Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations Academic research paper on "Mathematics"

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Travelling wave solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations

M. Arshad a, Aly Seadawyb,c'*, Dianchen Lua, Jun Wanga

a Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China b Faculty of Science, Taibah University, Al-Ula, Saudi Arabia c Mathematics Department, Faculty of Science, Beni-Suef University, Egypt

ARTICLE INFO

ABSTRACT

Article history:

Received 22 October 2016

Received in revised form 9 November 2016

Accepted 19 November 2016

Available online 25 November 2016

Keywords:

Traveling wave solutions Elliptic solutions

Generalized coupled Zakharov-Kuznetsov equation

Dispersive long wave equation Modified extended direct algebraic method

In this manuscript, we constructed different form of new exact solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations by utilizing the modified extended direct algebraic method. New exact traveling wave solutions for both equations are obtained in the form of soliton, periodic, bright, and dark solitary wave solutions. There are many applications of the present traveling wave solutions in physics and furthermore, a wide class of coupled nonlinear evolution equations can be solved by this method.

© 2016 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/4XI/).

Introduction

Nonlinear partial differential equations (PDEs) are involved nonlinear complex physical phenomena, which play a vital role in physical sciences. The generalized coupled Zakharov-Kuznetsov equations, dispersive long wave equation and generalized KdV equations are 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, plasma physics, hydrodynamics and many more [1-5]. The generalized Zakharov-Kuznetsov and dispersive long wave equations appear in many areas of applied sciences and engineering. Exact travelling wave solutions of nonlinear Coupled PDEs can perform a significant part in the understanding of these physical phenomena because various phenomena in nature such as vibration and self reinforcing solitary waves are described by them. So, the investigation of traveling wave solution turned into a main vital and major task, which plays an important role in nonlinear science. In last some decades, both mathematician and physicist have made much important effort in this area and demonstrated various useful techniques, such as the

* Corresponding author at: Faculty of Science, Taibah University, Al-Ula, Saudi Arabia.

E-mail address: Aly742001@yahoo.com (A. Seadawy).

Hirota's bilinear scheme [6], inverse scattering scheme [1], Back-lund transform method [6], homogeneous balance scheme [7], Painlev expansion [8], the mapping method and extended mapping method [9,10], Exp-function method [11], rational expansion method [12] and many more [13-15].

The Zakharov-Kuznetsov equation was initially derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions [16] and the dispersive long wave equation can be traced back to the works of Kaup [17], Broer [18], Kupershmidt [19], Martinez [20], etc and (1 + 1)-dimensional dispersive long wave equation is called the classical Boussinesq equation. A good understanding about the solutions of these PDEs is very useful for civil engineers, coastal, etc to concern the nonlinear water wave model in a coastal design, harbor and so on. Thus, searching more types of travelling wave solutions of these equations are of fundamental curiosity in uid dynamics. Many articles have been devoted to find the numerical and exact solutions of these equations. Different forms of exact travelling wave solutions of the ZK equation, the modified ZK equation and the generalized forms of these equations have obtained by using different methods [12,21-27]. Many numerical schemes such as Adomian decomposition method [28,29], homotopy perturbation method [30], homotopy analysis method [28,31], differential transform method, reduced differential method [32-36] and many other have been applied to obtain the numerical solutions in different

http://dx.doi.org/10.1016/j.rinp.2016.11.043 2211-3797/© 2016 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/).

form of ZK and dispersive long wave equations. The study about solutions, structures, interaction and further properties of soliton abstracts give much more attention and various meaningful results are successfully obtained [37-44].

In this paper, we further extend the ansatz equation in more general form in the modified extended direct algebraic method to construct travelling wave solutions of generalized coupled ZK equations and dispersive long wave equation. As a result, some new and more general exact travelling wave solutions are obtained.

This article is ordered as follows. An introduction is given in S ection "Introduction". In Section ''The generalized couple ZK equations", we construct some new exact soliton and elliptic solutions in general form of the generalized coupled ZK equations. New generalized form of travelling wave and elliptic function solutions of the dispersive long wave equation are got in Section ''The dispersive long wave equations". Lastly, the conclusion is given in Section ''Conclusion".

The generalized couple ZK equations

The Couple ZK equations in generalized form are as:

Ut + Uxxx + UyyX - 6UUX - Vx = 0,

Vt + SVxxx + kVyyx + gVx - 6lVVx - aUx = 0.

Consider the traveling wave solution of Eq. (1)

U(x,y, t) = U({) = £ V(x,y, t) = V({) = J2(2)

i=-m j=-n

/'(£) = \l Co + Ci/ + C2/2 + C3/3 + C4/4 + C5/5 + c6/6,

n = kix + fey + xt (3)

where a,, bj, c0, c1, c2, c3, c4, c5, c6, k1, k2 and m are arbitrary constants and m, n are positive integers, which are determined later. The parameters m and n are usually obtained by balancing the highest order linear term with the nonlinear terms of highest order in the resulting equations. Substituting Eq. (2) into Eq. (1), then integrating w.r.t. n and taking integration constant to zero, we get

(fc3 + k1k]) U" + (x - 3k1U)U - k2V = 0,

(sk] + kk1 k2) V" + (x + gk2 - 3ik2 V)V - ak1 U = 0.

