Arshada, Jun Wanga Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China Mathematics Department, Faculty of Science, Taibah University, Al-Ula, Saudi Arabia Mathema
Trang 16
7
8 Dianchen Lua, A.R Seadawyb,c,⇑ , M Arshada, Jun Wanga
Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China
Mathematics Department, Faculty of Science, Taibah University, Al-Ula, Saudi Arabia
Mathematics Department, Faculty of Science, Beni-Suef University, Egypt
12
15 Article history:
16 Received 21 November 2016
17 Received in revised form 21 January 2017
18 Accepted 5 February 2017
19 Available online xxxx
20 Keywords:
21 Modified extended direct algebraic method
22 Solitons
23 Solitary wave solutions
24 Jacobi and Weierstrass elliptic function
25 solutions
26 Three dimensional extended
Zakharov-27 Kuznetsov dynamical equation
28 (3 + 1)-Dim modified
KdV-Zakharov-29 Kuznetsov equation
30
3 1
a b s t r a c t
32
In this paper, new exact solitary wave, soliton and elliptic function solutions are constructed in various
33 forms of three dimensional nonlinear partial differential equations (PDEs) in mathematical physics by
34 utilizing modified extended direct algebraic method Soliton solutions in different forms such as bell
35 and anti-bell periodic, dark soliton, bright soliton, bright and dark solitary wave in periodic form etc
36 are obtained, which have large applications in different branches of physics and other areas of applied
37 sciences The obtained solutions are also presented graphically Furthermore, many other nonlinear
evo-38 lution equations arising in mathematical physics and engineering can also be solved by this powerful,
39 reliable and capable method The nonlinear three dimensional extended Zakharov-Kuznetsov dynamica
40 equation and (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation are selected to show the
41 reliability and effectiveness of the current method
42
Ó 2017 The Author Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license
43 (http://creativecommons.org/licenses/by-nc-nd/4.0/)
44 45
46 Introduction
47 Nonlinear PDEs involve nonlinear complex physical phenomena,
48 which plays a vital role in plasma physics and many other aeras of
49 applied sciences The generalized KdV-Zakharov-Kuznetsov
equa-50 tions are an important models for numerous physical phenomena
51 as well as waves in nonlinear LC circuit by way of mutual
induc-52 tance among neighboring inductors, shallow and stratified internal
53 waves, ion-acoustic waves in plasma physics, applications in space
54 environments and astrophysical, nonlinear optic, hydrodynamic
55 and many more [1–5] Moreover, the electrostatic solitary waves
56 have been determined in many areas such as solar, wind, earths
57 magnetotail and polar magnetosphere [2] etc The KdV equations
58 have several two dimensional weak variations Solitons and solitary
59 waves represent one of the famous and motivating features of
non-60 linear phenomena spacially in extended equations, which have
61 many important properties The Zakharov-Kuznetsov equation is
62 one of the two well studied canonical two dimensional extensions
63
of KdV equation [6] Nonlinear extended Z-K equations are utilized
64
to discuss the nonlinear dust ion-acoustic waves in magnetized two
65 ion-temperature dusty plasmas, propagations of the low frequency
66 ion-acoustic wave in a thick Quantum magneto-plasmas etc [7–9]
67 Recently, the authors in [10] derived nonlinear three dimensional
68 extended zakharov-Kuznetsov dynamical equation in a magnetized
69 two ion-temperature dusty plasma by using the theory of reductive
70 perturbation.
71 The modified KdV model can be drived for the elaboration of
72 ion-acoustic perturbation in plasma with components of two
neg-73 ative ions of different temperature [11] In the case of weakly two
74 dimensional variations of the modified KdV equation [11] , the
75 modified KdV-ZK model occurs [12–14] , which is an important
76 model arises in different branches of physics such as plasma
phy-77 sics, nonlinear optics fluid dynamics, theoretical physics quantum
78 mechanics and mathematical physics to analize the main
proper-79 ties of non-linear propagation of various physical phenomena.
80 Recently, many researchers have been giving much attention on
81 the study of constructing the solitons and solitary wave solutions
82
of nonlinear PDEs [15–18] , which occur in mathematical physics.
