOCAL FRACTIONAL BOUSSINESQ EQUATION IN FRACTAL DOMAIN. Fractals, Vol. 25, No. 4, (2017) 1740006 (7 pages)

15] Feng GAO; Xiao-Jun YANG; Yu-Feng ZHANG, EXACT TRAVELING WAVE SOLUTIONS FOR A NEW NON-LINEAR HEAT TRANSFER EQUATION. Thermal Science, 21(4) (2017).

16] Seadawy, Aly R., Dianchen Lu, and Mostafa MA Khater. "Bifurcations of traveling wave solutions for Dodd-Bullough-Mikhailov equation and coupled Higgs equation and their applications." Chinese Journal of Physics 55.4 (2017): 1310-1318.

17] Jonu Lee and Rathinasamy Sakthivel, Exact Travelling Wave Solutions of a Variety of Boussinesq-Like Equations, Chinese Journal of Physics 52 (2014) 939-957.

Changbum Chun and Rathinasamy Sakthivel, Homotopy perturbation technique for solving two-point boundary value problems - comparison with other methods, Computer Physics communications, 181 (2010), 1021-1024.

Jonu Lee and Rathinasamy Sakthivel, Direct approach for solving nonlinear evolution and two-point boundary value problems, Pramana Journal of Physics 81 (2013) 893-909.

[20] Seadawy, Aly R., Dianchen Lu, and Mostafa MA Khater. "Bifurcations of solitary wave solutions for the three dimensionalZakharov-Kuznetsov-Burgers equation and Boussinesq equation with dual dispersion." OptikInternational Journal for Light and Electron Optics (2017).

[21] Lu, Dianchen, Aly R. Seadawy, and Mostafa MA Khater. "Bifurcations of new multi soliton solutions of the van der Waals normal form for fluidized granular matter via six different methods." Results in Physics 7 (2017): 2028-2035.

[22] Seadawy, Aly R. "Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma." Computers & Mathematics with Applications 67.1 (2014): 172-180.

[23] Zhou, Yubin, Mingliang Wang, and Yueming Wang. "Periodic wave solutions to a coupled KdV equations with variable coefficients." Physics Letters A 308.1 (2003): 31-36. ^^^

[24] Aly R. Seadawy, Dianchen Lu &Mostafa M. A. Khater."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.", European Journal of Computational Mechanics(2017)

[25] Khater, Mostafa MA, Aly R. Seadawy, and Dianchen Lu. "Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and Time-fractional Cahn-Allen equation." Results in Physics 7 (2017): 2325-2333. ^ ^^^

[26] Konno, Kimiaki, and Hitoshi Oono. "New coupled integrabledispersionless equations." Journal of the Physical Society of Japan63.2 (1994): 377-378. ^

[27] Konno, Kimiaki, and Hiroshi Kakuhata. "Interaction among growing, decaying and stationary solitons for coupled integrabledispersionless equations." Journal of the Physical Society of Japan64.8 (1995): 27072709. ^^^

[28] Konno, Kimiaki, and Hiroshi Kakuhata. "Interaction among growing, decaying and stationary solitons for coupled integrabledispersionless equations." Journal of the Physical Society of Japan64.8 (1995): 27072709.

[29] Souleymanou, Abbagari, et al. "Traveling wave-guide channels of a new coupled integrabledispersionless system." Communications in Theoretical Physics 57.1 (2012): 10.

[30] Khalique, ChaudryMasood. "Exact solutions and conservation laws of a coupled integrabledispersionless system." Filomat 26.5 (2012): 957-964. ^^^

[31] Hon, Y. C., and E. G. Fan. "A series of exact solutions for coupled Higgs field equation and coupled Schrodinger-Boussinesq equation." Nonlinear Analysis: Theory, Methods & Applications 71.7 (2009): 35013508. ^^ J

[32] Hase, Yoko, and Junkichi Satsuma. "An N-soliton solution for the nonlinear Schrodinger equation coupled to the Boussinesq equation." Journal of the Physical Society of Japan 57.3 (1988): 679-682.

[33] Rostamy, Davood, et al. "The first integral method for solving Maccari's system." Applied Mathematics 2.02 (2011): 258. ~

[34] Zhao, Hong. "Applications of the generalized algebraic method to special-type nonlinear equations." Chaos, Solitons& Fractals 36.2 (2008): 359-369.

[35] M. Arshad, Aly Seadawy and Dianchen Lu, 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, The European Physical Journal Plus, (2017) 132: 371.

[36] Jabbari, A., H. Kheiri, and A. Bekir. "Exact solutions of the coupled Higgs equation and the Maccari system using He's semi-inverse method and-expansion method." Computers & Mathematics with Applications 62.5 (2011): 2177-2186.

[37] Bekir, Ahmet. "New exact travelling wave solutions of some complex nonlinear equations." Communications in Nonlinear Science and Numerical Simulation 14.4 (2009): 1069-1077.

[38] Maccari, Attilio. "The Kadomtsev-Petviashvili equation as a source of integrable model equations." Journal of Mathematical Physics37.12 (1996): 6207-6212.

[39] J.H. Choi, H. Kim, R. Sakthivel, Exact travelling wave solutions of reaction-diffusion models of fractional order, Journal of Applied Analysis and Computation 7 (2016), 236-248.

[40] Yel, Gülnur, Haci Mehmet Baskonus, and HasanBulut. "Novel archetypes of new coupled Konno-Oono equation by using sine-Gordon expansion method." Optical and Quantum Electronics 49.9 (2017): 285.