Fig. 1. Travelling wave solutions of Eq. (11) with various different forms are plotted: (a) Periodic solitary waves and (b) contour plot of u2, (c) solitary waves and (d) contour plot of v2.

By using balancing principle, we suppose the solution of Eq. (4) are as:

Ufô-^ + ^ + f + ttaf + aa + at/ + a2/2 + a3/ + a4 /4 , V (f)-^ + + ¥ + ¥ + b0 + bi/ + b2/2 + b3/3 + b4/4.

Substituting Eq. (5) into Eq. (4) setting the coefficients of to zero yields a systems of algebraic equations in a_4 ,a_3,a_2,

a_1, a0 , a1, a2 , a3 , a4 , b_4 , b_3, b_2, b_1, b0 , b1 ,b2, b3 , b4 , k ,k2, x ,a ,1 ,g ,1

and d. The systems of algebraic equations possesses the following solutions cases:

Case 1. C1 = C3=5 = 0 , C0 = = and C6 = Jk

(i) a_4 = a_3 = 0 , a_2 = 16c22(7kc4+k2), a_1 = 0 , a0 = 4c^k\ + k^

a1 = a2 = a3 = a4 = 0 , b_4 = b_3 = 0 , b_2 =

16c2k2(dk2+1k1 )

b_1 = 0 , b0 = 4c2k2(912 +1k1) , b1 = b2 = b3 = b4 = 0 , g = AC2X2k\-ktàk-i 1 _ (dk2 + 1k1 _ k1)

d ak1l(d2 +k2) ak1i k1(dk1_1k2) 1 + 12(dk2+1k1) + i"2 + (kj+k^)i '

x_ 3k^(dk2 +1k1)_4c2k1(k;+k2)2i X = 3(k?+k2)i '

(ii) a_4 = a_3 = 0 , a_2 = , a_1 = 0 , a0 = |^ (k? + k^)

a1 = a2 = a3 = a4 = 0 , b_4 = b_3 = 0 , b_2 =

16c2 k2 ( dk2+1k1 )

b_1 = 0 , b0 =|c2k2 (912+1k1), b1 = b2 = b3 = b4 = 0 ,

_ _ 4c212k1+3adk1i 4c2k2(dk2+1k1 _k1)

d ak1l(d2 +k2) ak2i k1 ( dk1_1k2) 1 + 12(dk2+1k1) + kk2 + (k1+k2)i '

_ 4c2k1 ( k2 +k2)21+3k2(dk2+1k1)

(iii) a_4 = a_3 = 0 ' a_2

a_ 1=0 ,

(iv) a_4 = a_3 = 0, a_2 =,

a_1 = 0 , a0 = fc2( k1 + k^ a1 = a2 = a3 = a4 = 0 ,

b4 b3 0 b2

16c2k2 (dk2+1k1) 27c4 1 ,

b_1 = 0 , b0=, b1 = b2 = b3 = b4 = 0 ,

k2 (dk2+1k1 ) ( 3k2 (k2 (d+g1)+gk21+1k2k1 ) _4c2 (k2+k2 )1 ( _dk|_ (k_1)k2k1 +kf) )

3k1 (k2+k2) 12

3k2 (dk2+1k1)_4c2k1 (k2+k2) 1

X = 3 (k?+k2)1 '

If Case 1(i), Eq. (6) is utilized, then following form of new exact solutions are obtained of Eq. (1).

u (x y t)= 4C2(k?+k2) 2C2(kj+k2) (3+tanh2 (ey^f)) ' 9 9tanh2( e\/_3f) '

v 1 (x,y,t)=4«+ik1)_ 2c2"2№ +kk1)(3+tanljjj(, h2 < 0 , h4 > 0.

91 9^tanh2( ey7 f)

u fx V t) = 4c2(k2+k2) 2c2(k2+k2) (3-tan2 (ey^)) ^^ 9 9tan2 (ey^n) '

V2 (x y t)=4c2k2 (dk2+1k1) + 2c2 k2 (dk2 +1k1^3_tan2 ( ev71^2;) ' 91 91tan2 (^v/c2-i)

h2 > 0 , h4 < 0. (11)

u !x V t)= 4c2(k1+k2) M^) (3+coth2 ;))

3 ( , 9 9coth2( ey^i)

v3(x y t)=4c2k2(dk2+1k1) 2c2 k2 (dk2 +1k1 ) (3+coth2 (y

91coth2 ( ey_c2n)

h2 < 0 h4 > 0.