http://dx.doi.org/10.1016/j.rinp.2017.02.002
2211-3797/Ó 2017 The Author Published by Elsevier B.V
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
⇑ Corresponding author at: Mathematics Department, Faculty of Science, Taibah
University, Al-Ula, Saudi Arabia
E-mail address:Aly742001@yahoo.com(A.R Seadawy)
Results in Physics xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
Results in Physics
j o u r n a l h o m e p a g e : w w w j o u r n a l s e l s e v i e r c o m / r e s u l t s - i n - p h y s i c s
Please cite this article in press as: Lu D et al New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified
Trang 283 So, Many powerful methods have been discovered to construct the
84 solitons and solitary wave solutions such as inverse scattering
85 scheme [3] , direct algebraic method [18] , Bcklund transform
86 method [19] , the Hirota’s bilinear scheme [20] , exp( /ð g
Þ)-87 expension method [21] , extended tanh method [22] , auxiliary
88 equation method [23] , the mapping method and extended
map-89 ping method [24] , rational expansion method [25] , elliptic function
90 method [26] and many more [27,28] Many numerical schemes
91 also have been developed such as Adomian decomposition method
92 [29] , homotopy analysis method [29,30] , homotopy perturbation
93 method [31] , differential transform and reduced differential
trans-94 form method [32–35] etc to obtain the numerical solutions in
dif-95 ferent form of non-linear evaluation equations The study about
96 solutions, structures, interaction and further properties of soliton
97 gained much attention and various meaningful results are
success-98 fully derived [36–39]
99 In this work, the ansatz equation is further extended in the
100 modified extended direct algebraic method to construct soliton
101 and solitary wave solutions of three-dimensional EZK equation
102 and modified KdV-ZK equation Consequently, more general and
103 new exact solutions in soliton and solitary wave form are
con-104 structed [40–52]
105 This article is ordered as follows An introduction is given in
106 Section ‘Introduction’ The main steps of the modified extended
107 direct algebraic method are specified in Section ‘Description of
108 modified extended direct algebraic method’ We obtain general
109 new exact solutions in soliton, solitary wave and elliptic solutions
110 in different form of three-dimensional EZK equation and (3 +
1)-111 dimensional modified KdV-ZK equation in Section ‘Application of
112 the modified extended direct algebraic method’ Lastly, the
conclu-113 sion is given in Section ‘Conclusion’.
114 Description of modified extended direct algebraic method:
115 Let us assume a general non-linear evolution equation in x ; y; z
116 and t as
117
F u; u t; ux; uy; uz; uxx; uyy; uzz; uxy; uxz; uyz; uxxx; ¼ 0; ð1Þ
119
120 where the function u ðx; y; z; tÞ is unknown and F is a polynomial
121 function with respect to some functions OR specified variables,
122 which have non-linear and terms and highest order derivatives of
123 the unknown functions and can be reduced to a polynomial
func-124 tion by utilizing transformations in which the real variables x ; y; z
125 and t can be merge in a complex variable The key steps of this
126 method are as:
127 Step 1: Consider that Eq (1) has the following solution as:
128
uðx; y; z; tÞ ¼ uðnÞ ¼ Xm
j¼m
130
132
/0ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X6
i¼0
ci/i
v u
; and n ¼ k1x þ k2y þ k3z þ x t; ð3Þ
134
135 where aj; cii from 1 to 6 are arbitrary constants and k1; k2; k3
136 are wave lengths and x is frequency.
137 Step 2: By balancing the highest order non-linear term with the
138 highest order derivative term of Eq (1 ), and the series of
140 am; amþ1; ; a1; a0; a1; am; c0; c1; ; c6; k1; k2; k3; x
142 Step 3: Substituting Eqs (2) and (3) into Eq (1) and setting the
143 coefficients of powers of /j/ðjÞ to zero, yields a systems
144 of algebraic equations in parameters am; amþ1; ;
145
a1; a0; a1; am; c0; c1; c2 ; c6; k1; k2; k3; x The systems
146
of algebraic equations are solved by Marhematica, then
147 the values of parameters can be obtained.
148 Step 4: By substituting the values of parameters and / ðnÞ obtained
149
in previous step into Eq (2) , then the solutions of Eq (1)
150 can be constructed.
151 152 Application of the modified extended direct algebraic method:
153 The extended Zakharov-Kuznetsov (EZK) equation:
154 First, we consider the (3 + 1)-dimensional extended
Zakharov-155 Kuznetsov (EZK) equation is as:
156
utþ Auuxþ Buxxxþ C uxyyþ uxzz
159 where A ; B and C are arbitrary constants.
160 Consider the traveling wave solutions and transformation in
161 Eqs (2) and (3) , then Eq (4) reduces into ODE as
162
x u0ðnÞ þ Ak1uu0ðnÞ þ Bk3
1þ Ck1ðk2
2þ k2
3Þ
u000ðnÞ ¼ 0; ð5Þ 164
165
By using balance principle on Eq (5) gives m ¼ 2, we consider
166 the solution of Eq (5) is as:
167
uðnÞ ¼ a2 /2þ a1 / þ a0þ a1/ þ þa1/2: ð6Þ
169 170 Substituting Eq (6) into Eq (5) and setting the coefficients of
171 coefficient of /j/ðjÞto zero, yields a systems of algebraic equations
172
in a2; a1; a0; a1; a2; A; B; C; k1; k2; k3and x The systems of algebraic
173 equations are solved by Mathematica, which have possesses the
174 following solutions cases:
175 Case 1: c1¼ c3¼ c5¼ 0; c0¼ 8c 2
27c 4; c6¼ c 2
4c 2,
176
a2¼ 32c
2 Bk21þ Ck2
2þ Ck2 3
9Ac4 ; a1¼ a1¼ a2¼ 0;
