new solitary wave solutions of 3 1 dimensional nonlinear extended zakharov kuznetsov and modified kdv zakharov kuznetsov equations and their applications

RINP 565 No of Pages 11, Model 5G 20 February 2017 Results in Physics xxx (2017) xxx–xxx Contents lists available at ScienceDirect Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications Dianchen Lu a, A.R Seadawy b,c,⇑, M Arshad a, Jun Wang a 10 11 12 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a b c 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 a r t i c l e i n f o Article history: Received 21 November 2016 Received in revised form 21 January 2017 Accepted February 2017 Available online xxxx Keywords: Modified extended direct algebraic method Solitons Solitary wave solutions Jacobi and Weierstrass elliptic function solutions Three dimensional extended ZakharovKuznetsov dynamical equation (3 + 1)-Dim modified KdV-ZakharovKuznetsov equation a b s t r a c t In this paper, new exact solitary wave, soliton and elliptic function solutions are constructed in various forms of three dimensional nonlinear partial differential equations (PDEs) in mathematical physics by utilizing modified extended direct algebraic method Soliton solutions in different forms such as bell and anti-bell periodic, dark soliton, bright soliton, bright and dark solitary wave in periodic form etc are obtained, which have large applications in different branches of physics and other areas of applied sciences The obtained solutions are also presented graphically Furthermore, many other nonlinear evolution equations arising in mathematical physics and engineering can also be solved by this powerful, reliable and capable method The nonlinear three dimensional extended Zakharov-Kuznetsov dynamica equation and (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation are selected to show the reliability and effectiveness of the current method 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/) 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Introduction 47 Nonlinear PDEs involve nonlinear complex physical phenomena, which plays a vital role in plasma physics and many other aeras of applied sciences The generalized KdV-Zakharov-Kuznetsov equations are an important models for numerous physical phenomena as well as waves in nonlinear LC circuit by way of mutual inductance among neighboring inductors, shallow and stratified internal waves, ion-acoustic waves in plasma physics, applications in space environments and astrophysical, nonlinear optic, hydrodynamic and many more [1–5] Moreover, the electrostatic solitary waves have been determined in many areas such as solar, wind, earths magnetotail and polar magnetosphere [2] etc The KdV equations have several two dimensional weak variations Solitons and solitary waves represent one of the famous and motivating features of nonlinear phenomena spacially in extended equations, which have many important properties The Zakharov-Kuznetsov equation is 48 49 50 51 52 53 54 55 56 57 58 59 60 61 ⇑ Corresponding author at: Mathematics Department, Faculty of Science, Taibah University, Al-Ula, Saudi Arabia E-mail address: Aly742001@yahoo.com (A.R Seadawy) one of the two well studied canonical two dimensional extensions of KdV equation [6] Nonlinear extended Z-K equations are utilized to discuss the nonlinear dust ion-acoustic waves in magnetized two ion-temperature dusty plasmas, propagations of the low frequency ion-acoustic wave in a thick Quantum magneto-plasmas etc [7–9] Recently, the authors in [10] derived nonlinear three dimensional extended zakharov-Kuznetsov dynamical equation in a magnetized two ion-temperature dusty plasma by using the theory of reductive perturbation The modified KdV model can be drived for the elaboration of ion-acoustic perturbation in plasma with components of two negative ions of different temperature [11] In the case of weakly two dimensional variations of the modified KdV equation [11], the modified KdV-ZK model occurs [12–14], which is an important model arises in different branches of physics such as plasma physics, nonlinear optics fluid dynamics, theoretical physics quantum mechanics and mathematical physics to analize the main properties of non-linear propagation of various physical phenomena Recently, many researchers have been giving much attention on the study of constructing the solitons