U4 (X, y ,t)=4c2(k;2+k2 ) + 2c2(k1+k2)(:

3_cot2( ey Jj-n)

9cot2 ( eyfn)

k1 (k1+k2) 2c2(k1+ki) 0 a0 = 9k51(dk2+ Ak,)3 +-3-, a1 = a2 = 0

b 16c2k2(dk2+Ak1) ^ = (

27c41 ' " 9kJ (dk2+1k1)

a3 = a4 = b_4 = b_3 = b_1 = b1 = b2 = b3 = b4 = 0 ,

+3ak1k2 ( k2 +k2)(dk2+AkO 1-,

_ 2k1P1

k2 ( dk2+1k1)

g = where

3k51 ( dk2+1k1)3 ( kî+k2)1 ' 3kj (dk2+1k1)^ akn2 ( k2+kj)2_kj (dk2+1k1)^ +2 ( dk2 +1k | k! +k1 +k 2 k!

31=2 (kî +k2 ) ( dk2+1k1) ,

P1 ^ k2 ( k1+k2) 12 ( dk2 + 1k1 k2 + k2) ( dk2+Ak1 ) _ ;

V4 (X,y ,t) =4c2"2 ( dk2+1k1) + 2c2 k2 (dk2 +1k1) (3_cot2(ey/Cjn) ^ 91 9^cot2 (ey^-n)

h2 > 0 h4 < 0.

where, f = Z^x + k2y + mt , m ^ 2 3(fe2+fe2l)'i '

Fig. 1(a) and (c) signify the evolution of the periodic bright and bright solitary wave solutions of Eq. (11) of the generalized coupled ZK Eq. (1), with c2 = _0.5 , c4 = 0.5 , k1 = 1, k2 = 0.5 , e = _1,1 = 2 , d = 1.5 , k = 1 and y = 1. A contour plots Figs. 1(b) and (d) are a collection of level curves drawn on different set of intervals. The Mathematica command Contour Plot draws contour plots of functions of two variables.

Similarly, we obtained the new more exact solutions in soliton-like from other sub-cases of case 1.

3k7 ( dk2 +1k1 )

6c2"5 ( dk2 )

Case 2. c0 = ci = c5=6 = 0, a—4 = a—3 = a—2 = a—1 = a0 = 0 , a1 = c3 ^k1 + k^, a2 = 2c4 k2 + k^, a3 = a4 = 0 ,

b-4 = b-3 = b-2 = b-1 = b0 = 0 , b1 =3c3fê-^M^2

-4 = b—3 = b—2 &2 = b3 = b4 = 0 ,

(3c3-16c2c4)k1 (k2+k2) , _dk2 „ ___32ac4k2_

X 4c4 k1' 1 27(c3-4c2 c3c4)2k1(k2+k2)3'

(9c4-84c2 c4c2+192c2 cj) k1 (k2+k2 ) 2 -16acjk2

12c4(4c2c4-c3)k2 (k2+k2)

Substituting Eq. (14) into Eq. (5), we have obtained the following soliton-like solutions of Eq. (1) are as:

U1 (x , y , t) - 2c2^echffirn)+2* ( ffi+$ (^D-^hpi

v 1(x.y ; t) ^ 3c2c3k12(ct4^-(:^-)c4k^+hk(;i);2se>c)h(pcrn> ; <2 > 0, a > 0.

v2(xy ;t)--3c2c3k12(e433k-:(:2ffi-)c4s^+hk(2p?2rse>c)h(pcirn>; > 0 .a > 0. u3(x,y, t) - ^tSffi2'+2* ffi+$) ffiffi2,

±3c2c3k^c3—4c2c4)(k1+k2) csc^pm)

v3(x ,y ,t)- 2 з2::k:(ffic:c;ch(ln>) , c2 >0 a <

U4(x ,y ,t) - (1 ± tanh (pS n)) + 2c2(kc2+k2) (1 ± tanh (pS f))2,

V4(x , y ,t)--3c2cзkl(c2-4c2c4)(:t+g)2(1;tanh^, C2 > 0 ,a - 0.

U5(x ,y ,t) - -i^ciil+i) (1 ±coth n)) + (1 ±coth (pSn))2 ,

v5(x, y ,t)--3clcзkl(c3-4clc4)(k|t+kk|)l(l;co'h(ffi)), C2 > 0 ,a - 0.

where, f - k1x + k2y + xt, x - (3cз3-16c244c>k1(k1+k2).

Fig. 2(a) and (c) signify the evolution of the bright solitary wave solutions of Eq. (15) of the generalized coupled ZK Eq. (1), with c2 - 0.5 , c3 - 2, c4 - 1, k1 - 1, k2 - 1 and y - 1. A contour plots Figs. 2(b) and (d) are a collection of level curves drawn on some intervals.