x ¼ k1 a0A þ 4Bc2k21þ 4c2Ck22þ 4c2Ck23
179 Substituting Eq (7) into Eq (6) along with the solutions of Eq.
180
(3) , the following solutions of Eq (4) are obtained:
181
u11ðnÞ ¼ a0þ 4c2 Bk
2
1þ Ck2
2þ Ck2 3
3 þ tanh2 ffiffiffiffiffiffiffiffi c2
3
q n
3Atanh2 ffiffiffiffiffiffiffiffi c2
3
q n
184
u12ðnÞ ¼ a0 4c2 Bk
2
1þ Ck2
2þ Ck2 3
3 tan2 ffiffiffiffiffiffiffiffi c2
3
q n
3A tan2 ffiffiffiffic2
3
q n
187
u13ðnÞ ¼ a0þ 4c2 Bk
2
1þ Ck2
2þ Ck2 3
3 þ coth2 ffiffiffiffic2
3
q n
3Acoth2 ffiffiffiffiffiffiffiffi c2
3
q n
190
u14ðnÞ ¼ a0 4c2 Bk
2
1þ Ck2
2þ Ck2 3
3 cot2 ffiffiffiffic2
3
q n
3Acot2 ffiffiffiffi
c2
3
q
n
193 where ; n ¼ k1x þ k2y þ k3z þ x t ; x ¼ k1 a0A þ 4Bc2k2þ 4c2Ck2
194 þ4c2Ck23Þ:
Trang 3195 Figs 1 (a) and (c) signify the evolution of the dark and periodic
196 bright solitary waves solutions of Eqs (8) and (9) of the EZK Eq (4)
197 at c2¼ 1:25; c4¼ 2; k1¼ 0:75; k2¼ 1:5; k3¼ 1; ¼ 1; A ¼ 2; B ¼ 1;
198 C ¼ 0:5; a0¼ 1:5; y ¼ 1; z ¼ 1 and c2¼ 1; c4¼ 2; k1¼ 0:5; k2¼ 1:5;
199 k3¼ 1:5; ¼ 1; A ¼ 2; B ¼ 1; C ¼ 1; a0¼ 1; y ¼ 1; z ¼ 1 respectively A
200 contour plots Figs 1 (b) and (d) are a collection of level curves
201 drawn on same set of intervals The command of Contour Plot on
202 mathematica draws ContourPlot of two variable functions The
203 points on contours join at same height on the surface The
204 sequence of equally spaced values of the functions is to have
cor-205 responding contours by default.
206 Case 2: c0¼ c1¼ c3¼ c5¼ c6¼ 0,
207
a2¼ a1¼ a1¼ 0;
a0¼ 4c2k1 Bk
2
1þ C k2
2þ k2 3
þ x
a2¼ 12c4 Bk
2
1þ C k2
2þ k2 3
209
210 The following solitary wave and soliton solutions of Eq (4) are
211 constructed by substituting Eq (12) into Eq (6) as:
212
u21ðnÞ ¼ x
Ak1 4c2 Bk
2
1þ C k2
2þ k2 3
A 1 3csc2 ffiffiffiffiffiffiffiffiffi
c2 p n
;
214
215
u22ðnÞ ¼ x
Ak1
4c2 Bk
2
1þ C k2
2þ k2 3
A 1 3csch2 ffiffiffiffiffiffiffiffiffi
c2
;
217
218
u23ðnÞ ¼ x
Ak1 4c2 Bk
2
1þ C k2
2þ k2 3
A 1 3 sec2 ffiffiffiffiffiffiffiffiffi
c2 p n
;
220
221
u24ðnÞ ¼ x
Ak1 4c2 Bk
2
1þ C k2
2þ k2 3
A
1 12c2c4exp 2 ffiffiffiffiffi
c2
p n
1 c2c4exp 2 ffiffiffiffiffi
c2
p n
!
; c2> 0; c4– 0; ð16Þ
223 224 where ; n ¼ k1x þ k2y þ k3z þ x t :
225 Case 3: c0¼ c1¼ c4¼ c5¼ c6¼ 0,
226
a2¼ a1¼ 0; a1¼ 3c3 Bk
2
1þ C k2
2þ k2 3
A ; a2¼ 0;
x ¼ k1 a0A þ c2 Bk21þ C k2
2þ k2 3
:
ð17Þ
228 229 The following solitary wave soliton solutions of Eq (4) are
con-230 structed by substituting Eq (17) into Eq (6) as:
231
u31ðnÞ ¼ a0þ 3c2 Bk
2
1þ C k2
2þ k2 3
A 1 þ tan2 1
2
ffiffiffiffiffiffiffiffiffi
c2 p x
;
234
u32ðnÞ ¼ a0þ 3c2 Bk
2
1þ C k2
2þ k2 3
A 1 tanh2 1
2
ffiffiffiffiffi
c2 p n
;
237 where; n ¼ k1x þ k2y þ k3z þ x t; x ¼ k1 a0A þ c2 Bk2þ C k2
þ k2
: 238 Case 4: c1¼ c3¼ c5¼ c6¼ 0 and c0¼ c 2
4c 4,
239
a2¼ 3c
2 Bk21þ C k2
2þ k2 3
Ac4 ; a1¼ 0;
a0¼ x þ 4c2k1 Bk21þ C k2
2þ k2 3
Ak1 ; a1¼ 0;
a2¼ 12c4 Bk
2
1þ Ck2
2þ Ck2 3
Fig 1 Traveling wave solutions of Eq.(8)with different forms are plotted: (a) dark solitary wave and (b) contour plot of u11
Please cite this article in press as: Lu D et al New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified
Trang 4a2¼ 3c
2 Bk21þ C k2
2þ k2 3
Ac4 ; a1¼ a1¼ a2¼ 0;
a0¼ x þ 4c2k1 Bk21þ C k2
2þ k2 3
244
245
a2¼ a1¼ a1¼ 0; a0¼ x þ 4c2k1 Bk21þ C k2
2þ k2 3
a2¼ 12c4 Bk
2
1þ Ck2
2þ Ck2 3
247