and solitary wave solutions 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/) 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 RINP 565 No of Pages 11, Model 5G 20 February 2017 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 D Lu et al / Results in Physics xxx (2017) xxx–xxx So, Many powerful methods have been discovered to construct the solitons and solitary wave solutions such as inverse scattering scheme [3], direct algebraic method [18], Bcklund transform method [19], the Hirota’s bilinear scheme [20], exp(À/ðgÞ)expension method [21], extended method [22], auxiliary equation method [23], the mapping method and extended mapping method [24], rational expansion method [25], elliptic function method [26] and many more [27,28] Many numerical schemes also have been developed such as Adomian decomposition method [29], homotopy analysis method [29,30], homotopy perturbation method [31], differential transform and reduced differential transform method [32–35] etc to obtain the numerical solutions in different form of non-linear evaluation equations The study about solutions, structures, interaction and further properties of soliton gained much attention and various meaningful results are successfully derived [36–39] In this work, the ansatz equation is further extended in the modified extended direct algebraic method to construct soliton and solitary wave solutions of three-dimensional EZK equation and modified KdV-ZK equation Consequently, more general and new exact solutions in soliton and solitary wave form are constructed [40–52] This article is ordered as follows An introduction is given in Section ‘Introduction’ The main steps of the modified extended direct algebraic method are specified in Section ‘Description of modified extended direct algebraic method’ We obtain general new exact solutions in soliton, solitary wave and elliptic solutions in different form of three-dimensional EZK equation and (3 + 1)dimensional modified KdV-ZK equation in Section ‘Application of the modified extended direct algebraic method’ Lastly, the conclusion is given in Section ‘Conclusion’ Description of modified extended direct algebraic method: 115 Let us assume a general non-linear evolution equation in x; y; z and t as 117 119 À Á F u; ut ; ux ; uy ; uz ; uxx ; uyy ; uzz ; uxy ; uxz ; uyz ; uxxx ; ẳ 0; 1ị 126 where the function uðx; y; z; tÞ is unknown and F is a polynomial function with respect to some functions OR specified variables, which have non-linear and terms and highest order derivatives of the unknown functions and can be reduced to a polynomial function by utilizing transformations in which the real variables x; y; z and t can be merge in a complex variable The key steps of this method are as: 127 Step 1: Consider that Eq (1) has the following solution as: 120 121 122 123 124 125 128 ux; y; z; tị ẳ unị ẳ 130 131 132 134 135 136 137 138 139 140 141 142 143 144 m X aj / j nị; 2ị jẳm v u uX / nị ẳ t ci /i ; and n ẳ k1 x ỵ k2 y ỵ k3 z ỵ xt; 3ị iẳ0 where aj ; ci i from to are arbitrary constants and k1 ; k2 ; k3 are wave lengths and x is frequency Step 2: By balancing the highest order non-linear term with the highest order derivative term of Eq (1), and the series of coefficients am ; amỵ1 ; ; aÀ1 ; a0 ; a1 ; am ; c0 ; c1 ; ; c6 ; k1 ; k2 ; k3 ; x are parameters can be obtained Step 3: Substituting Eqs (2) and (3) into Eq (1) and setting the coefficients of powers of / j /ðjÞ to zero, yields a systems of algebraic equations in parameters am ; amỵ1 ; ; 146 147 148 149 150 151 152 The extended Zakharov-Kuznetsov (EZK) equation: 153 First, we consider the (3 + 1)-dimensional extended ZakharovKuznetsov (EZK) equation is as: 154 ut ỵ Auux ỵ Buxxx ỵ C uxyy ỵ uxzz ẳ 0; 4ị where A; B and C are arbitrary constants Consider the traveling wave solutions and transformation in Eqs (2) and (3), then Eq (4) reduces into ODE as   xu0 nị ỵ Ak1 uu0 nị þ Bk31 þ Ck1 ðk22 þ k23 Þ u000 ðnÞ ¼ 0; ð5Þ By using balance principle on Eq (5) gives m ¼ 2, we consider the solution of Eq (5) is as: unị ẳ a2 /2 ỵ a1 ỵ a0 ỵ a1 / ỵ ỵa1 /2 : / 6ị 155 156 158 159 160 161 162 164 165 166 167 169 Substituting Eq (6) into Eq (5) and setting the coefficients of 170 coefficient of / j /ðjÞ to zero, yields a systems of algebraic equations in aÀ2 ; aÀ1 ; a0 ; a1 ; a2 ; A; B; C; k1 ; k2 ; k3 and x The systems of algebraic equations are solved