Case 3: c0 - c1 - c3 - c5-6 - 0,

(i) a-4 - a-3 - a-2 - a-1 - 0 , at, - 4c^k1 + , a1 - 0 , a2 - 2c4 ^k2 + k^, a3 - a4 - 0 ,

b-4 - b-3 - b-2 - b-1 - 0 , be - ^ ,

b - b3 - b4 - 0, x - ^f^1^k2)2,

2c4(k2+k2 ) ¡CC4—2b2C 2b2c4k2 (k2+k2)

, b2;M—2c4dk2 g_ 4c4k1 (k1+k2 ) (ac4—^^H^ (4c2 (ki+kl ) l—1 )

k - 2c4k1k2 , g - " - - ;

(ii) a—4 - a—3 - a—2 - a—1 - a0 - a1 - 0 , a2 - 2c^k2 + k^,

a3 - a4 - 0 , b—4 - b—3 - b—2 - b—1 - b0 - b1 - b3 - b4 - 0 ,

- 2c4(k1+k2) ' _ 4c4k1 (kj1+k2 )2 («4+2^ ç, >—b2 k;(4c, (k 1+k2 ) 1 ) b2|—2c4dk;

2b|C4k|(k|+k|)

2c4k1 k2 •

Substituting Eq. (20) into Eq. (5), we have obtained the following solutions of Eq. (1) are as:

U1 (x,y , t) - 424+2i — 2d (k2 + k2)csc2(pffin, v 1 (x,y 't)-|||C| — ^csc2^), c2 < 0 ,Q > 0.

U2(x,y , t) - 424+21 — 2c^k1 + k2)csch2(V^f) , V2(x ,y , t) - l||C| — ^csch2(Vcffia c2 < 0 ,c4 > 0 ,

U3(x ,y 't)-:cl(k^+k|)+ ^W+ffiffi ' 3 (dC4e(W>—n)2 '

V3 (x , y,t) + --2-2--2 •

3c4 (c2c4e(W)—n)2

where, f - k1x+k2y+xt , x - b2k2+::c;;:i2|+k2)2.

Similarly, we obtained the new exact solutions of Eq. (1) by using Eq. (21).

Case 4: c1 - c3 - c5-6 - 0,

(i) a—4 - a—3 - a—2 - a—1 - 0 , a„ - ^X^V^F3^ , a1 - 0 , a2 - 2c^k2 + k2^,

a3 - a4 - 0 b—4 - b—3 - b—2 - b—1 - 0

be - b2C2V'|(i—^ b1 - b3 - b4 - 0,

k1 (ki+k|)l(8b|C^ c2—3C0 c4+4acj)— b2k2 (4A/c2—3^4^; +k2)|+1) g - 2b2C4k2 (k2+k2) ,

, b2|—2c4dk2 x_ b2k2—8k1 c^c2 — 3C0C4(k2+k2) k - 2C4k1k2 , x - 2C4(k|+k|) '

.... _ 2(^2) (c2+V ffi — 33 C0 C4)

(ii) a—4 - a—3 - a—2 - a—1 - 0 , a0 - ——2A 3 2-'-,

a1 - 0 , a2 - 2c^^ + k2^,

a3 - a4 - 0 b—4 - b—3 - b—2 - b—1 - 0 be -b2(C2>, b1 - b3 - b4 - 0,

4c4k1 (k2 ■ k2 j2(ac4 8b^/c2 3c0c4 ) ■ b2k2(4^/c2—3c0c4(k2 +k2)|—1)

- 2b2C4 k2 (k2+k2 ) ,

_ b2|—2c4dk2 b2k2+8k1C^ c2—3CoC4(kj+k|)l ;

:C4dk2 2C4k1k2 ,

2c4(k2+k2)

(iii) a_4 = a_3 = a_2 = a_1 = 0 ,

_ 4bjcîc4k1kj (kj+kj)212_bjk512+2p2;c4k1 ( kj+kj)p

a° = Gbfc^ ,

ai = 0 , a2 = 2c4 ^k1 + k2), a3 = a4 = b_4 = b_3

= b_2 = b_1 = 0 , b1 = b3 = b4 =

2b2 k2 ( k2 +k2 ) 1 (2ac2 k1 +k2 ) _b2 c2 k2 1)+^

_4p^c4k1 (kj+kj)p_b4kj12 x = ,

_ _4«b2c2k1 k2 (k1 +k2)212_2P^p ( b;k; 1+2c4k1 (k1 +k2))+62fci 12

g = ■

bc = _

b21_2c4dk22

2c4k1k2 ,

P = (k2 + k2) a + 6^0^k2 + k^ 1) _ jbjCj(k1 + k^ 1) ;

2k7 ( ¿"2+J"1)4 \

(iii) a_4 = a_3 = a_2 = a_1 = 0 a0 = 6 I k21(S+S + + "O I,

-2<-3i + ,i2 1

c2"5 ( ik2+ik1)3

a2 = a3 = a4 = 0, b0 = -

-2ak1 ( k1+k2)k2 (dk2+Ak1)-

( k1+k|

6k2 ( dk2+Ak1)2

b_4 = b_3 = b_2 = b_1 = 0 , b1 = c3"2 y*1> ,

U U U A k1P2--k1^

b2 = b3 = b4 = 0 X = k5„(dk2+A,1V ,

_ ak1 (k1 +k| )2 kj 12 (dk2 + 1k1 )2 _k7 (dk2 +1k1 )4+p2 (dk3+(1+1)k2k1+k1 ) = »6 (ki+k2 }p(dk2+1k1)3 ,

g = _ where

P2 ^ fc2(fc? + k2) 12(<5k2 + 1k1 )5 ((c2 _3^3)k2 + k^(dk2 + 1k1) _4a);

