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PhysicsEngineering 10 (2011)2845-2850

Abstract

In this work, the (G7 G) -expansion method with the aid of Maple is applied to construct more general exact solutions of the

coupled Boussinesq equations, where the French scientist Joseph Valentin Boussinesq (1842-1929) described in the 1870's model equations for the propagation of long waves on the surface of water with a small amplitude. Each of the obtained solutions, namely hyperbolic function solutions, trigonometric function solutions and rational function solutions contain an explicit linear function of the variables in the considered equation. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.

© 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11

Keyword: Boussinesq systems, (G'/ G) -expansion method, hyperbolic function solutions, trigonometric function solutions, rational function solutions.

1. Introduction

Partial differential equations (PDEs) describe various nonlinear phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, mechanics, biology and mathematical finance. Their solution spaces are infinite dimensional and contain diverse solution structures. It is of significant importance to solve nonlinear PDEs from both theoretical and practical points of view. Due to the nonlinearity of differential equations and the high dimension of space variables, it is a difficult job for us to determine whatever exact solutions to nonlinear PDEs. Particularly, various methods have been utilized to explore different kinds of solutions of physical models described by nonlinear PDEs. One of the basic physical problems for those models is to obtain their travelling wave solutions(also known as solitons). In mathematics and physics, a soliton is a self reinforcing solitary wave, a wave packet or pulse, that maintains its shape while it travels at constant speed. Solitons are caused by a cancelation of nonlinear and dispersive effects in the medium. The term "dispersive effects" refers to a property of certain systems where the speed of the waves varies according to frequency. Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described by John Scott Russell (1808-1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the

* Corresponding author. Tel.: +98 451 4471544; fax: +98 451 5514701. E-mail address: abazari-r@uma.ac.ir, abazari.r@gmail.com.

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.04.473

"Wave of Translation" (also known as travelling wave solutions or solitons)[1]. The soliton solutions are typically obtained by means of the inverse scattering transform [2] and owe their stability to the integrability of the field equations.

It has been a successful idea to generate exact solutions of nonlinear wave equations by reducing PDEs into ordinary differential equations (ODEs). Many approaches to exact solutions in the literature follow such an idea, which contain the tanh-function method [3], the sech-function method [4], the homogeneous balance method [5, 6] the extended tanh-function method [7], the sine-cosine method [8], the tanh-coth method [9], the Jacobi elliptic function method [10], the exp-function method [11], the F-expansion method [12], the mapping method [13], and the extended F-expansion method [14]. Given an ODE of differential polynomial type, either constant-coefficient or variable-coefficient, one can always adopt computer algebra systems to search for rational solutions pretty systematically. This is one of the main reasons why those reduction methods work well. But, most of the methods may sometimes fail or can only lead to a kind of special solution and the solution procedures become very complex as the degree of nonlinearity increases.

Recently, the (G'l G) -expansion method, firstly introduced by Wang et al. [15], has become widely used to search for various exact solutions of NLEEs [15]-[21]. The main idea of this method is that the traveling wave solutions of non--linear equations can be expressed by a polynomial in (G7 G), where G = G(|) satisfies the second order linear ordinary differential equation G"(£,) + AG'(£) + juG(^) = 0, where ^ = kx + Wt and k,ffl are arbitrary constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and the non--linear terms appearing in the given non-linear equations.

The aim of this paper is to apply the (G'/ G) -expansion method to find new hyperbolic and trigonometric solutions of the following coupled Boussinesq equation

Here, the independent variable, x, is proportional to distance in the direction of propagation while t is proportional to elapsed time. The dependent variables n and u have the following physical interpretation. The

quantity n( x, t) + n0 corresponds to the total depth of the liquid at the point x at time t, where tf0 is the height

of the undisturbed water depth. The variable u (x, t) represents the horizontal velocity at the point

(x, y) = (x, 6t]0 ) (y is the vertical coordinate, with y = 0, corresponding to the channel bottom or sea bed) at

time t. Also, the quantity T = ri(pgn02), is the Bond number where r is the surface tension coefficient, p is

the density of water and g is the acceleration due to gravity. Thus u is the horizontal velocity field at the height

Orj0, where d is a fixed constant in the interval [0,1].