by Mathematica, which have possesses the following solutions cases: 171 aÀ2 ¼ À   2 32c22 Bk1 ỵ Ck2 ỵ Ck3 8c22 27c4 ; c6 ẳ c24 , 4c2 7ị Substituting Eq (7) into Eq (6) along with the solutions of Eq (3), the following solutions of Eq (4) are obtained: u11 nị ẳ a0 ỵ 4c2 Bk1 ỵ   q  ỵ ỵ  c32 n  qffiffiffiffiffiffiffiffi  ; 3Atanh  À c32 n Ck2 173 174 176 ; aÀ1 ¼ a1 ¼ a2 ¼ 0; 9Ac4   x ¼ Àk1 a0 A ỵ 4Bc2 k21 ỵ 4c2 Ck22 ỵ 4c2 Ck23 :  172 175 178 179 180 181 Ck3 c2 < 0; c4 > 0; ð8Þ ð9Þ ð10Þ ð11Þ  189 190  qffiffiffiffi    2 4c2 Bk1 ỵ Ck2 ỵ Ck3 cot2  c32 n u14 nị ẳ a0 q  ; c2 3Acot2 n c2 > 0; c4 < 0; 186 187  qffiffiffiffi    2 2 4c2 Bk1 ỵ Ck2 ỵ Ck3 þ coth  c32 n  qffiffiffiffiffiffiffiffi  ; u13 nị ẳ a0 ỵ 3Acoth  c32 n c2 < 0; c4 > 0; 183 184    q  2 4c2 Bk1 ỵ Ck2 þ Ck3 À tan2  À c32 n  q  ; u12 nị ẳ a0 3A tan2  c32 n c2 > 0; c4 < 0; where 145 Application of the modified extended direct algebraic method: Case 1: c1 ¼ c3 ¼ c5 ¼ 0; c0 ¼ 114 116 aÀ1 ; a0 ; a1 ; am ; c0 ; c1 ; c2 ; c6 ; k1 ; k2 ; k3 ; x The systems of algebraic equations are solved by Marhematica, then the values of parameters can be obtained Step 4: By substituting the values of parameters and /ðnÞ obtained in previous step into Eq (2), then the solutions of Eq (1) can be constructed 192 where; n ¼ k1 x þ k2 y þ k3 z þ xt; x ¼ k1 a0 A ỵ 4Bc2 k1 ỵ 4c2 Ck2 193 ỵ4c2 Ck3 ị: 194 2 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx Fig Traveling wave solutions of Eq (8) with different forms are plotted: (a) dark solitary wave and (b) contour plot of u11 206 Figs 1(a) and (c) signify the evolution of the dark and periodic bright solitary waves solutions of Eqs (8) and (9) of the EZK Eq (4) at c2 ¼ À1:25; c4 ¼ 2; k1 ¼ 0:75; k2 ¼ 1:5; k3 ¼ 1;  ¼ 1; A ¼ 2; B ¼ 1; C ¼ 0:5; a0 ¼ 1:5; y ¼ 1; z ¼ and c2 ¼ 1; c4 ¼ À2; k1 ¼ 0:5; k2 ¼ 1:5; k3 ¼ 1:5;  ¼ 1; A ¼ 2; B ¼ 1; C ¼ 1; a0 ¼ 1; y ¼ 1; z ¼ respectively A contour plots Figs 1(b) and (d) are a collection of level curves drawn on same set of intervals The command of Contour Plot on mathematica draws ContourPlot of two variable functions The points on contours join at same height on the surface The sequence of equally spaced values of the functions is to have corresponding contours by default Case 2: c0 ¼ c1 ¼ c3 ¼ c5 ¼ c6 ¼ 0, 209 aÀ2 ¼ aÀ1 ¼ a1 ¼ 0;    2 4c2 k1 Bk1 ỵ C k2 ỵ k3 ỵ x ; a0 ẳ Ak1    2 12c4 Bk1 ỵ C k2 ỵ k3 : a2 ẳ A 195 196 197 198 199 200 201 202 203 204 205 207 210 211 212 u21 nị ẳ 214 215 217 x Ak1 c2 < 0; c4 > 0; À u23 nị ẳ Ak1 c2 < 0; c4 > 0;    2 4c2 Bk1 ỵ C k2 ỵ k3 A A ! pffiffiffiffiffi 12c2 c4 exp ð2 c2 nÞ pffiffiffiffiffi ; c2 > 0; c4 – 0; ð1 À c2 c4 exp c2 nịị 16ị where; n ẳ k1 x þ k2 y þ k3 z þ xt: Case 3: c0 ¼ c1 ¼ c4 ¼ c5 ¼ c6 ¼ 0, aÀ2 ¼ aÀ1 ¼ 0; a1 ¼ À A     x ẳ k1 a0 A ỵ c2 Bk21 ỵ C k22 ỵ k23 : 12ị 3c2 Bk1 ỵC  k2 ỵ A k3  225 226 ; a2 ẳ 0; 17ị 228    p ỵ tan2 c2 x ; c2 < 0; ð18Þ ð19Þ    where; n ẳ k1 x ỵ k2 y ỵ k3 z þ xt; x ¼ Àk1 a0 A þ c2 Bk1 þ C k2 þ k3 pffiffiffiffiffiffiffiffiffi Á À sec2 ð Àc2 nÞ ; 2  : c22 Case 4: c1 ¼ c3 ¼ c5 ¼ c6 ¼ and c0 ¼ 4c4 , aÀ2 ¼ À 3c22 230 231 233 236 237 238 239    2 Bk1 ỵ C k2 ỵ k3 ; aÀ1 ¼ 0; Ac    x þ 4c2 k1 Bk21 þ C k22 þ k23 a0 ¼ À ; a1 ¼ 0; Ak1   2 12c4 Bk1 ỵ Ck2 ỵ Ck3 ; a2 ¼ À A 229 234    2   3c2 Bk1 ỵ C k2 ỵ k3  p u32 nị ẳ a0 ỵ c2 n ; À A c2 > 0; pffiffiffiffiffiffiffiffiffi Á À 3csc ð Àc2 nÞ ; 223 224    2 3c3 Bk1 ỵ C k2 ỵ k3 u31 nị ẳ a0 ỵ    2  x 4c2 Bk1 þ C k2 þ k3  pffiffiffiffiffiffiffiffiffi À 3csch c2 nị ; u22 nị ẳ A Ak1 ð14Þ c2 < 0; c4 > 0; x À  ð13Þ 218 220 A x Ak1 The following solitary wave soliton solutions of Eq (4) are constructed by substituting Eq (17) into Eq (6) as: The following solitary wave and soliton solutions of Eq (4) are constructed by substituting Eq (12) into Eq (6) as:    2 4c2 Bk1 ỵ C k2 ỵ k3 u24 nị ẳ 221    2 4c2 Bk1 ỵ C k2 ỵ k3 20ị ð15Þ 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 241 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx 242 aÀ2 ¼ À 244 a0 ¼ À    2 3c22 Bk1 ỵ C k2 ỵ k3 ; Ac   x ỵ 4c2 k1 Bk21 ỵ C k22 ỵ k23 249 250  a2 ẳ a1 ¼ a1 ¼ 0; a0 ¼ À   2 12c4 Bk1 ỵ Ck2 ỵ Ck3 a2 ẳ À : A  cot2 rffiffiffiffiffiffiffiffiffi  c2 À n  6c2  Ak   2 Bk1 ỵ C k2 ỵ k3 A r  c2 À n c2 < 0; c4 > 0; 24ị tan where; n ẳ k1 x ỵ k2 y ỵ k3 z ỵ xt: Similarly, one can find the soliton-like solutions of Eq (4) from Eqs (21) and (22)   Case 5: c2 ¼ c4 ¼ c5 ¼ c6 ¼ 0, 2 a0 ¼ x Ak1 ; a1 ẳ 3c3 Bk1 ỵ Ck2 ỵ Ck3 A ; 262 a2 ẳ 0: 263 