Bifurcations of solitary wave solutions for

Results in Physics 9 (2018) 142–150

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Bifurcations of solitary wave solutions for (two and three)-dimensional nonlinear partial differential equation in quantum and magnetized plasma by using two different methods Mostafa M.A. Khater a, Aly R. Seadawy b,c,⇑, Dianchen Lu a,* a b c

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia Mathematics Department, Faculty of Science, Beni-Suef University, Egypt

a r t i c l e

i n f o

Article history: Received 18 December 2017 Accepted 5 February 2018 Available online 22 February 2018 Keywords: Two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma Extended 0 simple equation method Novel GG -expansion method Traveling wave solutions Solitary wave solutions

a b s t r a c t In this research, we study new two techniques that called the extended simple equation method and the 0 novel GG -expansion method. The extended simple equation method depend on the auxiliary equation d/ ¼ a þ k/ þ l/2 which has three ways for solving depends on the specific condition on the paramedn ters as follow: When ðk ¼ 0Þ this auxiliary equation reduces to Riccati equation, when ða ¼ 0Þ this auxiliary equation reduces to Bernoulli equation and when ða – 0; k – 0; l – 0Þ we the general solutions 0 of this equation while thenovel GG -expansion method depends also on similar auxiliary equaauxiliary G0 G0 2 G0 0 which depend also on the value of ðk2 4ðv 1ÞlÞ and the specific tion G ¼ l þ k G þ ðv 1Þ G condition on the parameters as follow: When ðk ¼ 0Þ this auxiliary equation reduces to Riccati equation, when ðl ¼ 0Þ this auxiliary equation reduces to Bernoulli equation and when ðk2 – 4ðv 1ÞlÞ we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear KadomtsevPetviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified ZakharovKuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions. Ó 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

Introduction In this research, we treat with two important basic equations that called two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. Both of these equations were concluded by using the reductive perturbation method for the ion dynamics of nonlinear ion-acoustic equations [1]:

⇑ Corresponding authors at: Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia (Aly R. Seadawy); Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China ( Dianchen Lu). E-mail addresses: [email protected] (A.R. Seadawy), [email protected] (D. Lu).

8 nt þ r ðnuÞ ¼ 0; > > > < ut þ ðu rÞu ¼ rw þ u ^ Xex ; 2 > > r w ¼ ne n; > : ne ¼ expw;

ð1:1Þ

where n is the non-dimensional ion number density, u is the ion fluid velocity, wðx; y; tÞ is the electrostatic potential, X is the uniform external magnetic field, ex is the electron number density, x is direction of the wave propagation and ge is the unit vector. Obtaining both of our equations forced us to choice the suitable scaling for above mentioned variables: For obtaining two dimensional nonlinear KadomtsevPetviashvili Burgers equation in quantum plasma, we have to use

pffiffiffi

1 ¼ ðx VtÞ; n ¼ y; s ¼

pffiffiffi3

t;

where is the small parameter characterizing the typical amplitude of the waves. Expand the density, fluid velocity and electrical

https://doi.org/10.1016/j.rinp.2018.02.010 2211-3797/Ó 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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M.M.A. Khater et al. / Results in Physics 9 (2018) 142–150

potential in power series as [2]. Misra and Sahu get the two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma [3]:

! @ @w @w @3u @2w @2w þ Au þ B 3 C 2 þ D 2 ¼ 0; @1 @s @1 @1 @1 @n where A ¼ p4ffiffi3, B ¼

pffiffi 3ð49H2 Þ , 72

C ¼ k20 and D ¼

ð1:2Þ

pffiffi 3 . 6

For obtaining three-dimensional nonlinear modified ZakharovKuznetsov equation of ion-acoustic waves in a magnetized plasma [4], we have to use

p ffiffiffi 4

p ffiffiffi 4

p ffiffiffi 4

1 ¼ ðx tÞ; n ¼ y; f ¼ z; s ¼

ffiffiffi3 p 4

t:

Using the singular perturbation methods applied to weakly nonlinear classical waves for three-dimensional magnetized plasma, expand the density, electrostatic potential, fluid velocity by the small parameter as [4]. Munro and Parkes obtained threedimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma [5] as

! pffiffiffiffi @w @ 3 w @ @ 2 w @ 2 w @w @w ¼ 0; 16 c þ þ þ þ 30 w @s @1 @ 1 @ 13 @ 1 @n2 @f2

ð1:3Þ

where c is arbitrary real constant. for more details about our both equation, we advise for readers [6–18] Both of these equations formed as a nonlinear partial differential equation. The nonlinear partial differential equations describe complex physical phenomena in various fields such as wave propagation in solid state physics, plasma physics, population ecology, plasma wave, civil engineering, thermodynamics, fluid mechanics, soil mechanics, infectious disease epidemiology, condensed matter physics, neural networks, quantum mechanics, nonlinear optics and so on. This kind of equation has a vital role in finding exact and solitary traveling wave solutions. For two century ago, many of mathematics and physics try to discover the physical meaning and study the stability of these models. So that, they discovered many of methods to get numerical and exact traveling wave solutions. Under specific conditions on the parameters in the exact traveling wave solution, we obtain solitary traveling wave solutions that discovered by John Scott Russell in 1834. The solitary wave solutions have many different kinds. These kinds of solitary wave solutions serve the investigation of the physical meaning of this kind of equations and also, explaining how to make the most of these models. In the recent country, many numerical and exact methods were discovered to serve abovementioned goals. For example: The variational iteration method (VIM.), the fractional iteration method (FIM.), the Padé approximation, Adomian decomposition method, differential transform 0 0 method (DTM), GG -expansion method, Modified GG -expansion G0 G0 1 method, Novel G -expansion method, G ; G -expansion method, 0 extended GG -expansion method, Extended Jacobian Elliptic Function Expansion Method, a Riccati-Bernoulli Sub- ODE method, the sub equation method, the improved sub-equation method, the EXP-function method, the first integral method, the modified simple equation method, Kudryashov method, The extended Kudryashov method, The generalized Kudryashov method, the expð/ðnÞÞ-expansion method, tanh-function method, Modified extended tanh-function method, Exponential rational function method and so on [19–29]. The basic point of this research is to stratify the extended simple equation method for two dimensional nonlinear KadomtsevPetviashvili Burgers equation in quantum plasma and threedimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma The remnant of this paper is systematized as follows: In Section ‘‘An extended simple