(iv) a_4 = a_3 = a_2 = a_1 = 0 ,

-4bjc2c4k1k2(k1+k2 )212+bjkj12+2P2c4 k1 (k1+k2 )p 6b|c4k1 k2 (k1+k2)12 ,

a2 = 2c4 (k1 + k2 ), a1 = a3 = a4 = 0 , b_4

= b_3 = b_2 = b_1 = b1 = b3 = b4 = 0 , b _ Vjp-2b2k2 (k2 +k2 ^^a^ k1 (k2 +k2 ) _b2 c2k21)

6b2c4k2(k2 +k2)12 b4kj12+^V2c4k1 (k1+k2)p 2b2 c4k2(k1+k2)12 ,

_ _4ab2 c4 k1 k2 (k2 +k2 ) 212+2p^p (b2 k21+2c4 k1 (k2 +k2 ))+b4k212

*_b21_2c4dk2

1 = 2c4k1 k2 ,

p ^ y_13b2k4 (k1 + k2) (c4 (a + 6^0, (k2 + k2) ) _ 2^2 (k2 + k2'

(iv) a_4 = a_3 = a_2 = a_ = 0, a0 =-a1 = 2 c3(k2 + kj)'

c2 (k1 +k2 )1

2k2 (dk2 +1k1) p3 k1 (k1 +k 2 ) k5 (dk2 +kk1)3

c2k5(dk2 +a"1 ) -2akl (k2+k|)k|(dk2+Ik1

a2 = a3 = a4 = 0 , b0 =-

b_4 = b_3 = b_2 = b_ = 0 , b1 = +1k1',

k7 (ik2 +ik1 )4 k1P3 I 2 k2+k2-

b2 = b3 = b4 = 0 x = _ k51(dk2+ik1')3 ,

(dk| +1k1 k2 +k1 +k1 k2 )p3 k3 (dk2 +Ak1 )2 _a12 k1 (k1 +k2

g 1k2 (k2+k2)1(dk2+ 1k1)3 + '

P3 = y^k + k2) 12(dk2 + 1k1 )5 ((c2 _3c1c3)k2 (k + k^(dk2 + 1k1) _4a).

From Case 4, we obtained the new exact jacobi elliptic function solutions [45,46] of Eq. (1). Case 5: c4 = c5 =6 = 0,

A λ2+»2) (C2 \A2 3c1c3)

(i) a 4 = a 3 = a 2 = a 1 = 0 , a0 = 1 2;v 2 6V 2—, a1 = W k1 + k2) ,a2 = a3 = a4 = 0 ,

b 4=b 3 = b 2=b 1 = 0b0="2(c2 Vc2 6c1''3)(d-~—~,

b1 = cз"2(d2"11+1"l) , b2 = b3 = b4 = 0 ,

di1k1k2 (k2+k2)(kf dk| (1 1)k2k1^c2 3c1c3 +ak112 (k1+k2)2 g = k2(k1+k2 )1(dk2+1k1)

t2(d»2+l"1)2

Vc2 3c1c3k1 (k1+k2)1k2(dk2+1k1)

k21(dk2+1k1)

(ii) a_4 = a_3 = a_2 = a_1 = 0 , a0 = a1 = 3( 2 2) , a2 = a3 = a4 = 0 ,

^ cj-3clcз+c2)(fej+fej)

b_4 = b_3 = b_2 = b_1 = 0 , b0 = ^VCj-3C1C3+61j)"j(d"j~"l) ,

b1 = c3 t2(d>1+1k1), b2 = b3 = 0,

V c2 _3c1 c3 k21(dk2 +1k1 n гk2+(1-1)kl--1 +ak1

b4 = 0, g=-

Vcj-3ClCзkl(k1+kj)1-

k2 (dk2 +1k1 )

1(ik2+1k1)

k2(dk2+1k1)2

We also obtained the new exact travelling wave and elliptic solutions [45,46] of Eq. (1) from Case 5.

The dispersive long wave equations

The dispersive long wave equations are as:

Vt + VVx + Wx = 0 Wt +(vw)x + 3 Vxxx = 0

Consider the traveling wave solution of Eq. (33)

v(x, t) = V(n) = £ ai/(n), W(x, t) = W(n) ^bj/^) (34)

/'(n) ^ Vc0 + c1/ + c2/2 + c3/3 + c4/4 + cs/5 + c6/6 n = kx + 1t

where a,^ ,c0 , c^ c2 , c3, c4 , c5, c6 , k and 1 are arbitrary constants and m , n are positive integers, which are determined later. The parameters m and n are usually obtained by balancing the highest order linear term with the nonlinear terms of highest order in the resulting equations. Substituting Eq. (34) into Eq. (33), we get

+ kVV' + kW = 0 ' + k(VW)'+ k3 V''' = 0

By using balancing principle we get m = 1, n = 2, we suppose the solution of Eq. (36) is as:

d2k5 2d1k2k1 12k3k1

V (n)=af + ao + ai/ W ©=¥ + ^ + bo + bi/ + b2/2.