248 The following soliton-like solutions of Eq (4) are constructed by
249 substituting Eq (20) into Eq (6) as:
250
u41ðnÞ ¼ 6c2 Bk
2
1þ C k2
2þ k2 3
2
ffiffiffiffiffi
c2 2
r n
x þ 4c2k1 Bk21þ C k2
2þ k2 3
Ak1
6c2 Bk
2
1þ C k2
2þ k2 3
2
ffiffiffiffiffi
c2 2
r n
;
252
253
u42ðnÞ ¼ 6c2 Bk
2
1þ C k2
2þ k2 3
2
ffiffiffiffiffiffiffiffiffi
c2 2
r n
x þ 4c2k1 Bk21þ C k2
2þ k2 3
Ak1
þ 6c2 Bk
2
1þ C k2
2þ k2 3
2
ffiffiffiffiffiffiffiffiffi
c2 2
r n
c2< 0;c4> 0; ð24Þ
255
256 where ; n ¼ k1x þ k2y þ k3z þ x t :
257 Similarly, one can find the soliton-like solutions of Eq (4) from
258 Eqs (21) and (22)
259 Case 5: c2¼ c4¼ c5¼ c6¼ 0,
260
a2¼ 0; a1¼ 0; a0¼ x
Ak1; a1¼ 3c3 Bk
2
1þ Ck2
2þ Ck2 3
262
263 From Eq (25) , we construct the following new elliptic function
264 solution [40] of Eq (4) as:
265
u51ðnÞ ¼ x
Ak1 3c3 Bk
2
1þ ck2
2þ ck2 3
ffiffiffiffiffi
c3
p n
2 ; g2; g3
; c3> 0;
ð26Þ
267
268 where ; n ¼ k1x þ k2y þ k3z þ x t ; }isWeierstrassellipticfunctionand
269 g2¼ 4c 1
c 3; g3¼ 4c 0
c 3:
270 Case 6: c4 ¼ c5 ¼ c6 ¼ 0,
271
a2¼ a1¼ a2¼ 0; a1¼ 3c3 Bk
2
1þ C k2
2þ k2 3
x ¼ k1 a0A þ c2 Bk21þ C k2
2þ k2 3
ð27Þ
273
274 From Eq (27) , we can also construct the new exact Weierstrass
275 elliptic function solutions [40–42] of Eq (4)
276 Case 7: c1¼ c3¼ c5¼ c6¼ 0,
278 a2¼ a1¼ a1¼ 0;a2¼ 12c4 Bk
2þC k 2þk2
x ¼ k1 a0Aþ4c2 Bk2þC k 2þk2
;
ð28Þ
280
281 (ii):
282
a2¼ 12c0 Bk
2þC k 2þk2
A ;a1¼ a1¼ a2¼ 0;
x ¼ k1 a0Aþ4c2 Bk2þC k 2þk2
;
ð29Þ
284 285 (iii):
286
a2¼ 12c0 Bk
2
1þC k2
2þk2 3
A ; a1¼ a1¼ 0;
a2¼ 12c4 Bk
2
1þC k2
2þk2 3
x ¼ k1 a0Aþ4c2 Bk21þC k2
2þk2 3
:
ð30Þ
288 289 From Eqs (28)–(30) , one can constructed the new Jacobi elliptic
290 function solutions [45] of Eq (4) by choosing the value of
291
m ð0 6 m 6 1Þ according to the Table 1
292 The (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov
(mKdV-293 ZK) Equation
294 Now, we consider the (3 + 1)-dimensional mkdv-zk equation is
295 as:
296
utþ q u2uxþ uxxxþ uxyyþ uxzz¼ 0; ð31Þ 298
299 where q is a arbitrary constant and can be obtained as a model from
300 modified KdV in case of weakly two dimensional variations of the
301 modified KdV equation [11] naturally.
302 Consider the traveling wave solutions and transformation in
303 Eqs (2) and (3) , the (31) reduces into ODE as
304
x u0þ q k1u2u0þ k3
1þ k1k22þ k1k23
307
By using balancing principle on Eq (32) gets m ¼ 1 and
consid-308 ering the solution of Eq (32) is as:
309
312 Substituting Eq (33) into Eq (32) and setting the coefficients of
313 /i/ðiÞ to zero, yields a systems of algebraic equations in
314
a1; a0; a1; q ; k1; k2; k3and x The system of equations have the
fol-315 lowing solutions cases:
316 Case 1: c0¼ c1¼ c5¼ c6¼ 0,
317
a1¼ 0; a0¼ c3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2þ k2þ k2
2 q c4
v u
; a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2þ k2þ k2
q
v u
;
x ¼ k1 3c
2 8c2c4
k2þ k2þ k2
320 Substituting the values of Eq (34) into Eq (33) along with the
321 solutions of Eq (3) , the following solutions of Eq (31) in
soliton-322 like and solitary wave form are obtained:
323
u11ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
q
v u
c3 ffiffiffiffiffiffiffiffi 8c4
ffiffiffiffiffiffiffiffi 8c4
p
c2sech ffiffiffiffiffi c
2
p n
ffiffiffiffi
D
p
c4sech ffiffiffiffiffi
c2
p n
!