0 equation method and novel GG -expansion method[30–35]”, we give the description of the extended simple equation method. In Section ‘‘Application”, we use this method to get the exact solutions of (NLPDEs.) pointed out above. In Section ‘‘Discuss the results”, 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 make our paper is suitable for publication. In Section ‘‘Conclusion”, conclusions are given. An extended simple equation method and novel method [30–35]

G 0 G

-expansion

The general theorize suppose the common form of the nonlinear partial differential equation be in the following form:

Pðu; ut ; ux ; uy ; uz ; utt ; uxx ; . . .Þ ¼ 0;

ð2:1Þ

where P is a polynomial in uðx; y; z; tÞ and its partial derivatives in which the highest order derivatives and nonlinear terms are entangled. We show the basic point for this method as follows: Step 1. Using the transformation

uðx; y; tÞ ¼ uðnÞ;

n ¼ kx þ ly ct;

ð2:2Þ

where k; l; c are a real constants that will be determined later, to reduce Eq. (2.1) to the following ODE:

Fðu; u0 ; u00 ; u000 ; . . . ::Þ ¼ 0;

ð2:3Þ

where F is a polynomial in uðnÞ and its total derivatives. Step 2. the traveling wave solution of Eq. (2.3), that presumed by this method has the form:

uðnÞ ¼

8 N X > > > ai /i ðnÞ; > < i¼N

N > X > i > > ai ðd þ #ðnÞÞ ; :

ð2:4Þ

i¼N 0

ðnÞ where #ðnÞ ¼ GGðnÞ ,while ðd; ai Þ are constants to be deter-

mined, such that aN or aN – 0 and ð/ðnÞ & GðnÞÞ satisfies the following second order non linear ordinary differential equation(LODE):

8 0 < / ðnÞ ¼ a þ k/ðnÞ þ l/2 ðnÞ; 0 2 : G0 ðnÞ ¼ l þ k G0 ðnÞ þ ðv 1Þ G0 ðnÞ ; GðnÞ GðnÞ GðnÞ

ð2:5Þ

Step 3. Determine the value of N in (2.4) by balancing the highest order derivatives term and the nonlinear term. Step 4. Substitute Eq. (2.4) along Eq. (2.5) into Eq. (2.3) collecting h i i all the terms of the same power /i ðnÞ & ðd þ #ðnÞÞ where ði ¼ 0; 1; 2; 3; . . .Þ and equating them to zero, we get a system of algebraic equations, which can be solved by Maple or Mathematica to get the values of ai ; a; k and l. Step 5. Substituting these values and the solutions of Eq. (2.5) into Eq. (2.4) we obtain the exact solutions of Eq. (2.1). Application Here, we will apply the extended simple equation method described in Section ‘‘An extended simple equation method and 0 novel GG -expansion method [30–35]” to find the exact traveling wave solutions and the solitary wave solutions of two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum

144


plasma and three-dimensional nonlinear modified ZakharovKuznetsov equation of ion-acoustic waves in a magnetized plasma.

Case 2. When 4al > k2 and

Two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation

wðhÞ ¼

ws þ Aww1 þ Bw111 Cw11

1

þ Dwnn ¼ 0;

ð3:1Þ

where A; B; C are nonlinear coefficient. Using the wave transformation wðx; y; tÞ ¼ wðhÞ where h ¼ k1 þ ln þ xs and integrate the obtained equation with zero constant of integration, we obtain

2 2 Akw2 Ck w00 þ x þ Dl w0 ¼ 0:

ð3:2Þ

þ

wðhÞ ¼

a2 /2 ðhÞ

þ

a1 þ a0 þ a1 /ðhÞ þ a2 /2 ðhÞ: /ðhÞ

ð3:3Þ

a2 k2 a0 ¼ ; 4l 2

a¼ a1 ¼

k2 ; 4l a2 k

l

x ¼ Dl2 ; a2 ¼ a1 ¼ 0;

Exact and solitary traveling wave solution by using novel expansion method

ð3:8Þ

G0 G

wðhÞ ¼

a2 2

ðd þ wðhÞÞ

þ

a1 2 þ a0 þ a1 ðd þ wðhÞÞ þ a2 ðd þ wðhÞÞ : ðd þ wðhÞÞ ð3:9Þ

Substituting Eq. (3.9) and its derivative into Eq. (3.2) and collecting i

all term with the same power of ðd þ #ðhÞÞ where ði ¼ 6; 5; . . . ; 5; 6Þ. We get the system of algebraic equations by solving it, we obtain:

:

So that, the exact traveling wave solutions of Eq. (3.2):

wðhÞ ¼

ð3:7Þ

So that, by using Eq. (2.4) we get:

all term with the same power of /i ðnÞ where ði ¼ 6; 5; . . . ; 5; 6Þ. We get the system of algebraic equations by solving it, we obtain:

6Ckl2 ; a2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4al k2 tan 4al k2 2ðh þ CÞ þ k ;

0 12 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al k2 a2 @ 2 @ A þ 2 4al k cot ðh þ CÞ þ kA : 4l 2

Substituting Eq. (3.3) and its derivative into Eq. (3.2) and collecting

A¼

a2 4l 2

0 1 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al k2 a2 k2 a2 k @ 2 @ A þ 4al k cot wðhÞ ¼ ðh þ CÞ þ kA 4l2 2l2 2

Balance the highest order derivatives and nonlinear terms appear ing in Eq. (3.2) ) w00 &w2 ) N þ 2 ¼ 2N ) N ¼ 2. Exact and solitary traveling wave solution by using the extended simple equation method So that, by using Eq. (2.4) we get:

a2 k2 4l 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a l k a2 k @ ðh þ CÞÞ þ kA þ 2 4al k2 tanð 2 2l

Consider Two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation:

l < 0:

a2 k2 a2 k þ /ðhÞ þ a2 /2 ðhÞ: 4l2 l

ð3:4Þ

A ¼ 0; a1

Thus, the solitary wave solutions Eq. (3.2): When k ¼ 0, we get: For both cases, when al > 0 and al < 0:

C¼

x þ ðDÞl2 2

2k k ¼ a1 ¼ a2 ¼ 0:

;

v ¼ 1;

a2 ¼ a2 ;

a0 ¼ a0 ;

But, we can not use these values of the parameters for this equation because of its will eliminate the nonlinear term in Eq. (3.2) which 0 change of the equation. So that, Novel GG -expansion method fails to solve this equation. That prove not all methods can be applied for all nonlinear partial differential equations models.