With the aid of Maple, new explicit and exact travelling wave and solitary solutions for the Boussinesq systems (1) are obtained by using the (G'/G) -expansion method.

2. Application of the (G'lG) -expansion method for the equation (1)

In this section, we apply the proposed method to obtain new and more general exact solutions of Eq. (1), which arises in several physical applications including the propagation of long waves on the surface of water with a small amplitude. Let us assume the traveling wave solution of Eq. (1) in the form

where £ = kx + (t, and k ,( are constants. Substituting Eq. (2) into Eq. (1) and integrating once with respect to £ and setting the integration constants as zero, we obtain the following nonlinear ordinary differential system

n( x, t ) = y (£), u( x, t ) = U (£),

oV + kU + kUV +1 k 2a>V" = 0, 6

1 < - r2 _ 7 3t/// 1 ,2„r.//

oU + kV + -kU2 -TkV— k oU =0, 2 2

Suppose that the solution of the nonlinear ordinary differential system (3) can be expressed by a polynomial in (G'l G) as follows:

U (£ = (G) +«0,

V (£) = (G )i + fi.

where am ^ 0,fin ^ 0 and m, n are called the balance number, a,(i = 0,1,..., m) and fij,(j = 0,1,..., n) are constants to be determined later, G(£) satisfies a second order linear ordinary differential equation (LODE):

G"(|) + A.G' (|) + ^G (|) = 0, (5)

where ^ and ^ are arbitrary constants. By considering the homogeneous balance between the highest order

derivatives and nonlinear terms appearing in nonlinear ordinary differential system (3) we get m = n = 2. Substituting Eqs. (4) along with Eq. (5) into Eq. (3) and collecting all the terms with the same power of (G'lG) together, equating each coefficient to zero, yields a set of simultaneous algebraic equations for k, o, a, (i = 0,1,2), and fi, (j = 0,1,2), as follows:

1H (filX+2fi1v)ok2 +a0 (fi0 +1))+ofi0 = 0, (6)

T^(fi1l+2fi2^)k 3+ (a2^2+1a1A/u)ok 2-(fi0+1 a02)k-oa0 =0, (7)

(^ - fiffiok 2+(a1+a{jfi1+a1fi0 ))+ofi1 = 0, (8)

3 1 2 6 1 1 0 1 1 0 1

T(2fi1^+6fi2X/n+fi1?i) k 3+(3a2ljU+1 a(A2 +2^))ok 2-(a1a0 +fi1)k-oa1 =0, (9) 12 4

("2 ^1^+3 +3 fi2^) ok2+( +1)a2+a0A+aA )k+ofi2 = 0, (10)

T(3fi1A+4fi2A2+8fi2lu)k 3+((2i2 +4^)a2 +^a1A)ok 2-(fi2 +a2a0 ^■a12)k-oa2 = 0, (11)

(1 fi+5 fi2X) ok2 + (a2fix+ afi) k = 0, (12)

2t(5P2X+P1 ) k 3+(a1+5a2A)ok2- ka2a1 = 0, (13)

k 2ofi2+ka2P2 = 0, (14)

6Tk 3fi2+3 k 2oa-1 ka22 = 0, (15)

Solving (6)-(15) by use of Maple, we get the following results:

1 72k2t00 + la^a . 1

{ m = —-,4.2 1 1,A = ±

, ( = +6kT

2 ^ 1 72k2t00 + 7 a2 a ,a = ±- 0 11

7a (16)

a2 = ±6

1 72k T + 7a12a1

A = ±

2 n 6k2T _

—a,n2 =-.}

7a ' " 48 k2r61 2]

where a0 = 12t — 7, a1 =7 — 6t, and k ^ 0 and are free constant parameters. Therefore, substitute the case (16) in (4), we get

U (!) = ±6.

—k 2T( G )2 +a1( G ) ±-^-72k ^ + 7aa 7a G 1 G 168

6k 2T( G )2 ± 7

2 ,G\ 1 72k t00 + 7a2a

—a (—)+--0—

7a G 48 k t01

a G 2 y

Substituting the general solutions of ordinary differential equation (5) into Eq. (19), we obtain three types of

traveling wave solutions of Eq. (1) in view of the positive, negative or zero of A2 - 4^ .