;
326
u12ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
q
v u
c ffiffiffiffiffiffiffiffi3 8c4
ffiffiffiffiffiffiffiffi 8c4
p
c2sech ffiffiffiffiffi
c2
p n
ffiffiffiffi
D
p
þ c4sech ffiffiffiffiffi c
2
p n
!
;
Trang 5u13ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
þ k2
þ k2
q
v
u
c3
ffiffiffiffiffiffiffiffi 8c4
ffiffiffiffiffiffiffiffi 8c4
p
c2csch ffiffiffiffiffi c
2
ffiffiffiffiffiffiffi
D p
c4csch ffiffiffiffiffi c
2
!
;
331
332
u14ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
q
v
u
c3 ffiffiffiffiffiffiffiffi 8c4
ffiffiffiffiffiffiffiffi 8c4
p
c2csch ffiffiffiffiffi
c2
p n
ffiffiffiffiffiffiffi
D
p
þ c4csch ffiffiffiffiffi
c2
p n
!
;
334
335
u15ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
þ k2
þ k2
q
v
u
c3
ffiffiffiffiffiffiffiffi 8c4
ffiffiffi 2
p
c2
ffiffiffiffiffi
c4
p 1 tanh
ffiffiffiffiffi
c2
p
2 n
;
337
338
u16ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
q
v
u
c3 ffiffiffiffiffiffiffiffi 8c4
ffiffiffi 2
p
c2 ffiffiffiffiffi
c4
p 1 coth
ffiffiffiffiffi
c2 p
2 n
;
340
341 where ; n ¼ k1x þ k2y þ k3z þ x t ; D ¼ c2 4c2c4;
342 x ¼k 1ð3c 2 8c 2 c 4Þ ðk 2 þk 2 þk 2Þ
343 Case 2: c0¼ c1¼ c3¼ c5¼ c6¼ 0,
344
a1¼ a0¼ 0; a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v u
;
x ¼ c2k1 k21þ k2
2þ k2 3
346
347 We constructed the following soliton-like solutions of Eq (31)
348 from Eqs (3), (33) and (41) as:
349
u21ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6c2 k21þ k2
2þ k2 3
q
v u
csc ffiffiffiffiffiffiffiffiffi
c2 p n
ð Þ; c2< 0; c4> 0; ð42Þ
351
352
u22ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6c2 k21þ k2
2þ k2 3
q
v u
csch ffiffiffiffiffiffiffiffiffi
c2
ð Þ; c2< 0; c4> 0; ð43Þ
354
355
u23ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6c2 k21þ k2
2þ k2 3
q
v u
sec ffiffiffiffiffiffiffiffiffi
c2 p n
ð Þ; c2< 0; c4> 0; ð44Þ
357
358
u24ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k21þ k2
2þ k2 3
q
v
2exp ffiffiffiffiffi
c2
p n
1 c2c4exp 2 ffiffiffiffiffi
c2
p n
ð Þ ; 2> 0;
360
361 where ; n ¼ k1x þ k2y þ k3z þ x t ; x ¼ c2k1 k21þ k2
2þ k2 3
:
362 Case 3: c1¼ c3¼ c5¼ c6¼ 0; c4> 0andc0¼ c 2
4c 4
363
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3c
2 k21þ k2
2þ k2 3
2 q c4
v u
; a0¼ 0;
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v u
; x ¼ 4c2k1 k21þ k2
2þ k2 3
; ð46Þ
365 366
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3c
2 k21þ k2
2þ k2 3
2 q c4
v u
; a0¼ 0;
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v u
; x ¼ 2c2k1 k21þ k2
2þ k2 3
; ð47Þ
368 369
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3c
2 k21þ k2
2þ k2 3
2 q c4
v u
; a0¼ 0; a1¼ 0;
x ¼ c2k1 k21þ k2
2þ k2 3
372
a1¼ 0; a0¼ 0; a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v u
;
x ¼ c2k1 k21þ k2
2þ k2 3
375 Substituting Eq (46) into Eq (33) , the following solitary wave
solu-376 tions of Eq (31) are obtained:
377
u31ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3c2 k
2
1þ k2
2þ k2 3
q
v u
cot
ffiffiffiffiffi
c2 2
r n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3c2 k
2
1þ k2
2þ k2 3
q
v u
tan
ffiffiffiffiffi
c2 2
r n
; c2> 0; ð50Þ
379 380
u32ðnÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3c2 k21þ k2
2þ k2 3
q
v u
coth
ffiffiffiffiffiffiffiffiffi
c2 2
r n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3c2 k21þ k2
2þ k2 3
q
v u
tanh
ffiffiffiffiffiffiffiffiffi
c2 2
r n
; c2< 0; ð51Þ
382
383 where ; n ¼ k1x þ k2y þ k3z þ x t ; x ¼ 4c2k1 k21þ k2
2þ k2 3
:
384 From Eqs (47)–(49) , one can also construct the new soliton-like
385 and solitay wave solutions of Eq (31)
386 Case 4: c2¼ c4¼ c5¼ c6¼ 0,
Table 1
Jacobi elliptic functions
dnn
nsn¼ ðsnnÞ1; dcn ¼dnn
cnn
dnn
snn
Please cite this article in press as: Lu D et al New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified
Trang 6a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c0 k
2
1þ k2
2þ k2 3
q
v
u
; a0¼ c1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
2c0q
v u
;
a1¼ 0; x ¼ 3c
2k1 k21þ k2
2þ k2 3
8c0
389
390 From Eq (52) , we construct the following new Weierstrass elliptic
391 function solution [40] of Eq (31) as:
392