We get the trivial solution. When a ¼ 0, we get ðk ¼ 0Þ: We also get the trivial solution.

Three-dimensional nonlinear modified Zakharov-Kuznetsov equation

And the general solutions will be in the follows form: Case 1. When 4al > k2 and

l > 0:

0 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al k2 a2 k2 a2 k @ 2 @ A ðH þ CÞ kA wðhÞ ¼ þ 4al k tan 2 4l2 2l2 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al k2 a2 @ 2 ðh þ CÞA kA ; 4al k tan@ þ 2 2 4l 0

ð3:5Þ

0 1 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al k2 a2 k2 a2 k @ 2 @ A þ 4al k cot ðh þ CÞ kA wðhÞ ¼ 4l2 2l2 2 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al k2 a2 @ 2 þ 2 4al k cot@ ðh þ CÞA kA : 4l 2 0

ð3:6Þ

Consider three-dimensional nonlinear modified ZakharovKuznetsov equation

pffiffiffiffi 16 ws cw1 þ 30 ww1 þ w11 þ wnn þ wff ¼ 0;

ð3:10Þ

where c is positive real constant. Using the wave transformation wðx; y; z; tÞ ¼ wðhÞ where h ¼ k1 þ ln þ pf xs and integrate the obtained equation with zero constant of integration, we obtain

pffiffiffiffi3 3 2 2 w þ k þ kl þ kp w00 ¼ 0:

16ðx kcÞw þ 20

Using the transformation

v¼

ð3:11Þ

pffiffiffiffi w, we get:

3 2 2 16ðx kcÞv 2 þ 20v 3 þ 2 k þ kl þ kp v 02 þ vv 00 ¼ 0:

ð3:12Þ

Balance the highest order derivatives and nonlinear terms appear ing in Eq. (3.12) ) vv 00 &v 3 ) N þ N þ 2 ¼ 3N ) N ¼ 2.

145


Exact and solitary traveling wave solution by using the extended simple equation method So that, by using Eq. (2.4) we get the same formal solution of Eq. (3.2). Substituting Eq. (3.3) and its derivative into Eq. (3.12) and collecting all term with the same power of /i ðhÞ where ði ¼ 6; 5; . . . ; 5; 6Þ We get the system of algebraic equations by solving it, we obtain:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 a0 þ kp al þ k al k2 a0 þ 4la0 a þ 4xla ; c¼ l¼ ; kla 4kla a0 k la0 a2 ¼ a1 ¼ 0; a1 ¼ ; a2 ¼ :

a

Exact and solitary traveling wave solution by using novel expansion method

a0 k

a

/ðhÞ þ

la0 2 / ðhÞ: a

G

-

i

ð3:13Þ

collecting all term with the same power of ðd þ #ðhÞÞ where ði ¼ 6; 5; . . . ; 5; 6Þ. We get the system of algebraic equations by solving it, we obtain:

Thus, the solitary wave solutions Eq. (3.12): When k ¼ 0, we get:

Case I.: p¼

Case 1. When al > 0:

pffiffiffiffiffiffiffi la v ðhÞ ¼ a0 þ 0 tan2 ð alðh þ C ÞÞ; a

ð3:14Þ

la0 2 pffiffiffiffiffiffiffi cot ð alðh þ C ÞÞ: a

ð3:15Þ

v ðhÞ ¼ a0 þ

G0

So that, by using Eq. (2.4) we get the same formal solution of Eq. (3.2). Substituting Eq. (3.9) and its derivative into Eq. (3.12) and

a

So that, the exact traveling wave solutions of Eq. (3.2):

v ðhÞ ¼ a0 þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ! ð1þ lÞa0 k 4al k2 4al k2 3k2 @ v ðhÞ ¼ a0 1þ cot þ ðhþCÞA 4al 2al 2 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 a0 ð4al k2 Þ 2 @ 4al k cot ðhþCÞA: ð3:21Þ þ 4al 2

Case 2. When al < 0

v¼

1 kd 2l

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 2 3 2 3 2 2 2 k k2 d kl k2 d þ 4k lkd þ 4kl lkd 4k l2 4kl l2 þ 4cd k 4xd ; k

d þ kd l 2

d

2

;

ðck xÞðkd lÞ ; a2 ¼ a1 ¼ a0 ¼ 0; a1 ¼ 4 dðkd 2lÞ 2 ðck xÞ k2 d 2lkd þ l2 a2 ¼ 4 : 2 2 d ðkd 2lÞ

v ðhÞ ¼ a0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi la0 2 tanh alðh þ C Þ ; a

ð3:16Þ

So that, the exact traveling wave solution of Eq. (3.12) be in the follows from:

v ðhÞ ¼ a0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi la0 2 coth alðh þ C Þ : a

ð3:17Þ

v ðhÞ ¼

When a ¼ 0, we get In Both cases, when k > 0 and k < 0: We reject this case of solutions because of compensation in output gives undefined value which obtained by dividing on zero. And the general solutions will be in the follows form: Case 1. When 4al > k2 and

l > 0:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ! ð1 lÞa0 k 4al k2 4al k2 k2 þ v ðhÞ ¼ a0 1 tan@ ðhþCÞA 4al 2al 2 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4al k2 a0 ð4al k2 Þ 2@ tan þ ðhþCÞA; ð3:18Þ 4al 2

v ðhÞ ¼ a0

2

!