Substituting Eq. (37) into Eq. (36)setting the coefficients of /'/(1) to zero yields a systems of algebraic equations in a—1, a0 , a1, b—2, b—1, b0 , b1, b2, k and The systems of algebraic equations possesses the following solutions cases:

Case 1: c0 - c1 - c5-6 - 0

(i) a_i = 0, ao , ai =-2^,b_2 = b_i = 0 ,

, c2k2-4c2c4k2 b0 = 3 12c.->

b1 = -3c3k2 , b2 = -fc4k2.

(ii) a-1 = 0, ao = ^p-T4, ai = ^, b-2 = b-i = 0

, C2k2-4C2C4k2

b0 =-Î5C4-,

bi = -33Сзк2 , b2 = C4k2.

Substituting Eq. (38) into Eq. (37), we have obtained the following soliton-like solutions of Eq. (33) are as:

v i(x ,t) = Wi(x ,t) =

v 2 (X ,t) =

w2(x ,t)

-p3c3k2-6pc4i 4 pC4"c2ksech(pC2g) б/ P3(PD-C4sech(pci{)) '

c2k2-4c2c4k2 2c3 c2k2sech(pcr2'n) 8c4c2k2sech2(pc2g) _ > 0 д > 0 12c4 3(ps-c4sech(pcïs)) з(pД-c4sech(pCfг))2 ' 2 ' '

-p3c3k2 -бу" 4 pc4c2ksech(pc2g)

c?k2-4c2c4k2 2c2c3k2sech(pcïj) 8c4c2k2sech2(v/) — ^—тт:--г -/. ...,./ „-, --;-;-;-/г, С2 > 0 , Д

3(c4sech(Pcîn)^v«) 3 (с4 sech ( pc2 n) +РД)2

C2 > 0 , Д > 0' (4i)

V3(X ,t) = W3(X ,t) =

-pj^k2 -бу" ± 4p/c4"c2kcsch(pc2n) 6pc4k P^P-ATCics^vcjn)) '

c|k2-4c2c4k2 ±2/ c2k2csch(p2n) 8c4c^k2csch2 (у/г) c > 0 д < 0

i2c4 3(V=ÄTC4csch(pin)) 3(p/STc4csch(p"in))2 ' C2 > 0 'Д < 0'

v4(x,t)=-^Cipc-6Mi+/ ± tanh(>f n)),

+^ (i ± tanh (p2n)) -^ (i ±tanh (P/n))2

c2 > 0 д

Fig. 3. Travelling wave solutions of Eq. (11) with various different forms are plotted: (a) Periodic solitary waves and (b) contour plot of u2, (c) solitary waves and (d) contour plot of v2.

U5 (x , t)- + ^ ( 1 ± coth n) ) ,

^ (1 ± coth (PC2n))

Ws(x ,t) =

c2k2-4c2c4k2

_ ^(1 ± coth(^fn)) , c2 > 0 ,A = 0.

where , n = kx + 1t , A = c2 _ 4c2c4.

Figs. 3(a) and (c) signify the evolution of the solitary wave solutions of Eq. (43) of the dispersive long wave Eq. (33), with c2 = 1, c3 = 2 , c4 = 1, k = 1 and 1 = 1.5. Fig. 3(b) and (d) are contour plots of V4 and W4 respectively of Eq. (43).

Similarly, we obtained the new exact soliton-like solutions of Eq. (33) by taking Case 1(ii).

Case 2: c0 = c1 = c3 = c5 = c6 = 0,

(i) a_1 = 0 , a0 = _1, a1 = _^Pp, b_2 = b_1 = 0 ,

bo = -

b = 0 , b2 - -

(ii) a-1 = 0 , ao = -k, a1 =

. c2 k2 . . 2c4k2 bo -—b1 = 0 , b2 =--^

b-2 = b-1 = 0

Substituting Eq. (45) into Eq. (37), we have obtained the following soliton-like solutions of Eq. (33) are as:

(x ,t)-- + ^ csc (p-^n),

W^t)--^ + ^£3^csc2(p-C2 n), C2 < 0, C4 > 0.

v 2 (x,t) = -£ - ^ csch(Viin) ,

W2(x ,t) = -zf + 2c3k-csch2n), C2 < 0 , C4 > 0.

v !x f\ — l ^ 4c2p5ike<VC2n) v3 (x,t)-- + V33(c2£4e(2^n)-1);

W3(x,y ,t)--£23L -

_ 8c2c4k2 e'2^

3(c2c4e<2Viiin)-1)2

where n = kx + 1t.

Figs. 4(a) and (c) signify the evolution of the dark solitary wave and bright solitary solutions of Eq. (47) of the dispersive long wave Eq. (33), with c2 = _1, c4 = k = 1 and 1 = 1. Figs. 4 (b) and (d) are contour plots of V1 and W1 respectively of Eq. (47). we can also obtained new exact soliton-like solutions from Case 2(ii).