u41ðnÞ ¼ c1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
2c0q
v u
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c0 k
2
1þ k2
2þ k2 3
q
v u
} pffiffiffiffic3n
2 ; g2; g3
394 395 where ; n ¼ k1x þ k2y þ k3z þ x t ;
} is Weierstrass elliptic function and g2¼ 4c1
c3 ; g3¼ 4c0
c3 :
397
Fig 2 Traveling wave solutions of Eq.(9)with different forms are plotted: (a) periodic bright solitary wave and (b) contour plot of u12
Fig 3 Traveling wave solutions of Eq.(13)with different forms are plotted: (a) periodic solitary wave and (b) contour plot of u21at c2¼ 4; c4¼ 3; k1¼ 1; k2¼ 1:5; k3¼ 2;
x¼ 1:5; A ¼ 3; B ¼ 2; C ¼ 0:75; y ¼ 1; z ¼ 1
Trang 7398 Case 5: c5¼ c6¼ 0,
399
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c0 k
2
1þ k2
2þ k2 3
q
v
u
; a0¼ c1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
2c0q
v u
;
a1¼ 0; x ¼ 3c
2 8c0c2
k1 k21þ k2
2þ k2 3
401
402
a1¼ 0; a0¼ c3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
2 q c4
v u
;
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v u
;
x ¼ 3c
2 8c2c4
k1 k21þ k2
2þ k2 3
Fig 4 Traveling wave solutions of Eq.(19)with different forms are plotted: (a) bright solitary wave and (b) contour plot of u32at c2¼ 1; c3¼ 1; k1¼ 0:5; k2¼ 1:5; k3¼ 1;
A¼ 3; B ¼ 2; C ¼ 2; a0¼ 1; y ¼ 1; z ¼ 1
Fig 5 Traveling wave solutions of Eq.(26)with different forms are plotted: (a) periodic traveling wave and (b) contour plot at c0¼ 0; c1¼ 1; c2¼ 1; k1¼ 0:75;
k2¼ 1:5; k3¼ 0:5;x¼ 1; A ¼ 3; B ¼ 2; C ¼ 2; y ¼ 1; z ¼ 1
Please cite this article in press as: Lu D et al New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified
Trang 8405 Case 6: c4¼ c5¼ c6¼ 0,
406
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c0 k
2
1þ k2
2þ k2 3
q
v
u
; a0¼ c1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
2 q c0
v u
;
a1¼ 0; x ¼ 8c0c2 3c
2
k1 k21þ k2
2þ k2 3
408
409 We can also construct the new Weierstrass elliptic function
410 solutions [40–44] of Eq (31) from Eqs (54)–(56) of Cases 5 and 6.
411 Case 7: c1¼ c3¼ c5¼ c6¼ 0,
412
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c0 k
2
1þ k2
2þ k2 3
q
v u
; a0¼ 0;
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v u
;
x ¼ c2þ 6 ffiffiffiffiffiffiffiffiffi c0c4
p
ð Þk1 k21þ k2
2þ k2 3
Fig 6 Traveling wave solutions of Eq.(35)with different forms are plotted: (a) bright soliton and (b) contour plot of u11 at c2¼ 1; c3¼ 2; c4¼ 2; k1¼ 0:5; k2¼ 1:5;
k3¼ 1;q¼ 2; y ¼ 1; z ¼ 1
Fig 7 Traveling wave solutions of Eq.(44)with different forms are drawn: (a) periodic dark soliton and (b) contour plot of u23 at c2¼ 2; c4¼ 1; k1¼ 0:5; k2¼ 1:5;
k3¼ 0:75;q¼ 2; y ¼ 1; z ¼ 1
Trang 9a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c0 k
2þ k2þ k2
q
v
u
; a0¼ 0; a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2þ k2þ k2
q
v u
;
x ¼ 6 p ffiffiffiffiffiffiffiffiffi c0c4 c
2
ð Þk1 k21þ k2
2þ k2 3
417
418
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c0 k
2
1þ k2
2þ k2 3
q
v
u
; a0¼ 0; a1¼ 0;
x ¼ c2k1 k21þ k2
2þ k2 3
420
421
a1¼ 0; a0¼ 0; a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v u
;
x ¼ c2k1 k21þ k2
2þ k2 3
423 424 From Eqs (57)–(60) , one can obtained the new Jacobi elliptic
425 function solutions [45] of Eq (31) by choosing the value of
426
m ð0 6 m 6 1Þ according to the Table 1
Fig 8 Traveling wave solutions of Eq.(50)with different forms are drawn: (a) periodic dark soliton and (b) contour plot of u23 at c2¼ 0:5; c4¼ 1; k1¼ 0:5; k2¼ 1:5;
k3¼ 0:75;q¼ 2; y ¼ 1; z ¼ 1
Fig 9 Traveling wave solutions of Eq.(53)with different forms are plotted: (a) periodic bright and dark soliton and (b) contour plot at c0¼ 0:1; c1¼ 1; c3¼ 1; k1¼ 0:75;
k2¼ 1:5; k3¼ 0:5;q¼ 2; y ¼ 1; z ¼ 1
Please cite this article in press as: Lu D et al New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified
Trang 10427 Case 8: c0¼ c1¼ c2¼ c5¼ c6¼ 0,
428
a1¼ 0; a0¼ c3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
2 q c4
v u
;
a1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v
u
; x ¼ 3c
2k1 k21þ k2
2þ k2 3
8c4 : ð61Þ
430
431 We obtained the new solitary wave solution of Eq (31) by
sub-432 stituting Eq (61) into Eq (33) is as.