k þ 1 4al

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 lÞa0 k 4al k2

0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4al k2 @ ðhþCÞA cot 2 1

2al 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a0 ð4al k2 Þ 2 @ 4al k þ ðhþCÞA: cot 2 4al

Case 2. When 4al > k2 and

ð3:19Þ

l < 0:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ! ð1þ lÞa0 k 4al k2 4al k2 3k2 þ ðhþCÞA v ðhÞ ¼ a0 1þ tan@ 2al 2 4al 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4al k2 a0 ð4al k2 Þ 2@ ðhþCÞA; ð3:20Þ tan þ 2 4al

4ðck xÞðkd lÞ kd l 2 ðd þ #ðhÞÞ ðd þ #ðhÞÞ : dðkd 2lÞ dðkd 2lÞ

ð3:22Þ Thus, the solitary traveling wave solution of Eq. (3.12) be in the form: When X ¼ k2 4kl þ 4l > 0 and ðkðv 1Þ – 0Þ or ðlðv 1Þ – 0Þ:

v ðhÞ ¼

4ðck xÞðkd lÞ dðkd 2lÞ

pffiffiffiffi i sechð XhÞÞÞ

ðk þ

d

pffiffiffiffi pffiffiffiffi 1 ðk þ Xðtanhð XhÞ 2ðv 1Þ

kd l 1 d 2ðv 1Þ dðkd 2lÞ

2

pffiffiffiffi pffiffiffiffi pffiffiffiffi Xðtanhð XhÞ i sechð XhÞÞÞ ;

4ðck xÞðkd lÞ v ðhÞ ¼ dðkd 2lÞ

"

pffiffiffiffi 1 d k þ X tanh 2ðv 1Þ

pffiffiffiffi kd l 1 d k þ X tanh 2ðv 1Þ dðkd 2lÞ

v ðhÞ ¼


ð3:23Þ

" d

X

2

h

pffiffiffiffi !!!2 #

X

2

; ð3:24Þ

h

pffiffiffiffi 1 k þ X coth 2ðv 1Þ

pffiffiffiffi kd l 1 d k þ X coth 2ðv 1Þ dðkd 2lÞ

pffiffiffiffi !!!

pffiffiffiffi !!!

X

2

h

pffiffiffiffi !!!2 3 X 5; ð3:25Þ h 2

146


pffiffiffiffi pffiffiffiffi 4ðck xÞðkd lÞ 1 v ðhÞ ¼ d k þ Xðcothð XhÞ dðkd 2lÞ 2ðv 1Þ pffiffiffiffi kd l 1 d cschð XhÞÞ 2ðv 1Þ dðkd 2lÞ 2

pffiffiffiffi pffiffiffiffi pffiffiffiffi ; ð3:26Þ ðk þ Xðcothð XhÞ cschð XhÞÞÞ 4ðck xÞðkd lÞ 1 v ðhÞ ¼ d ð2k dðkd 2lÞ 4ðv 1Þ pffiffiffiffi ! pffiffiffiffi !!! pffiffiffiffi X X þ X tanh h coth h 4 4 kd l 1 d ð2k 4ðv 1Þ dðkd 2lÞ ! pffiffiffiffi pffiffiffiffi !!!2 3 pffiffiffiffi X X 5; h cothð hÞ þ X tanh 4 4

! pffiffiffiffi 2l coshð XhÞ pffiffiffiffi pffiffiffiffi pffiffiffiffi dþ pffiffiffiffi X sinhð XhÞkcoshð XhÞi X !2 3 pffiffiffiffi kd l 2l coshð XhÞ pffiffiffiffi pffiffiffiffi pffiffiffiffi 5; dþ pffiffiffiffi dðkd2lÞ X sinhð XhÞkcoshð XhÞi X

4ðck xÞðkd lÞ v ðhÞ ¼ dðkd2lÞ

ð3:32Þ ! pffiffiffiffi 2l sinhð XhÞ pffiffiffiffi pffiffiffiffi pffiffiffiffi X coshð XhÞksinhð XhÞi X !2 3 pffiffiffiffi kd l 2l sinhð XhÞ pffiffiffiffi pffiffiffiffi pffiffiffiffi 5; ð3:33Þ dþ pffiffiffiffi dðkd2lÞ X coshð XhÞksinhð XhÞi X

v ðhÞ ¼

ð3:27Þ

kd l 1 dþ k 2ðv 1Þ dðkd 2lÞ

ð3:28Þ

4ðck xÞðkd lÞ dðkd2lÞ

"

d

" pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 4ðck xÞðkd lÞ 1 k þ Xðtanð XnÞ dþ dðkd 2lÞ 2ðv 1Þ pffiffiffiffiffiffiffiffi kd l 1 dþ secð XhÞÞÞ 2ðv 1Þ dðkd 2lÞ 2

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ðk þ Xðtanð XhÞ secð XhÞÞÞ ; ð3:36Þ

v ðhÞ ¼

kd l 1 dþ 2ðv 1Þ dðkd 2lÞ

v ðhÞ ¼

0

ð3:29Þ

pffiffiffi 1 2l cosh 2X h 4ðck xÞðkd lÞ 4@ pffiffiffi pffiffiffi A v ðhÞ ¼ d þ pffiffiffiffi dðkd 2lÞ X sinh 2X h k cosh 2X h pffiffiffi 0 12 3 2l cosh 2X h kd l @ pffiffiffi A 7 d þ pffiffiffiffi 5; ð3:30Þ pffiffiffi X dðkd 2lÞ X sinhð 2 hÞ k cosh 2X h 20

pffiffiffi 1 X 2 l sinh h 2 4ðck xÞðkd lÞ 4@ pffiffiffi pffiffiffi A v ðhÞ ¼ d þ pffiffiffiffi dðkd 2lÞ X cosh 2X h k sinh 2X h pffiffiffi 0 12 3 2l sinh 2X h kd l @ pffiffiffi pffiffiffi A 7 d þ pffiffiffiffi 5; ð3:31Þ X dðkd 2lÞ X cosh 2 h k sinh 2X h 20

or

pffiffiffiffiffiffiffiffi !!! pffiffiffiffiffiffiffiffi 1 X kþ X coth h 2ðv 1Þ 2 pffiffiffiffiffiffiffiffi !!!2 3 pffiffiffiffiffiffiffiffi kd l 1 X 5; ð3:35Þ h d kþ X coth 2ðv 1Þ 2 dðkd2lÞ

v ðhÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 11 XðA2 þ B2 Þ þ A X coshð XhÞ AA pffiffiffiffi þ A sinhð XhÞ þ B