From Eq (25), we construct the following new elliptic function solution [40] of Eq (4) as: 264 ð25Þ   2 p  x 3c3 Bk1 ỵ ck2 ỵ ck3 c3 n } u51 nị ẳ ; g ; g ; c3 > 0; A Ak1 ð26Þ 267 268 269 270 271 273 274 275 276 277 278 12c4  A  2 Bk1 ỵ C k2 ỵ k3  A x ẳ k1 a0 A ỵ 4c2 Bk1 ỵ C  where; n ẳ k1 x ỵ k2 y þ k3 z þ xt; }isWeierstrassellipticfunctionand g ¼ À 4cc31 ; g ¼ À 4cc30 : Case 6: c4 ¼ c5 ¼ c6 ¼ 0,    2 3c3 Bk1 ỵ C k2 ỵ k3 aÀ2 ¼ aÀ1 ¼ a2 ¼ 0; a1 ¼ À     A x ¼ Àk1 a0 A þ c2 Bk21 þ C k22 þ k23 ; ð27Þ From Eq (27), we can also construct the new exact Weierstrass elliptic function solutions [40–42] of Eq (4) Case 7: c1 ¼ c3 ¼ c5 ¼ c6 ¼ 0,    2 12c4 Bk1 ỵ C k2 þ k3 (i:) aÀ2 ¼ aÀ1 ¼ a1 ¼ 0;a2 ¼ À ; ð28Þ    A 2 ; x ẳ k1 a0 A ỵ 4c2 Bk1 þ C k2 þ k3 281 282 284 ; aÀ1 ẳ a1 ẳ 0; ; 2 k2 ỵ k3 285 286 ð30Þ  : 288 From Eqs (28)–(30), one can constructed the new Jacobi elliptic function solutions [45] of Eq (4) by choosing the value of mð0 m 1Þ according to the Table 289 The (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov (mKdVZK) Equation 292 as: 290 291 293 294 295 ut ỵ qu2 ux ỵ uxxx ỵ uxyy ỵ uxzz ẳ 0; A   x ỵ 4c2 k1 Bk21 ỵ C k22 ỵ k23 a2 ẳ 0; aÀ1 ¼ 0; 265    2 12c0 Bk1 ỵ C k2 ỵ k3 29ị Now, we consider the (3 + 1)-dimensional mkdv-zk equation is ð23Þ    2 6c2 Bk1 ỵ C k2 ỵ k3 ỵ 255 259 a2 ẳ c2 > 0; c4 > 0; À 260 aÀ2 ¼ À ; r  c2 u41 nị ẳ n cot A    x ỵ 4c2 k1 Bk21 þ C k22 þ k23 À Ak    2 r  6c2 Bk1 ỵ C k2 ỵ k3 c2 tan2 n ; A u42 nị ẳ 258 (iii): 22ị    2 6c2 Bk1 ỵ C k2 ỵ k3 253 256 Ak1  ;aÀ1 ¼ a1 ¼ a2 ¼ 0; A    2 ; a0 A þ 4c2 Bk1 þ C k2 þ k3 The following soliton-like solutions of Eq (4) are constructed by substituting Eq (20) into Eq (6) as: 252 257  ð21Þ x þ 4c2 k1 Bk21 þ C k22 þ k23 248 x ¼ Àk1 ; Ak1 245 247 (ii): aÀ2 ¼ À aÀ1 ¼ a1 ¼ a2 ¼ 0;     2 12c0 Bk1 ỵ C k2 þ k3 ð31Þ where q is a arbitrary constant and can be obtained as a model from modified KdV in case of weakly two dimensional variations of the modified KdV equation [11] naturally Consider the traveling wave solutions and transformation in Eqs (2) and (3), the (31) reduces into ODE as   xu0 ỵ qk1 u2 u0 ỵ k31 ỵ k1 k22 ỵ k1 k23 u000 ẳ 0: 32ị By using balancing principle on Eq (32) gets m ¼ and considering the solution of Eq (32) is as: a1 unị ẳ ỵ a0 ỵ a1 /: / 296 298 299 300 301 302 303 304 306 307 308 309 ð33Þ 311 Substituting Eq (33) into Eq (32) and setting the coefficients of 312 /i /ðiÞ to zero, yields a systems of algebraic equations in aÀ1 ; a0 ; a1 ; q; k1 ; k2 ; k3 and x The system of equations have the following solutions cases: Case 1: c0 ¼ c1 ¼ c5 ¼ c6 ¼ 0, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     u u u k2 ỵ k2 ỵ k2 u 6c4 k2 ỵ k2 þ k2 3 t c3 t ; a1 ẳ ặ ; a1 ẳ 0; a0 ẳ ặ q 2qc4   Á 2 k1 3c23 À 8c2 c4 k1 ỵ k2 ỵ k3 xẳ : 34ị 8c4 313 Substituting the values of Eq (34) into Eq (33) along with the solutions of Eq (3), the following solutions of Eq (31) in solitonlike and solitary wave form are obtained: 320 c2 > 0; D > 0; 325 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u p p ! u k2 ỵ k2 ỵ k2 t 8c4 c2 sechð c2 nÞ c3 ; ặ p ặ p u11 nị ẳ pffiffiffiffiffi q 8c4 D À c4 sechð c2 nÞ ð35Þ 315 316 317 319 321 322 323 326 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u p p ! u k2 ỵ k2 þ k2 t 8c4 c2 sechð c2 nị c3 ặ p ầ p u12 nị ẳ ; p q 8c4 D ỵ c4 sech c2 nị c2 > 0; D > 0; 314 ð36Þ 280 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 328 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx Table Jacobi elliptic functions S No 329 c4 c2 À1 À m m cnn snn; cdn ¼ dnn Àm2 À1 2m2 À À m2 À1 À m2 À m2 m2 À m2 dnn À m2 2m2 ncn ẳ cnnị1 m 2m Àm2 À1 À m2 À m2 snn scn ẳ cnn m2 m2 1ị 2m2 À 1 2Àm 1Àm 10 2m2 À 2 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u p p ! u k2 ỵ k2 ỵ k2 t c3 8c4 c2 cschð c2 nÞ ; u13 nị ẳ ặ p ặ p p q 8c4 ÀD À c4 cschð c2 nÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u p p ! u k2 ỵ k2 þ k2 t c3 8c4 c2 cschð c2 nị ; ặ p ầ p u14 nị ẳ p q 8c4 D ỵ c4 csch c2 nị ð37Þ 334 c2 > 0; D < 0; 335 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  u  pffiffiffi  pffiffiffiffiffi ! u k ỵ k2 ỵ k2 t c3 c2 2c2 ; n u15 nị ẳ ặ p ầ p ặ q c4 8c4 38ị c2 > 0; D ¼ 0; 338 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  p  p ! u k ỵ k2 ỵ k2 t c3 c2 2c p p ặ ầ u16 nị ẳ n ; ặ coth q c4 8c4 342 343 344 346 347 348 349 351 352 354 ð40Þ À 4c2 c4 ; q ð41Þ We constructed the following soliton-like solutions of Eq (31) from Eqs (3), (33) and (41) as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  u6c2 k ỵ k2 ỵ k2 t pffiffiffiffiffiffiffiffiffi csc ð Àc2 nÞ; c2 < 0; c4 > 0; 42ị v  u  u6c2 k ỵ k2 ỵ k2 p t csch c2 nị; c2 < 0; c4 > 0; u22 nị ẳ ặ 43ị u21 nị ẳ ầ q q u23 nị ẳ ặ 358 u24 nị ẳ ặ v  u  u6c2 k ỵ k2 ỵ k2 t q pffiffiffiffiffiffiffiffiffi sec ð Àc2 nÞ; c2 < 0; c4 > 0; vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi u uÀ6c4 k2 þ k2 þ k2 t q pffiffiffiffiffi 2c2 exp ð c2 nÞ pffiffiffiffiffi ; À c2 c4 exp c2 nị csn ẳ cnn snn a1 v   u u 3c2 k2 ỵ k2 ỵ k2 t ; ẳặ 2q c v   u u 6c4 k2 ỵ k2 ỵ k2 t ; a1 ẳ Ç À q vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 3c2 k2 þ k2 þ k2 t ; ¼Ỉ À 2q c vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 6c4 k2 ỵ k2 ỵ k2 t ; a1 ẳ ặ q a1 v   u u 3c2 k2 ỵ k2 ỵ k2 t ; ẳặ 2q c  v   u u 6c4 k2 ỵ k2 ỵ k2 t ; ¼ a0 ¼ 0; a1 ẳ ặ   x ẳ c2 k1 k21 ỵ k22 ỵ k23 : snn sdn ẳ dnn dsn ¼ dnn snn 362 363 a0 ¼ 0;   x ẳ 4c2 k1 k21 ỵ k22 ỵ k23 ; ð44Þ > 0; 360 c4 – 0; 361   2 where; n ¼ k1 x ỵ k2 y ỵ k3 z ỵ xt; x ẳ c2 k1 k1 ỵ k2 ỵ k3 : 45ị 46ị 365 366 a0 ẳ 0;   x ẳ 2c2 k1 k21 ỵ k22 ỵ k23 ; 47ị 368 369 a0 ¼ 0; a1 ¼ 0;  x ¼ c2 k1 k21 ỵ k22 ỵ k23 ; Case 2: c0 ¼ c1 ¼ c3 ¼ c5 ¼ c6 ¼ 0, 355 357 39ị c2 > 0; D ẳ 0; a1 ndn ẳ dnnị m2 m2 1ị a1 337 where; n ẳ k1 x ỵ k2 y ỵ k3 z ỵ xt; D ẳ k 3c2 8c c k2 ỵk2 ỵk2 x ẳ 8c4 Þ4ð Þ : nsn ¼ ðsnnÞÀ1 ; dcn ¼ dnn cnn c2 c2 > 0; D < 0; c23 cnn Case 3: c1 ¼ c3 ¼ c5 ¼ c6 ¼ 0; c4 > 0andc0 ¼ 4c24 332 341 /ðnÞ 331 340 c0 48ị 372 v   u u 6c4 k2 ỵ k2 ỵ k2 t ; a1 ẳ 0; a0 ẳ 0; a1 ẳ ặ q   x ẳ c2 k1 k21 ỵ k22 ỵ k23 : ð49Þ Substituting Eq (46) into Eq (33), the following solitary wave solutions of Eq (31) are obtained: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u r  u 3c2 k2 ỵ k2 ỵ k2  t c2 Ỉ cot u31 nị ẳ ặ n q v   u r  u 3c2 k2 ỵ k2 ỵ k2  t c2 Ỉ tan n ; Ç À q vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  rffiffiffiffiffiffiffiffiffi  u3c2 k ỵ k2 ỵ k2  t c2 ặ coth u32 nị ẳ ặ n q vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  rffiffiffiffiffiffiffiffiffi  u3c2 k ỵ k2 ỵ k2  t c2 ặ n ; ầ q where; n ẳ k1 x ỵ k2 y ỵ k3 z ỵ xt; x ¼ À4c2 k1  371 c2 > 0; ð50Þ 374 375 376 377 379 380 c2 < 0; 51ị  2 k1 ỵ k2 ỵ k3 : From Eqs (47)–(49), one can also construct the new soliton-like and solitay wave solutions of Eq (31) Case 4: c2 ¼ c4 ¼ c5 ¼ c6 ¼ 0, 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 382 383 384 385 386 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx Fig Traveling wave solutions of Eq (9) with different forms are plotted: (a) periodic bright solitary wave and (b) contour plot of u12 Fig Traveling wave solutions of Eq (13) with different forms are plotted: (a) periodic solitary wave and (b) contour plot of u21 at c2 ¼ À4; c4 ¼ 3; k1 ¼ 1; k2 ¼ 1:5; k3 ¼ 2; x ¼ 1:5; A ¼ 3; B ¼ 2; C ¼ 0:75; y ¼ 1; z ¼ 387 aÀ1 389 390 391 392 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 6c0 k2 ỵ k2 ỵ k2 t ; ẳặ a1 ẳ 0; q x¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 u c1 t k1 ỵ k2 ỵ k3 a0 ẳ Æ ; 2c0 q   2 3c21 k1 k1 ỵ k2 ỵ k3 8c0 : 52ị From Eq (52), we construct the following new Weierstrass elliptic function solution [40] of Eq (31) as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   v   u u 2 u u 6c0 k2 ỵ k2 ỵ k2 t c1 t k1 ỵ k2 ỵ k3 ặ u41 nị ẳ ặ 2c0 q q ; Â pffiffiffiffi c3 n } ; g2 ; g3 c3 > 0; 53ị where; n ẳ k1 x ỵ k2 y ỵ k3 z ỵ xt; 394 395 4c1 4c0 } is Weierstrass elliptic function and g ¼ À ; g3 ¼ À : c3 c3 397 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx Fig Traveling wave solutions of Eq (19) with different forms are plotted: (a) bright solitary wave and (b) contour plot of u32 at c2 ¼ 1; c3 ¼ 1; k1 ¼ 0:5; k2 ¼ 1:5; k3 ¼ 1; A ¼ 3; B ¼ 2; C ¼ 2; a0 ¼ 1; y ¼ 1; z ¼ Fig Traveling wave solutions of Eq (26) with different forms are plotted: (a) periodic traveling wave and (b) contour plot at c0 ¼ 0; c1 ¼ À1; c2 ¼ 1; k1 ¼ 0:75; k2 ¼ 1:5; k3 ¼ 0:5; x ¼ 1; A ¼ 3; B ¼ 2; C ¼ 2; y ¼ 1; z ¼ 398 399 Case 5: c5 ¼ c6 ¼ 0, a1 v   u u 6c0 k2 ỵ k2 ỵ k2 t ; ẳầ q À 401 a1 ¼ 0; x¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 u c1 t k1 ỵ k2 ỵ k3 a0 ẳ ầ ; 2c0 q  Á  2 3c21 À 8c0 c2 k1 k1 ỵ k2 ỵ k3 8c0 ; 54ị vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 u c3 t k1 ỵ k2 ỵ k3 ; a1 ẳ 0; a0 ẳ ầ 2qc4 v   u u 6c4 k2 ỵ k2 ỵ k2 t ; a1 ẳ ầ q   Á 2 3c3 À 8c2 c4 k1 k1 ỵ k2 ỵ k3 xẳ : 8c4 402 55ị 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 404 