433
u81ðnÞ ¼ c3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 k
2
1þ k2
2þ k2 3
2 q c4
v u
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6c4 k
2
1þ k2
2þ k2 3
q
v
3
c2n2 4c4
!
; c4> 0; ð62Þ
435
436 where ; n ¼ k1x þ k2y þ k3z þ x t ; x ¼3c 2 k 1ðk 2 þk 2 þk 2Þ
438 In this paper, some general new exact solutions in the form of
439 soliton, solitary wave, elliptic function and Weiertrass elliptic
func-440 tion solutions of three-dimensional EZK and (3 + 1)-dimensional
441 modified KdV-ZK equations are constructed by utilizing modified
442 extended Direct algebraic method These solitions and other
443 solutions in which many are new and derived in explicit form, have
444 many applications and useful in different areas of physics,
engi-445 neering and other fields of applied sciences These general
solu-446 tions can provide a useful help for researchers to study and
447 understand the physical interpretation of system This method
448 has several advantages such as the calculations are simple and
449 straightforward, gives more general solutions then other existing
450 methods, the reduction in the size of computational work and
con-451 sistency gives its wider applicability ( Figs 2–9 ).
453 [1] Ablowitz MJ, Clarkson PA Solitons, nonlinear evolution equations and inverse
454 scattering New York: Cambridge Univ Press; 1991.
455 [2] Shukla PK, Mamun AA Introduction to dusty plasma physics Bristol, U
456 K.: Institute of Physics Publishing; 2002
457 [3] Korteweg DJ, de Vries G On the change of form of long waves advancing in a
458 rectangular canal and on a new type of long stationary waves Philos Mag
459 1895;39:422–43
460 [4] Kochanov MB, Kudryashov NA, Sinel’shchikov DI Non-linear waves on shallow
461 water under an ice cover Higher order expansions J Appl Math Mech
463 [5] El-Tantawy SA, Moslem WM Nonlinear structures of the Korteweg-de Vries
464 and modified Korteweg-de Vries equations in non-maxwellian
electron-465 positron-ion plasma: solitons collision and rouge waves Phys Plasmas
466 2014;21:052112.
467 [6] Kadomtsev BB, Petviashvili VI On the stability of solitary waves in weakly
468 dispersing media Sov Phys Dokl 1970;15:539–41.
469 [7] Liu ZM, Duan WS, He GJ Efeects of dust size distribution on dust acoustic
470 waves in magnetized two-ion-temperature dusty plasmas Phys Plasmas
471 2008;15:083702.
472 [8] Seadawy AR Stability analysis for two-dimensional ion-acoustic waves in
473 quantum plasmas Phys Plasmas 2014;21:052107.
474 [9] Seadawy AR Nonlinear wave solutions of the three-dimensional
Zakharov-475 Kuznetsov-Burgers equation in dusty plasma Physica A 2015;439:124–31.
476 [10] Seadawy AR, Lu Dianchen Ion acoustic solitary wave solutions of
three-477 dimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a
478 magnetized two-ion-temperature dusty plasma Result Phys 2016;6:560–93
479 [11] Demontis F Exact solutions of the modified Korteweg-de Vries equation
480 Theoret Math Phys 2011;168(1):886–97
481 [12] Islam MH, Khan K, Akbar MA, Salam MA Exact traveling wave solutions of
482 modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation
483 Springer Plus 2014;3(105):1–9
484 [13] Naher H, Abdullah FA, Akbar MA Generalized and improved (G’/G)-expansion
485 method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation
486 Plos One 2013;8(5):1–7
487 [14] Naher H, Abdullah FA, Akbar MA New traveling wave solution of higher
488 dimensional nonlinear partial differential equation by the Exp-function
489 method J Appl Math 2012;2012:1–14 Article ID 575387
490 [15] Zhang BG, Liu ZR, Xiao Q New exact solitay wave and multiple soliton
491 solutions of quantum Zakharov-Kuznetsov equation Appl Math Comput
492 2010;217:392–402
493 [16] Wang GW, Xu TZ, Johnson S, Biswas A Solitons and Lie group analysis to an
494 extended quantum Zakharov-Kuznetsov equation Asterophys Space Sci
495 2014;349:317–27
496 [17] Pakzad HR Soliton energy of the Kadomtsev-Petviashvili equation in warm
497 dusty plusma with variable dust charge, two-temperature ions, and
498 nonthermal electrons Asterophys Space Sci 2010;326:69–75
499 [18] Seadawy AR, El-Rashidy K Travelling wave solutions for some coupled
500 nonlinear evolution equations by using the direct algebraic method Math
501 Comput Modelling 2013;57:1371–9
502 [19] Yu X, Gao YT, Sun ZY, Liu Y N-soliton solutions, bucklund transformation and