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 112 3 pffiffiffiffi pffiffiffiffi XðA2 þ B2 Þ þ A X coshð XhÞ 7 pffiffiffiffi þ BAA 5; @k þ A sinh Xh

dþ pffiffiffiffi

4ðck xÞðkd lÞ 1 dþ ðk dðkd 2lÞ 2ðv 1Þ pffiffiffiffiffiffiffiffi !!! pffiffiffiffiffiffiffiffi kd l 1 X þ X tanh dþ h 2ðv 1Þ dðkd 2lÞ 2 pffiffiffiffiffiffiffiffi !!!2 3 pffiffiffiffiffiffiffiffi X 5; ð3:34Þ k þ X tanh h 2

4ðck xÞðkd lÞ 1 v ðhÞ ¼ dþ ðk dðkd 2lÞ 2ðv 1Þ

"

where A, B are arbitrary real constants and A2 þ B2 > 0. When X ¼ k2 4kl þ 4l < 0 and ðkðv 1Þ – 0Þ ðlðv 1Þ – 0Þ:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 11 XðA2 þ B2 Þ A X coshð XhÞ AA pffiffiffiffi þ A sinhð XhÞ þ B

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 112 3 pffiffiffiffi pffiffiffiffi XðA2 þ B2 Þ A X coshð XhÞ 7 pffiffiffiffi þ BAA 5; þ A sinh Xh

4ðck xÞðkd lÞ dðkd2lÞ

v ðhÞ ¼

4ðck xÞðkd lÞ 1 v ðhÞ ¼ dþ ðk dðkd 2lÞ 2ðv 1Þ

"

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 4ðck xÞðkd lÞ 1 ðd ðk þ Xðcotð XhÞ dðkd 2lÞ 2ðv 1Þ

pffiffiffiffiffiffiffiffi cscð XhÞÞÞÞ ðk þ

v ðhÞ ¼

kd l 1 d 2ðv 1Þ dðkd 2lÞ

2

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ; Xðcotð XhÞ cscð XhÞÞÞ


pffiffiffiffiffiffiffiffi þ X tan

dþ

ð3:37Þ

1 ð2k 4ðv 1Þ

pffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffi !!!! X X h cot h 4 4

pffiffiffiffiffiffiffiffi kd l 1 dþ 2k þ X tan 4ðv 1Þ dðkd 2lÞ pffiffiffiffiffiffiffiffi !!!!2 3 X 5; h cot 4

pffiffiffiffiffiffiffiffi ! X h 4

ð3:38Þ

147


4ðck xÞðkd lÞ 1 v ðhÞ ¼ dþ ðk dðkd 2lÞ 2ðv 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 11 XðA2 B2 Þ A X cosð XhÞ AA pffiffiffiffiffiffiffiffi þ A sinð XhÞ þ B kd l 1 dþ ðk 2ðv 1Þ dðkd 2lÞ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 112 XðA2 B2 Þ A X cosð XhÞ AA 7 pffiffiffiffiffiffiffiffi þ 5; A sinð XhÞ þ B

v ðhÞ ¼

4ðck xÞðkd lÞ dðkd 2lÞ "

! pffiffiffiffiffiffiffiffi 2l sinð XhÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi d þ pffiffiffiffiffiffiffiffi X cosð XhÞ ksinð XhÞ X !2 3 pffiffiffiffiffiffiffiffi kd l 2l sinð XnÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 5; d þ pffiffiffiffiffiffiffiffi dðkd 2lÞ X cosð XhÞ ksinð XhÞ X

ð3:44Þ

ð3:39Þ

4ðck xÞðkd lÞ 1 ðk dþ dðkd 2lÞ 2ðv 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 11 XðA2 B2 Þ þ A X cosð XhÞ AA pffiffiffiffiffiffiffiffi þ A sinð XhÞ þ B kd l 1 dþ 2ðv 1Þ dðkd 2lÞ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 112 XðA2 B2 Þ þ A X cosð XhÞ AA 7 pffiffiffiffiffiffiffiffi @k þ 5; ð3:40Þ A sinð XhÞ þ B

v ðhÞ ¼

pffiffiffiffiffi 20 1 2l cos 2 X h 4ðck xÞðkd lÞ 4@ pffiffiffiffiffi A v ðhÞ ¼ d pffiffiffiffiffiffiffiffi pffiffiffiffiffi dðkd 2lÞ X sin X h þ k cos X h 2

2

pffiffiffiffiffi 12 3 X 2 l cos h 2 kd l @ pffiffiffiffiffi A 7 d pffiffiffiffiffiffiffiffi pffiffiffiffiffi 5; ð3:41Þ dðkd 2lÞ X sin 2 X h þ k cos 2 X h 0

where A, B are arbitrary real constants and A2 B2 > 0. When l ¼ 0 and kðv 1Þ – 0, we have:

4ðck xÞðkd lÞ kk d dðkd 2lÞ ðv 1Þðk þ coshðkhÞ sinhðkhÞÞ 2 # kd l kk ; ð3:45Þ d ðv 1Þðk þ coshðkhÞ sinhðkhÞÞ dðkd 2lÞ

v ðhÞ ¼

4ðck xÞðkd lÞ kðcoshðkhÞ þ sinhðkhÞÞ d dðkd 2lÞ ðv 1Þðk þ coshðkhÞ þ sinhðkhÞÞ 2 # kd l kðcoshðkhÞ þ sinhðkhÞÞ d ; ðv 1Þðk þ coshðkhÞ þ sinhðkhÞÞ dðkd 2lÞ

v ðhÞ ¼

ð3:46Þ 4ðck xÞðkd lÞ 1 kd l d dðkd 2lÞ ðv 1Þh þ C dðkd 2lÞ 2 # 1 d ; ð3:47Þ ðv 1Þh þ C

v ðhÞ ¼

where C, k are arbitrary constants. Case II.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 2 2 2 3 2 2 2 4cl2 k 4cl2 l k þ 16c2 d k þ 4xl2 k þ 4xl2 l k 32xcd k 8ca1 dk þ 16x2 d þ 8xa1 d þ a21 ; p¼ 2 l 4k c 4xk 1

pffiffiffiffiffi 20 1 2l sin 2 X h 4ðck xÞðkd lÞ 4@ pffiffiffiffiffi A d þ pffiffiffiffiffiffiffiffi pffiffiffiffiffi v ðhÞ ¼ dðkd 2lÞ X cos X h k sin X h 2

2

pffiffiffiffiffi 0 12 3 2l sin 2 X h kd l @ pffiffiffiffiffi A 7 d þ pffiffiffiffiffiffiffiffi pffiffiffiffiffi 5; dðkd 2lÞ X cos X h k sin X h 2

a2 ¼

ð3:42Þ

2

4ðck xÞðkd lÞ v ðhÞ ¼ dðkd 2lÞ "

v ¼ 1;

a0 ¼ a1 ¼ a2 ¼ 0;

! pffiffiffiffiffiffiffiffi 2l cosð XhÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi d pffiffiffiffiffiffiffiffi X sinð XhÞ þ kcosð XhÞ X

!2 3 pffiffiffiffiffiffiffiffi kd l 2l cosð XhÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 5; ð3:43Þ d pffiffiffiffiffiffiffiffi dðkd 2lÞ X sinð XhÞ þ kcosð XhÞ X

a1 2 ; 4ðck xÞ

k¼4

lðck xÞ ; 4cdk 4dx a1

a1 ¼ a1 :

So that, the exact traveling wave solution of Eq. (3.12) be in the follows from:

v ðhÞ ¼ a1

ðd þ #ðhÞÞ

1

a1 2 ðd þ #ðhÞÞ : 4ðck xÞ

ð3:48Þ

Thus, the solitary traveling wave solution of Eq. (3.12) be in the X ¼ k2 4kl þ 4l > 0 and ðkðv 1Þ – 0Þ or

form: When ðlðv 1Þ – 0Þ:

2

v ðhÞ ¼ a1 4 d

pffiffiffiffi 1 k þ X tanh 2ðv 1Þ

pffiffiffiffi !!!1

X

2

h

pffiffiffiffi a1 1 d k þ X tanh 2ðv 1Þ 4ðck xÞ

pffiffiffiffi !!!2 3 X 5; h 2 ð3:49Þ

148


2

pffiffiffiffi 1 v ðhÞ ¼ a1 4 d k þ X coth 2ðv 1Þ

2

pffiffiffiffi !!!1

X

2

pffiffiffiffi a1 1 d k þ X coth 2ðv 1Þ 4ðck xÞ

!1 pffiffiffiffi 2l coshð XhÞ pffiffiffiffi pffiffiffiffi pffiffiffiffi X sinhð XhÞ k coshð XhÞ i X

v ðhÞ ¼ a1 4

h pffiffiffiffi !!!2 3 X 5; h 2

d þ pffiffiffiffi

!2 3 pffiffiffiffi a1 2l coshð XhÞ 5; pffiffiffiffi pffiffiffiffi pffiffiffiffi d þ pffiffiffiffi 4ðck xÞ X sinhð XhÞ k coshð XhÞ i X

ð3:50Þ 2

"

1 pffiffiffiffi pffiffiffiffi pffiffiffiffi 1 ðk þ Xðtanhð XhÞ i sechð XhÞÞÞ 2ðv 1Þ 2 # pffiffiffiffi pffiffiffiffi pffiffiffiffi a1 1 ; d ðk þ Xðtanhð XhÞ i sechð XhÞÞÞ 2ðv 1Þ 4ðck xÞ

v ðhÞ ¼ a1

ð3:58Þ

d

d þ pffiffiffiffi

!2 3 pffiffiffiffi a1 2l sinhð XhÞ 5; pffiffiffiffi pffiffiffiffi pffiffiffiffi d þ pffiffiffiffi 4ðck xÞ X coshð XhÞ k sinhð XhÞ i X

ð3:51Þ "

1 pffiffiffiffi pffiffiffiffi pffiffiffiffi 1 ðk þ Xðcothð XhÞ cschð XhÞÞÞ 2ðv 1Þ 2 # pffiffiffiffi pffiffiffiffi pffiffiffiffi a1 1 d ðk þ Xðcothð XhÞ cschð XhÞÞÞ ; 4ðck xÞ 2ðv 1Þ

v ðhÞ ¼ a1

!1 pffiffiffiffi 2l sinhð XhÞ pffiffiffiffi pffiffiffiffi pffiffiffiffi X coshð XhÞ k sinhð XhÞ i X

v ðhÞ ¼ a1 4

d

ð3:52Þ 2

v ðhÞ ¼ a1 4

d

pffiffiffiffi 1 2k þ X tanh 4ðv 1Þ

pffiffiffiffi !