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx Fig Traveling wave solutions of Eq (35) with different forms are plotted: (a) bright soliton and (b) contour plot of u11 at c2 ¼ 1; c3 ¼ 2; c4 ¼ À2; k1 ¼ 0:5; k2 ¼ 1:5; k3 ¼ 1; q ¼ 2; y ¼ 1; z ¼ Fig Traveling wave solutions of Eq (44) with different forms are drawn: (a) periodic dark soliton and (b) contour plot of u23 at c2 ¼ À2; c4 ¼ 1; k1 ¼ 0:5; k2 ¼ 1:5; k3 ¼ 0:75; q ¼ 2; y ¼ 1; z ¼ 405 Case 6: c4 ¼ c5 ¼ c6 ¼ 0, 406 aÀ1 408 409 410 v   u u 6c0 k2 ỵ k2 ỵ k2 t ; ẳặ a1 ẳ 0; vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 u c1 t k1 ỵ k2 ỵ k3 a0 ẳ ặ ; À q 2qc0  À Á  2 8c0 c2 3c21 k1 k1 ỵ k2 ỵ k3 xẳ : 56ị 8c0 We can also construct the new Weierstrass elliptic function solutions [40–44] of Eq (31) from Eqs (54)–(56) of Cases and Case 7: c1 ¼ c3 ¼ c5 ¼ c6 ¼ 0, aÀ1 v   u u 6c0 k2 ỵ k2 ỵ k2 t ; ẳặ q 411 412 a0 ¼ 0; vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 6c4 k2 ỵ k2 ỵ k2 t ; a1 ẳ ầ q   p x ẳ c2 ỵ c0 c4 ịk1 k21 ỵ k22 ỵ k23 ; ð57Þ 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 414 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx Fig Traveling wave solutions of Eq (50) with different forms are drawn: (a) periodic dark soliton and (b) contour plot of u23 at c2 ¼ 0:5; c4 ¼ 1; k1 ¼ 0:5; k2 ¼ 1:5; k3 ¼ 0:75; q ¼ 2; y ¼ 1; z ¼ Fig Traveling wave solutions of Eq (53) with different forms are plotted: (a) periodic bright and dark soliton and (b) contour plot at c0 ¼ 0:1; c1 ¼ À1; c3 ¼ 1; k1 ¼ 0:75; k2 ¼ 1:5; k3 ¼ 0:5; q ¼ 2; y ¼ 1; z ¼ 415 417 418 420 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 6c0 k2 þ k2 þ k2 t aÀ1 ¼ Æ À ; a0 ¼ 0; q   pffiffiffiffiffiffiffiffiffi x ẳ c0 c4 c2 ịk1 k21 ỵ k22 ỵ k23 ; v   u u 6c0 k2 ỵ k2 ỵ k2 t ; a1 ẳ ặ q   x ẳ c2 k1 k21 ỵ k22 ỵ k23 ; v   u u 6c4 k2 ỵ k2 ỵ k2 t a1 ẳ ặ ; q 58ị a1 ¼ 0; a0 ¼ 0; 421 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 6c4 k2 ỵ k2 ỵ k2 t ; a1 ẳ ầ   x ẳ c2 k1 k21 ỵ k22 ỵ k23 : q 60ị 423 a0 ẳ 0; a1 ẳ 0; 59ị From Eqs (57)–(60), one can obtained the new Jacobi elliptic function solutions [45] of Eq (31) by choosing the value of mð0 m 1Þ according to the Table 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 424 425 426 RINP 565 No of Pages 11, Model 5G 20 February 2017 10 427 428 430 431 432 433 435 D Lu et al / Results in Physics xxx (2017) xxx–xxx Case 8: c0 ¼ c1 ¼ c2 ¼ c5 ¼ c6 ¼ 0, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 u c3 t k1 þ k2 þ k3 À ; aÀ1 ¼ 0; a0 ¼ Ỉ 2qc4 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     u 2 u 6c4 k2 ỵ k2 ỵ k2 3c23 k1 k1 ỵ k2 ỵ k3 t ; xẳ : a1 ẳ ặ 8c4 q ð61Þ We obtained the new solitary wave solution of Eq (31) by substituting Eq (61) into Eq (33) is as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u 2 u c3 t k1 ỵ k2 ỵ k3 u81 nị ẳ Æ 2q c vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 6c4 k ỵ k2 ỵ k2 t Ỉ À q ! 4c3 c23 n2 À 4c4 ; 3c23 k1 k21 ỵk22 ỵk23 ị c4 > 0; 62ị 436 where; n ẳ k1 x ỵ k2 y þ k3 z þ xt; x ¼ 437 Conclusion 438 451 In this paper, some general new exact solutions in the form of soliton, solitary wave, elliptic function and Weiertrass elliptic function solutions of three-dimensional EZK and (3 + 1)-dimensional modified KdV-ZK equations are constructed by utilizing modified extended Direct algebraic method These solitions and other solutions in which many are new and derived in explicit form, have many applications and useful in different areas of physics, engineering and other fields of applied sciences These general solutions can provide a useful help for researchers to study and understand the physical interpretation of system This method has several advantages such as the calculations are simple and straightforward, gives more general solutions then other existing methods, the reduction in the size of computational work and consistency gives its wider applicability (Figs 2–9) 452 References 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 [1] Ablowitz MJ, Clarkson PA Solitons, nonlinear evolution equations and inverse scattering New York: Cambridge Univ Press; 1991 [2] Shukla PK, Mamun AA Introduction to dusty plasma physics Bristol, U K.: Institute of Physics Publishing; 2002 [3] Korteweg DJ, de Vries G On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves Philos Mag 1895;39:422–43 [4] Kochanov MB, Kudryashov NA, Sinel’shchikov DI Non-linear waves on shallow water under an ice cover