503 lax pair for a generalized variable coefficient fifth-order Korteweg de Vries
504 equation Phys Scr 2010;81:045402
505 [20] Hirota R Exact Solutions of the Korteweg-de Vries equation for multiple
506 collisions of solitons Phys Rev Lett 1971;27:1192–4
507 [21] Alam MN, Hafez MG, Akbar MA, Roshid HO Exact travelling wave solutions to
508 the (3+1)-dimensional mKdV-ZK and the (2+1)-dimensional Burgers equations
509 via exp(/ðgÞ)-expension method Alexa Eng J 2015;54:635–44
510 [22] Abdou MA The extended tanh-method and its applications for solving
511 nonlinear physical models Appl Math Comput 2007;190:988–96
512 [23] Guiqiong Xu, Li Zhibin Exact travelling wave solutions of the
whitham-broer-513 kaup and broer-kaup-kupershmidt equations Chaos Solitons Fractals
514 2005;24:549–66
515 [24] Wen XY Construction of new exact rational form non-travelling wave
516 solutions to the (2+1)-dimensional generalized Broer-Kaup system Appl
517 Math Comput 2010;217:1367–75
518 [25] Zenga Xin, Wang Deng-Shan A generalized extended rational expansion
519 method and its application to (1+1)-dimensional dispersive long wave
520 equation Appl Math Comput 2009;212:296–304
521 [26] Yao Yuqin Abundant families of new traveling wave solutions for the coupled
522 Drinfl’d-sokolov-wilson equation Chaos Solitons Fractals 2005;24:301–7
523 [27] Peng YZ Exact solutions for some nonlinear partial differential equations Phys
524 Lett A 2003;314:401–8
525 [28] Seadawy AR, El-Rashidy K Traveling wave solutions for some coupled
526 nonlinear evolution equations Math Comput Modelling 2013;57:1371–9
527 [29] Ugurlua Yavuz, Kaya Dogan, Inanb Ibrahim E Comparison of three
semi-528 analytical methods for solving (1 + 1)-dimensional dispersive long wave
529 equations Comput Math Appl 2011;61:1278–90
530 [30] Dinarvand S, Khosravi S, Doosthoseini A, Rashidi MM The homotopy analysis
531 method for solving the Sawada-Kotera and Laxs fifth-order KdV equations Adv
532 Theor Appl Mech 2008;1:327–35
533 [31] Biazar J, Badpeima F, Azimi F Application of the homotopy perturbation
534 method to Zakharov-Kuznetsov equations Comput Math Appl
535 2009;58:2391–4
536 [32] Ravi ASV, Kanth K Aruna, Differential transform method for solving linear and
537 non-linear systems of partial differential equations Phys Lett A
538 2008;372:6896–8
539 [33] Rashidi MM, Erfani E Traveling wave solutions of WBK shallow water
540 equations by differential transform method Adv Theor Appl Mech
541 2010;3:263–71
542 [34] Keskin Y, Oturanc G Reduced differential transform method for partial
543 differential equations Int J Nonlinear Sci Numer Simul 2009;10:741–9
544 [35] Abazari R, Abazari M Numerical simulation of generalized HirotaSatsuma
545 coupled KdV equation by RDTM and comparison with DTM Commun
546 Nonlinear Sci Numer Simul 2012;17:619–29
547 [36] Helal MA, Seadawy AR Variational method for the derivative nonlinear
548 Schrodinger equation with computational applications Phys Scr 2009;80
549 Article ID 035004
550 [37] Helal MA, Seadawy AR Exact soliton solutions of a D-dimensional nonlinear
551 Schrodinger equation with damping and diffusive terms Z Angew Math Phys
552 2011;62:839–47
553 [38] Seadawy AR New exact solutions for the KdV equation with higher order
554 nonlinearity by using the variational method Comput Math Appl
555 2011;62:3741–55
556 [39] Helal MA, Seadawy AR Benjamin-Feir instability in nonlinear dispersive
557 waves Comput Math Appl 2012;64:3557–68
558 [40] Nickel J, Serov VS, Schurmann HW Some elliptic travelling wave solution to
559 the Novikkov-Veselov equation Proc Prog Electromagnet Res Symp
560 2006;61:323–31
561 [41] Schurmann HW Travelling-wave solutions of the cubic-quintic nonlinear
562 schrodinger equation Phys Rev E 1996;54:4312–20
563 [42] Nickel J Elliptic solutions to a generalized BBM equation Phys Lett A
564 2007;364:221–6
565 [43] Schurmann HW, Serov VS, Nickel J Superposition in nonlinear wave and
566 evolution equations Int J Theor Phys 2006;45:1057–73
567 [44] Nickel J, Schurmann HW Comments on Exact solutions of the derivative
568 nonlinear Schrodinger equation for a nonlinear transmission line Phys Rev E
569 2007;75:038601
570 [45] Yomba E Jacobi elliptic function solutions of the generalized
Zakharov-571 Kuznetsov equation with nonlinear dispersion and t-dependent coefficients
572 Phys Lett A 2010;374:1611–5