X

4

pffiffiffiffi !!!!1

h coth

pffiffiffiffi a1 1 d 2k þ X tanh 4ðv 1Þ 4ðck xÞ

X

4

v ðhÞ ¼ a1 4

pffiffiffiffiffiffiffiffi 1 k þ X tanh 2ðv 1Þ

pffiffiffiffi !!!!2 3 X 5; h coth h 4 4

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffi pffiffiffiffi 111 XðA2 þ B2 Þ þ A X coshð XhÞ 1 @k þ AA pffiffiffiffi 2ðv 1Þ Asinh Xh þ B

or

pffiffiffiffiffiffiffiffi !!!1 X h 2

pffiffiffiffiffiffiffiffi a1 1 dþ k þ X tanh 2ðv 1Þ 4ðck xÞ

pffiffiffiffiffiffiffiffi !!!2 3 X 5; h 2 ð3:60Þ

2

v ðhÞ ¼ a1 4

0

d

pffiffiffiffiffiffiffiffi 1 k þ X coth 2ðv 1Þ

pffiffiffiffiffiffiffiffi !!!1 X h 2

pffiffiffiffiffiffiffiffi a1 1 d k þ X coth 2ðv 1Þ 4ðck xÞ

pffiffiffiffiffiffiffiffi !!!2 3 X 5; h 2 ð3:61Þ

"

1 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ðk þ Xðtanð XhÞ secð XhÞÞÞ 2ðv 1Þ 2 # pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi a1 1 ; dþ ðk þ Xðtanð XhÞ secð XhÞÞÞ 2ðv 1Þ 4ðck xÞ

v ðhÞ ¼ a1

dþ

ð3:62Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 112 3 pffiffiffiffi pffiffiffiffi XðA2 þ B2 Þ þ A X coshð XhÞ a1 @ 1 7 pffiffiffiffi dþ @k þ þ BAA 5; 2ðv 1Þ 4ðck xÞ Asinh Xh

ð3:55Þ

20 v ðhÞ ¼ a1 64@d þ pffiffiffiffi

pffiffiffi 11 2l cosh 2X h pffiffiffi pffiffiffi A X sinh 2X h k cosh 2X h pffiffiffi 0 12 3 2l cosh 2X h a1 @d þ pffiffiffiffi pffiffiffi pffiffiffi A 7 5; ð3:56Þ X 4ðck xÞ X sinh 2 h k cosh 2X h

20 v ðhÞ ¼ a1 64@d þ pffiffiffiffi

dþ

X

ð3:54Þ

v ðhÞ ¼ a1 64@d þ

2

pffiffiffiffi !

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 111 XðA2 þ B2 Þ A X coshð XhÞ 1 6@ @ AA v ðhÞ ¼ a1 4 d þ k þ pffiffiffiffi 2ðv 1Þ Asinh Xh þ B 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 pffiffiffiffi pffiffiffiffi 112 XðA2 þ B2 Þ A X coshð XhÞ a1 @ 1 7 @ A A pffiffiffiffi dþ k þ 5; 2ðv 1Þ 4ðck xÞ Asinh Xh þ B

20

where A, B are arbitrary real constants and A2 þ B2 > 0. When X ¼ k2 4kl þ 4l < 0 and ðkðv 1Þ – 0Þ ðlðv 1Þ – 0Þ:

h

ð3:53Þ 20

ð3:59Þ

pffiffiffi 11 2l sinh 2X h pffiffiffi pffiffiffi A X cosh 2X h k sinh 2X h pffiffiffi 0 12 3 X 2 l sinh h 2 a1 @d þ pffiffiffiffi pffiffiffi pffiffiffi A 7 5; ð3:57Þ 4ðck xÞ X cosh 2X h k sinh 2X h

"

1 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ðk þ Xðcotð XhÞ cscð XhÞÞÞ 2ðv 1Þ 2 # pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi a1 1 ; d ðk þ Xðcotð XhÞ cscð XhÞÞÞ 2ðv 1Þ 4ðck xÞ

v ðhÞ ¼ a1

d

ð3:63Þ 2

v ðhÞ ¼ a1 4

dþ

pffiffiffiffiffiffiffiffi 1 2k þ X tan 4ðv 1Þ

pffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffi !!!!1 X X h cot h 4 4

pffiffiffiffiffiffiffiffi a1 1 dþ 2k þ X tan 4ðck xÞ 4ðv 1Þ

pffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffi !!!!2 3 X X 5; h cot h 4 4

ð3:64Þ 20


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 111 2 XðA B2 Þ A X cosð XhÞ 1 @k þ AA pffiffiffiffiffiffiffiffi 2ðv 1Þ Asinð XhÞ þ B

3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 12 XðA2 B2 Þ A X cosð XhÞ a1 1 A 7 pffiffiffiffiffiffiffiffi dþ @k þ 5; 4ðck xÞ 2ðv 1Þ Asinð XhÞ þ B

ð3:65Þ

149


20


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 111 XðA2 B2 Þ þ A X cosð XhÞ 1 @k þ AA pffiffiffiffiffiffiffiffi 2ðv 1Þ Asin Xh þ B

3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 112 XðA2 B2 Þ þ A X cosð XhÞ a1 @ 1 7 @ A A pffiffiffiffiffiffiffiffi dþ k þ 5; 4ðck xÞ 2ðv 1Þ Asinð X hÞ þ B

ð3:66Þ

20 pffiffiffiffiffi 11 X 2 l cos h 2 pffiffiffiffiffi A v ðhÞ ¼ a1 64@d pffiffiffiffiffiffiffiffi pffiffiffiffiffi X sin 2 X h þ k cos 2 X h pffiffiffiffiffi 0 12 3 2l cos 2 X h a1 @d pffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi A 7 5; 4ðck xÞ X sin X h þ k cos X h 2

2

ð3:67Þ 20 pffiffiffiffiffi 11 X 2 l sin h 2 6@ pffiffiffiffiffi A v ðhÞ ¼ a1 4 d þ pffiffiffiffiffiffiffiffi pffiffiffiffiffi X cos 2 X h k sin 2 X h pffiffiffiffiffi 0 12 3 2l sin 2 X h a1 @d þ pffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi A 7 5; X 4ðck xÞ X cos h k sin X h 2

2

ð3:68Þ 2

v ðhÞ ¼ a1 4

!1 pffiffiffiffiffiffiffiffi 2l cosð XhÞ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi d pffiffiffiffiffiffiffiffi X sinð XhÞ þ k cosð XhÞ X

!2 3 pffiffiffiffiffiffiffiffi a1 2l cosð XhÞ 5; pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi d pffiffiffiffiffiffiffiffi 4ðck xÞ X sinð X hÞ þ k cosð XhÞ X ð3:69Þ