When A — 4u, =--> 0, using the general solutions of ordinary differential equation (5), we obtain

hyperbolic function solution uH ( x, t ) and T]H (x, t ) of coupled Boussinesq equation (1) as follows:

uH ( x, t ) = ±

a(Q2—cl)

(C2 sinh(1^T!) + C cosh(2 J — a!))2

n ( x, t ) =

0(C2—C2)

2 (C2 sinh(1 J — ^!) + C1 cosh(2 J — ^!))2

where 00 = 12t — 7,01 =7 — 6t,! = k(x + 6t

t), and k ^ 0 and C1, C2, are arbitrary constants. It is

easy to see that the hyperbolic solutions (18)-(19) can be rewritten at C2 > C2, as follows

uH ( x, t ) = ±

(6t — 7)

12t— 7 , 2. k -^(tanh (—

x — 61

-2-Tt) — p ) — 1), (20)

49 — 42t h

7 — 6T

, , 3 12t — 7 . 2, k

n(x') = rT7(tanh (" 2K

while at C12 < C2, one can obtain

T7( x—61

Tt) — p ) — 1), 49 — 42t h

\ _i_ 3 12t — 7 . 2. k

uH( x,t ) = ±:jT4-T^(coth (— 2 Al

(6t — 7) ' V

x — 61

-2-Tt) — p ) — 1), (22)

49 — 42t h

7 — 6T

, , 3 12t — 7t 2. k

'h(x,t^TY(coth (—^

■1, c

12t — 7

(x — 6.-Tt) — p ) — 1),

V 49 — 42t h

where p = tanh 1 (—-), and k are arbitrary constants. Now, when A — 4^ =

2(2 — 2q

< 0, we obtain

trigonometric function solution uT (x, t) and nT (x, t) of coupled Boussinesq equation (1) as follows:

uT ( x, t ) = ±

40(C2 — C1)

(c2Si„(iJ—JL !)+c,C0S(II—A !))2

n ( x, t ) =

°(C2 — C12)

2(C2sin(li T !)+C1cos(ii T !))2

where 00 = 12t — 7,J1 = 7 — 6t,! = k(x + 6t.

-1), and k ^ 0 and C , C, are arbitrary constants.

Similarity, the trigonometric solutions (24)-(25) can be rewritten at C2 > C2, and Q2 < C2, as follows

uT ( x, t ) = +

Vt ( x, t )

(6t — 7)

12t — 7 . 2. k

1 1 (tan (— 2 V

12t — 7

( x — 6.

I-2— Tt) — p ) + 1),

49 — 42t t

7 — 6T

3 12t — 7 . 2. k

2IT—7(tan (—^

TT* x—61

-2-Tt) — p ) + 1),

49 — 42t t

uT ( x, t ) = +

(6t — 7)

12t — 7 , 2, k —^(cot (—ÏM

12t — 7

( x — 6A

-Tt) — p ) + 1), (28)

49 — 42t t

7 — 6T

, , 3 12t — 7 . 2. k

n (x, t) = — 21T7(cot (—

—1, C1

x—61

Tt) — p ) +1),

49 — 42t t

respectively, where p = tan—1(—-), and k are arbitrary constants. Finally, when A2 — 4^ = 0, then, the T C2 rational function solutions to Eq. (1), obtained as follow:

urat ( X, t) = ±

(C2(+x +1 ) + C1)2

nmt(Xt)= . 2 (31)

(C2(+X +1 ) + C1)

where C1, C2, and k are arbitrary constants.

3. Conclusions

This study shows that the (G'l G)-expansion method is quite efficient and practically well suited for use in finding exact solutions for the coupled Boussinesq equation. Our solutions are in more general forms, and many known solutions to these equations are only special cases of them. With the aid of Maple, we have assured the correctness of the obtained solutions by putting them back into the original equation. We hope that they will be useful for further studies in applied sciences.

4. Acknowledgment

This work is partially supported by Grant-in-Aid from the Islamic Azad University, Young Researchers Club, Ardabil Branch, Iran.

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