Higher order expansions J Appl Math Mech 2013;77:25–32 [5] El-Tantawy SA, Moslem WM Nonlinear structures of the Korteweg-de Vries and modified Korteweg-de Vries equations in non-maxwellian electronpositron-ion plasma: solitons collision and rouge waves Phys Plasmas 2014;21:052112 [6] Kadomtsev BB, Petviashvili VI On the stability of solitary waves in weakly dispersing media Sov Phys Dokl 1970;15:539–41 [7] Liu ZM, Duan WS, He GJ Efeects of dust size distribution on dust acoustic waves in magnetized two-ion-temperature dusty plasmas Phys Plasmas 2008;15:083702 [8] Seadawy AR Stability analysis for two-dimensional ion-acoustic waves in quantum plasmas Phys Plasmas 2014;21:052107 [9] Seadawy AR Nonlinear wave solutions of the three-dimensional ZakharovKuznetsov-Burgers equation in dusty plasma Physica A 2015;439:124–31 [10] Seadawy AR, Lu Dianchen Ion acoustic solitary wave solutions of threedimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma Result Phys 2016;6:560–93 [11] Demontis F Exact solutions of the modified Korteweg-de Vries equation Theoret Math Phys 2011;168(1):886–97 [12] Islam MH, Khan K, Akbar MA, Salam MA Exact traveling wave solutions of modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation Springer Plus 2014;3(105):1–9 [13] Naher H, Abdullah FA, Akbar MA Generalized and improved (G’/G)-expansion method for (3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation Plos One 2013;8(5):1–7 439 440 441 442 443 444 445 446 447 448 449 450 8c4 : [14] Naher H, Abdullah FA, Akbar MA New traveling wave solution of higher dimensional nonlinear partial differential equation by the Exp-function method J Appl Math 2012;2012:1–14 Article ID 575387 [15] Zhang BG, Liu ZR, Xiao Q New exact solitay wave and multiple soliton solutions of quantum Zakharov-Kuznetsov equation Appl Math Comput 2010;217:392–402 [16] Wang GW, Xu TZ, Johnson S, Biswas A Solitons and Lie group analysis to an extended quantum Zakharov-Kuznetsov equation Asterophys Space Sci 2014;349:317–27 [17] Pakzad HR Soliton energy of the Kadomtsev-Petviashvili equation in warm dusty plusma with variable dust charge, two-temperature ions, and nonthermal electrons Asterophys Space Sci 2010;326:69–75 [18] Seadawy AR, El-Rashidy K Travelling wave solutions for some coupled nonlinear evolution equations by using the direct algebraic method Math Comput Modelling 2013;57:1371–9 [19] Yu X, Gao YT, Sun ZY, Liu Y N-soliton solutions, bucklund transformation and lax pair for a generalized variable coefficient fifth-order Korteweg de Vries equation Phys Scr 2010;81:045402 [20] Hirota R Exact Solutions of the Korteweg-de Vries equation for multiple collisions of solitons Phys Rev Lett 1971;27:1192–4 [21] Alam MN, Hafez MG, Akbar MA, Roshid HO Exact travelling wave solutions to the (3+1)-dimensional mKdV-ZK and the (2+1)-dimensional Burgers equations via exp(À/ðgÞ)-expension method Alexa Eng J 2015;54:635–44 [22] Abdou MA The extended tanh-method and its applications for solving nonlinear physical models Appl Math Comput 2007;190:988–96 [23] Guiqiong Xu, Li Zhibin Exact travelling wave solutions of the whitham-broerkaup and broer-kaup-kupershmidt equations Chaos Solitons Fractals 2005;24:549–66 [24] Wen XY Construction of new exact rational form non-travelling wave solutions to the (2+1)-dimensional generalized Broer-Kaup system Appl Math Comput 2010;217:1367–75 [25] Zenga Xin, Wang Deng-Shan A generalized extended rational expansion method and its application to (1+1)-dimensional dispersive long wave equation Appl Math Comput 2009;212:296–304 [26] Yao Yuqin Abundant families of new traveling wave solutions for the coupled Drinfl’d-sokolov-wilson equation Chaos Solitons Fractals 2005;24:301–7 [27] Peng YZ Exact solutions for some nonlinear partial differential equations Phys Lett A 2003;314:401–8 [28] Seadawy AR, El-Rashidy K Traveling wave solutions for some coupled nonlinear evolution equations Math Comput Modelling 2013;57:1371–9 [29] Ugurlua Yavuz, Kaya Dogan, Inanb Ibrahim E Comparison of three semianalytical methods for solving (1 + 1)-dimensional dispersive long wave equations Comput Math Appl 2011;61:1278–90 [30] Dinarvand S, Khosravi S, Doosthoseini A, Rashidi MM The homotopy analysis method for solving the Sawada-Kotera and Laxs fifth-order KdV equations Adv Theor Appl Mech 2008;1:327–35 [31] Biazar J, Badpeima F, Azimi F Application of the