Case 3: C' = c3 = c5=6 = 0,

l k/clk , , „ ,

(i) a-l = 0 , a0 =-k,al =—/3—, b-2 = b-l = 0 , Ьо

c2k2 . . 2c4k2

= -^, bl = 0 , bk =--; (50)

(ii) a-l = 0 , ao =-p al = 3Tp, b-2 = b-l = 0 , bo

c2k2 . . 2c4k2

= --, b' = 0 , bk =--; (51)

(iii)a-l = -2/3°k, ao = -p a' = 0 , b-2 = -3Cok2 , b-l

= 0 , bo = -kr,bl = bk = 0; (52)

2/COk i 2c° k2

(iv) a-l = /з , a° = -k, a' = 0 , b-2 =--, b-l

= 0 , b° = -3Ckk2 , b' = bk = 0; (53)

(v) ab-bl

(vi) ab-bl

(vii) ab-bl

(viii) ab

_ o Ъ — kyCoygC -C2C

— U , U° — 3

= 0 , b2 = - C4 k2;

_ k/cpc a _ u a _ 2/5* ъ — 2 c с2

= -^зг, a° = -a' = b-2 = -зc°k

TT , b-2 = -3 C

= ^,a° = — lu, a' = -Mp, b-2 = -Cok2 ,

= 0 , b° =-

= 0 , bk = -C4k2;

_ VC0k ^ — ■ ^ — VCS h — 2 r t2

a° = -ï, al = b-2 = -зc°k ,

= 0 , b° = 3 (-2VC°VQk2 - Ckfc2), = 0 , bk = -c4k2;

_ k/cok ^ — ■ ^ — MSc h — 2 r ь2 -^з^ a° — - c, a' =^3^ b-2 = -зc°k ,

= 0 , bo = 3 (2/CoQk2 - C2k2^,

b' = 0 , bk = -c4k2.

Case 4: c5 = c6 = 0

(i) a_i = 0, ao , ai =-2^, b_2 = b_i = 0,

, c3k2_4c2c4k2

b0 = 12Ï4— '

bi = _£ , b2 = _^.

(ii) a_i = 0 , a0 =p3^pc|PCâ , ai = ^ , b_2 = b_i = 0 ,

b„=Ci

c2k2_4c2c4k2

bi = _f , b2 = _32.

..... a _ 2kpcQ a _ _y3cik2_6ipc0 a _ 0

(iii) a_1 =--p-, a0 =-6VCP , a1 = 0 ,

b_2 = _2f2, b_i = _

b0 = bi = b2 = 0.

(iv) (iv) : a_i = ffi a0 = ai = 0 b_2 = b_i = _

. _ c2k2_4c0c2k2 b _ b _ 0 b0 =-Î2C0-, bi = b2 = 0.

Case 5: c4 = c5=6 = 0,

(i) a — 2pcQt a _ _y3cit2_6pcQi a _ 0 (i) a_i = —^0-, a0 =-67^-, ai = 0 ,

b_2 = _fC0k2 , b_i = Cik2 ,

b0 =bi = 0, b2 = 0.

3 1 b2

(П) a_i = ^,a0 = ^ ,ai = 0, b_2 = _fC0k2 , b_i = cik2 ,

b0 =bi = 0, b2 = 0.

We can also obtained the new exact travelling wave and elliptic solutions [45,46] of Eq. (33) from Case 3, Case 4 and Case 5.

Conclusion

In this study, we create the new exact travelling wave solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations, which are very important in physics and mathematics. The solutions of both equations are obtained in the explicit form. Many new exact solutions are obtained, which have important applications in applied sciences and might provide a useful help for researcher and physicists to study more complex physical phenomena.

References

[1] Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. New York: Cambridge Univ. Press; i99i.

[2] 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 i895;39:422-43.

[3] Kochanov MB, Kudryashov NA, Sinel'shchikov DI. Non-linear waves on shallow water under an ice cover. Higher order expansions. J App Math Mech 20i3;77:25-32.

[4] Seadawy AR. Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput Math Appl 20i4;67 (i):i72-80.

[5] Seadawy AR. Stability analysis for two-dimensional ionacoustic waves in quantum plasmas. Phys Plasmas 20i4;2i(5). Article ID052i07.

[6] Hirota R. Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett i97i;27:ii92-4.

Wang ML. Exact solutions for a compound KdV-Burgers equation. Phys Lett A 1996;213:279-87.

Weiss J, Tabor M, Carnevale G. The Painleve property and a partial differential equations with an essential singularity. Phys. Lett. A 1985;27:205-8. Zheng CL, Fang JP. New exact solutions and fractal patterns of generalized Broer-Kaup system via a mapping approach. Chaos Soliton Fract 2006;27:1321-7.

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.

Zhang Sheng, Tong Jing-Lin, Wang Wei. Exp-function method for a nonlinear ordinary differential equation and new exact solutions of the dispersive long wave equations. Comput Math Appl 2009;58:2294-9. 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.