Note that: All the obtained results have been checked with Maple program by putting them back into the original equation and found correct. Discuss the results In this research, we show a good comparison between our results that obtained by using extended simple equation method 0 and novel GG -expansion method and on the other hand the results that obtained by other researchers using the different methods. We sort our comparison for each models in the main following steps: Fristly: The result of the two dimensional nonlinear KadomtsevPetviashvili Burgers equation. 1. Discussion of the results that obtained in our research: We applied two different methods on this equation and we 0 found that novel GG -expansion method can not applied for this equation which proof and evidence that not all methods can be apply for all models and that clear in this model where we applied the extended simple equation method and obtained exact and solitrary wave solutions. 2. Discussion of the results that obtained in different researches vs that obtained in our research: However, Seadawy, Aly R. applied three different methods (extendied direct algebraic mapping method, extended Sech-tanh method and extended direct algebraic sech method) to two dimensional nonlinear KadomtsevPetviashvili-Burgers equation in quantum plasma, we get new form of ion solitary wave solution (3.5)–(3.8) for that he obtained in his research [2]. Secondly: The result of Three-dimensional nonlinear modified Zakharov-Kuznetsov equation:

2

!1 pffiffiffiffiffiffiffiffi 2l sinð XhÞ 4 ffiffiffiffiffiffiffi ffi p p ffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffi ffi v ðhÞ ¼ a1 d þ X cosð XhÞ k sinð XhÞ X

!2 3 pffiffiffiffiffiffiffiffi a1 2l sinð XhÞ 5; pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi d þ pffiffiffiffiffiffiffiffi 4ðck xÞ X cosð XhÞ k sinð XhÞ X ð3:70Þ

where A, B are arbitrary real constants and A2 B2 > 0. When l ¼ 0 and kðv 1Þ – 0, we have:

" d

1 kk ðv 1Þðk þ coshðkhÞ sinhðkhÞÞ 2 # a1 kk ; d ðv 1Þðk þ coshðkhÞ sinhðkhÞÞ 4ðck xÞ

v ðhÞ ¼ a1

ð3:71Þ " v ðhÞ ¼ a1 d

1

kðcoshðkhÞ þ sinhðkhÞÞ ðv 1Þðk þ coshðkhÞ þ sinhðkhÞÞ 2 # a1 kðcoshðkhÞ þ sinhðk hÞÞ ; d ðv 1Þðk þ coshðkhÞ þ sinhðkhÞÞ 4ðck xÞ ð3:72Þ

v ðhÞ ¼ a1

" d

1 ðv 1Þh þ C

1

2 # a1 1 ; d ðv 1Þh þ C 4ðck xÞ ð3:73Þ

where C, k are arbitrary constants.

1. Discussion of the results that obtained in our research: We applied two different methods on this equation and we obtained many form of ion solitary wave solutions and also we can note that both of method obtained a completely different forms of solution. So that, we get a large number of solitary wave solutions of this model. 2. Discussion of the results that obtained in different researches vs that obtained in our research: However, Seadawy, Aly R. applied two different methods (the extended direct algebraic function method and fractional direct algebraic function method) to the nonlinear three-dimensional modified Zakharov-Kuznetsov (mZK) equation, we get our solution (3.18) is similar to his solution h2 2 2 i pffiffiffiffiffiffiffiffiffiffi (21) in [4] when a0 ¼ k þlb2þp ða1 b1 a1 c1 a21 c1 Þ 1

2 ðð2al0:5k2 ÞÞ pffiffiffiffiffiffi 2 and all our solutions is 1 4ka l : & k ¼ ðð1lÞð

4alk ÞÞ

considered as a new form of solutions for this vitial model in electrostatic field. So that, it is shown that the extended simple equation method 0 and the novel GG -expansion method are very effective and powerful mathematical tools for obtaining the traveling and solitary wave solutions of many phenomena that can represent by nonlinear partial differential equation. Conclusion In this research, we succeed to apply the extended simple equa 0 0 tion method and novel GG -expansion method GG -expansion on

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two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We get many new form of solutions for both vital models in magntic and electric fields. This show how these methods is very effective, very simple and direct methods to apply them for many nonlinear evolution equations (NLEEs.). Competing interests This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. The authors did not have any competing interests in this research. Author’s contributions All parts contained in the research carried out by the authors through hard work and a review of the various references and contributions in the field of mathematics and the physical Applied. Acknowledgment The author thanks the referees for their suggestions and comments. References [1] Munro S, Parkes EJ. The stability of obliquely-propagating solitary wave solutions to a modified Zakharov-Kuznetsov equation. J Plasma Phys 2004;70 (5):543–52. [2] Seadawy Aly R. Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma. Math Methods Appl Sci 2017;40(5):1598–607. [3] Xing L, Ma Wen-Xiu, Masood Khalique Chaudry. A direct linear Backlund transformation of a (2 + 1) dimensional of Korteweg-de vries like model. Appl Math Lett 2015;50:37–42 . [4] Seadawy Aly R. Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. Comput Math Appl 2016;71(1):201–12. [5] Munro Susan, Parkes EJ. The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions. J Plasma Phys 1999;62(3):305–17. [6] Wazwaz Abdul-Majid. The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. Commun Nonlinear Sci Numer Simul 2008;13(6):1039–47. [7] Biswas Anjan, Zerrad Essaid. Solitary wave solution of the Zakharov-Kuznetsov equation in plasmas with power law nonlinearity. Nonlinear Anal: Real World Appl 2010;11(4):3272–4. [8] Zhen Hui-Ling et al. Dynamic behaviors and soliton solutions of the modified Zakharov-Kuznetsov equation in the electrical transmission line. Comput Math Appl 2014;68(5):579–88. [9] Li Biao, Chen Yong, Zhang Hongqing. Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation. Appl Math Comput 2003;146 (2):653–66. [10] Seadawy Aly R. Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput Math Appl 2014;67 (1):172–80. [11] Wazwaz Abdul-Majid. Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form. Commun Nonlinear Sci Numer Simul 2005;10(6):597–606. _ Generalized solitary and periodic wave solutions to a (2 + 1)[12] Aslan Ismail. dimensional Zakharov-Kuznetsov equation. Appl Math Comput 2010;217 (4):1421–9.

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