homotopy perturbation method to Zakharov-Kuznetsov equations Comput Math Appl 2009;58:2391–4 [32] Ravi ASV, Kanth K Aruna, Differential transform method for solving linear and non-linear systems of partial differential equations Phys Lett A 2008;372:6896–8 [33] Rashidi MM, Erfani E Traveling wave solutions of WBK shallow water equations by differential transform method Adv Theor Appl Mech 2010;3:263–71 [34] Keskin Y, Oturanc G Reduced differential transform method for partial differential equations Int J Nonlinear Sci Numer Simul 2009;10:741–9 [35] Abazari R, Abazari M Numerical simulation of generalized HirotaSatsuma coupled KdV equation by RDTM and comparison with DTM Commun Nonlinear Sci Numer Simul 2012;17:619–29 [36] Helal MA, Seadawy AR Variational method for the derivative nonlinear Schrodinger equation with computational applications Phys Scr 2009;80 Article ID 035004 [37] Helal MA, Seadawy AR Exact soliton solutions of a D-dimensional nonlinear Schrodinger equation with damping and diffusive terms Z Angew Math Phys 2011;62:839–47 [38] Seadawy AR New exact solutions for the KdV equation with higher order nonlinearity by using the variational method Comput Math Appl 2011;62:3741–55 [39] Helal MA, Seadawy AR Benjamin-Feir instability in nonlinear dispersive waves Comput Math Appl 2012;64:3557–68 [40] Nickel J, Serov VS, Schurmann HW Some elliptic travelling wave solution to the Novikkov-Veselov equation Proc Prog Electromagnet Res Symp 2006;61:323–31 [41] Schurmann HW Travelling-wave solutions of the cubic-quintic nonlinear schrodinger equation Phys Rev E 1996;54:4312–20 [42] Nickel J Elliptic solutions to a generalized BBM equation Phys Lett A 2007;364:221–6 [43] Schurmann HW, Serov VS, Nickel J Superposition in nonlinear wave and evolution equations Int J Theor Phys 2006;45:1057–73 [44] Nickel J, Schurmann HW Comments on Exact solutions of the derivative nonlinear Schrodinger equation for a nonlinear transmission line Phys Rev E 2007;75:038601 [45] Yomba E Jacobi elliptic function solutions of the generalized ZakharovKuznetsov equation with nonlinear dispersion and t-dependent coefficients Phys Lett A 2010;374:1611–5 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 RINP 565 No of Pages 11, Model 5G 20 February 2017 D Lu et al / Results in Physics xxx (2017) xxx–xxx 573 574 575 576 577 578 579 580 581 582 583 584 [46] Seadawy AR, Dianchen Lu Ion acoustic solitary wave solutions of threedimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma Results Phys 2016;6:590–3 [47] Aly R Seadawy, Travelling wave solutions of a weakly nonlinear twodimensional higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves Eur Phys J Plus 2017;132:29 [48] Arshad M, Seadawy Aly, Dianchen Lu, Wang Jun Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations Results Phys 2016;6:1136–45 [49] Selima Ehab S, Seadawy Aly R, Yao Xiaohua The nonlinear dispersive DaveyStewartson system for surface waves propagation in shallow water and its stability Eur Phys J Plus 2016;131:425 11 [50] Kamruzzaman Khan M, Akbar Ali Solving unsteady Korteweg de-Vries equation and its two alternatives Math Methods Appl Sci 2016;39 (10):2752–60 [51] Khan Kamruzzaman, Ali Akbar M, Rashidi MM, Zamanpour Isa Exact traveling wave solutions of an autonomous system via the enhanced (G /G)- expansion method Waves Random Complex Media 2015;25(3) [52] Shafiqul Islam Md, Khan Kamruzzaman, Ali Akbar M An analytical method for finding exact solutions of modified Korteweg-de Vries equation Results Phys 2015;5:131–5 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 KdV-Zakharov-Kuznetsov equations and their applications Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.002 585 586 587 588 589 590 591 592 593 594 ... method are as: 12 7 Step 1: Consider that Eq (1) has the following solution as: 12 0 12 1 12 2 1 23 12 4 12 5 12 8 uðx; y; z; tị ẳ unị ẳ 13 0 13 1 13 2 13 4 13 5 13 6 13 7 13 8 13 9 14 0 14 1 14 2 1 43 14 4 m X aj /... No of Pages 11 , Model 5G 20 February 2 017 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 10 0 10 1 10 2 1 03 10 4 10 5 10 6 10 7 10 8 10 9 11 0 11 1 11 2 1 13 D Lu et al / Results in Physics xxx (2 017 )... 30 2 30 3 30 4 30 6 30 7 30 8 30 9 ? ?33 Þ 31 1 Substituting Eq (33 ) into Eq (32 ) and setting the coefficients of 31 2 /i /ðiÞ to zero, yields a systems of algebraic equations in a? ?1 ; a0 ; a1 ; q; k1 ;