Peng YZ. Exact solutions for some nonlinear partial differential equations. Phys Lett A 2003;314(5-6):401-8.

Peng YZ. New exact solutions to a new Hamiltonian amplitude equation. J Phys SocJpn 2004;73(5):1156-8.

Seadawy AR, El-Rashidy K. Traveling wave solutions for some coupled nonlinear evolution equations. Math Comput Model 2013;57:1371-9. Zakharov VE, Kuznetsov EA. On three-dimensional solitons. Sov Phys 1974;39:285-8.

Kaup DJ. Finding eigenvalue problems for solving nonlinear evolution equations. ProgTheor Phys 1975;54:72-8.

Broer LJF. Approximate equations for long water waves. Appl Sci Res 1975;31:377-95.

Kupershmidt BA. Mathematics of dispersive water waves. Commun Math Phys 1985;99:51-73.

Martinez L. Schrodinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations. J Math Phys 1980;21:2342-9.

Huang W. A polynomial expansion method and its application in the coupled Zakharov-Kuznetsov equations. Chaos, Solitons Fractals 2006;29(2):365-71. Inc M. Exact solutions with solitary patterns for the Zakharov-Kuznetsov equations with fully nonlinear dispersion. Chaos, Solitons Fractals 2007;33 (5):1783-90.

Wazwaz AM. 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.

Zhao X, Zhou H, Tang Y, Jia H. Travelling wave solutions for modified Zakharov-Kuznetsov equation. Appl Math Comput 2006;181(1):634-48. Seadawy AR, El-Rashidy K. Water wave solutions of the coupled system Zakharov-Kuznetsov and generalized coupled KdV equations. Sci World J 2014;2014. Article ID 724759.

Gear JA. Strong interactions between solitary waves belonging to different wave modes. Stud Appl Math 1985;72(2):95-124.

Gao Y, Tang X-Y. A coupled variable coefficient modified KdV equation arising from a two-layer fluid system. Commun Theor Phys 2007;48(6):961-70. Ugurlua Yavuz, Kaya Dogan, Inanb Ibrahim E. Comparison of three semi-analytical methods for solving (1+1)-dimensional dispersive long wave equations. Comput Math Appl 2011;61:1278-90.

Kaya Dogan, El-Sayed Salah M. On the solution of the coupled Schrodinger KdV equation by the decomposition method. Phys Lett A 2003;313:82-8. Biazar J, Badpeima F, Azimi F. Application of the homotopy perturbation method to Zakharov-Kuznetsov equations. Comput Math Appl 2009;58(11-12):2391-4.

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.

Zhou JK. Differential transformation and its applications for electrical circuits. Wuhan, China: Huazhong Univ. Press; 1986 (in Chinese). Ravi Kanth ASV, Aruna K. Differential transform method for solving linear and non-linear systems of partial differential equations. Phys Lett A 2008;372:6896-8.

Rashidi MM, Erfani E. Traveling wave solutions of WBK shallow water equations by differential transform method. Adv Theor Appl Mech 2010;3:263-71.

Keskin Y, Oturanc G. Reduced differential transform method for partial differential equations. Inter J Nonlinear Sci Numer Simul 2009;10:741-9. Abazari Reza, Abazari Malek. Numerical simulation of generalized Hirota Satsuma coupled KdV equation by RDTM and comparison with DTM. Commun Nonlinear Sci Numer Simul 2012;17:619-29.

Helal MA, Seadawy AR. Variational method for the derivative nonlinear Schrodinger equation with computational applications. Phys Scr 2009;80(3). Article ID 035004.

Helal MA, Seadawy AR. Exact soliton solutions of a D-dimensional nonlinear

Schrodinger equation with damping and diffusive terms. Zeitschrift fur

Angewandte Mathematik und Physik 2011;62(5):839-47.

Seadawy AR. New exact solutions for the KdV equation with higher order

nonlinearity by using the variational method. Comput Math Appl 2011;62

(10):3741-55.

Helal MA, Seadawy AR. Benjamin-Feir instability in nonlinear dispersive waves. Comput Math Appl 2012;64(11):3557-68.

[41] Lu Xing. Madelung fluid description on a generalized mixed nonlinear Schrodinger equation. Nonlinear Dyn 2015;81:239-47.

[42] Lu Xing, Ma Wen-Xiu, Jun Yu, Lin Fuhong, Khalique Chaudry Masood. Envelope bright- and dark-soliton solutions for the Gerdjikov-Ivanov model. Nonlinear Dyn 2015;82:1211-20.

[43] Lu X, Ma WX, Yu J, Khalique CM, Linb F. Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrodinger equation. Commun Nonlinear Sci Numer Simul 2016;31:40-6.

[44] Seadawy AR, Lu Dianchen. Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results Phys 2016;6:590-3.

[45] Nickel J. Elliptic solutions to a generalized BBM equation. Phy Lett A 2007;364:221-6.

[46] Nickel J. Travelling wave solutions to the Kuramoto-Sivashinsky equation. Chaos, Solitons Fractals 2007;33:1376-82.