Cent. Eur. J. Phys. • 11(4) ^2013 • 518-525 DOI: 10.2478/s11534-013-0197-1

VERS ITA

Central European Journal of Physics

Computational study of some nonlinear shallow water equations

Research Article

Ali H. Bhrawy1,2, Mohamed A. Abdelkawy2*

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt

Received 09 December 2012; accepted 27 February 2013

Abstract:

PACS C200B): Keywords:

The shallow water equations have wide applications in ocean, atmospheric modeling and hydraulic engineering, also they can be used to model flows in rivers and coastal areas. In this article we obtained exact solutions of three equations of shallow water by using J-expansion method. Hyperbolic and triangular periodic solutions can be obtained from the J -expansion method. 02.30.Gp, 02.30.Jr, 02.70.-c, 02.70.Wz

J -expansion method • nonlinear physical phenomena • nonlinear shallow water equations • time dependent nonlinear system of shallow water © Versitasp. zo.o.

1. Introduction

The shallow water equations (SWEs) are a set of hyperbolic partial differential equations that describe the free-surface flows arising in shores, rivers, estuaries, tidal flats, coastal regions and also SWEs describe many other physical situations [1] such as storm surges, tidal fluctuations, tsunami waves, forces acting on off-shore structures and modeling the transport of chemical species. The SWEs, which are based on the conservation laws of mass and momentum, may be derived by integrating in depth the Reynolds averaged Navier-Stokes equations [2]. The

^E-mail: melkawy@yahoo.com

speed of shallow water waves (SWWs) is independent of wavelength or wave period and is controlled by the depth of water, also all SWW travel at the same speed [3]. In contrast, deep water waves of different length travel at different speeds (the long ones faster than the short ones).

The shallow water equations have been solved in many articles numerically and analytically. Variational iteration method (RVIM) is applied in [3] to compute the approximate solution for coupled Whitham-Broer-Kaup shallow water. While in [4], Benkhaldoun et al. proposed finite volume method for numerically solve shallow water equations on nonflat topography. In [5], the authors presented spectral Galerkin method for the numerical solution of the shallow water equations in two dimensions. The shallow water equations over irregular domains with wetting and drying are discussed by second-order Runge-Kutta

Springer

discontinuous Galerkln scheme In [6]. Moreover, the exact solutions for shallow water equations are obtained in several articles suing different methods such as generally projective Riccati equation method [7], Backlund transformation and Lax pairs [8], the tanh-coth, Exp-function and Hirota's methods [9, 10].

We, firstly, will study time dependent nonlinear system of shallow water (SW). The authors of [11-13] studied the coupled Whitham-Broer-Kaup (CWBK) equations that describe the dispersion of SWW with different scattering relations:

u, + uux + vx + puxx = 0,

v, + (uv)x + auxxx - pvxx = 0,

where u is the velocity and v the total depth, which can be used as a model for water waves. When a = 1 and p = 0, the CWBK equations are reduced to the approximate long-wave (ALW) equations in SW. When a = 0 and p = 0.5, the CWBK equations are reduced to the modified Boussinesq (MB) equations [14].

The second equation that we will study in this paper is the (1+1)-dimensional SWE in the following form [15]:

uxt + uxx - uxxxt - 4uxux, - 2uxxu, = 0. (1.2)

The third one, we will study in this paper is the (2+1)-dimensional SWE in the following form:

4ux, + uxxxy + 8uxyux + 4uxxuy =0. (1.3)

A large variety of physical, chemical, and biological phenomena is governed by nonlinear partial differential equations (NPDEs). The exact solutions of these NPDEs plays an important role in the study of nonlinear phenomena. In the past decades, many methods were developed for finding exact solutions of NPDEs as the inverse scattering method [16]- [18], Hirota's bilinear method [19], new similarity transformation method [20], homogeneous balance method [21, 22], the sine-cosine method [23, 25], tanh function methods [26]- [32], exp-function method [33]- [36], Ja-cobi and Weierstrass elliptic function method [37]- [39]... etc. In this article we find exact solutions of three SWEs using the G expansion method [40]-[44]. Kudryashov [45] proposed the GG-expansion method to find rational solutions of Schwarzian integrable hierarchies. Moreover, Wang et al. [46] developed this approach for the exact solution of nonlinear wave equations. The main advantages of the GG-expansion method, that the obtained solutions are more general solutions with some free parameters, while we can obtained other solution by taking different choice of the parameters. In addition

1. The G-expansion method is a powerful technique to integrate nonlinear differential equations without need of boundary conditions or initial trial function. In contrast, in all finite difference, finite element and spectral methods, it is necessary to have boundary or initial conditions,

2. The GG-expansion method gives a general solution without approximation, in contrast, other methods give solutions in a series form and it becomes essential to investigate the accuracy and convergence of analytical and numerical methods,

3. In case of the Painleve test of integrability fails, G-expansion method can be used as a powerful technique to integrate the NEEs,

4. The rational solutions are computed by using GG-expansion method and they could not be obtained by other methods.

2. Gg expansion method

This section is devoted to the study of implementing the GG expansion method for a given partial differential equation

C(u,ux,uy,ut,uxy,...) = 0, (2.1)

where u(x, y, t) is an unknown function in the independent variables x, y and t. In order to obtain the solution of Eq. (2.1), we combine the independent variables x, y and t into one particular variable through the new variable

Z = x + y + vt, u(x, y, t) = U(Z),

where v is the wave speed, using this variable, enables us to reduce Eq. (2.1) to the following ordinary differential equation (ODE)

G^U,U' ,U" ,U"' = 0. (2.2)

The ODE is integrated as long as all terms contain derivatives in Z, upon setting the constant of integration to zero, one is looking for although the non-zero case can be treated as well. At this stage, we search for the exact solutions which satisfy this ODE. For this purpose, U(Z) can be expressed as a finite series of CG,

N g' '

u(x,y,t) = U(Z) = y a<— , G" + vG' + AG = 0.

The parameter N can be determined by balancing the system of nonlinear algebraic equations in the parameters

linear term(s) of highest order with the nonlinear one(s), v, j, A and a,, i = 1, ••• , N. "We next solve the over-

where N is a positive integer, so that an analytic solution determined system of nonlinear algebraic equations by

in closed form may be investigated. Expressing the ODE using Mathematica program. (2.2) in terms of Eq. (2.3) and comparing the coefficients of each power of G in both sides, to get an over-determined

3. Nonlinear system of shallow water

In this section, we firstly implement the proposed method for solving the nonlinear system of SW:

u, + uux + v* + $uxx = 0, V, + (uv)x + auxxx - pvxx = 0,

as described in the introductory section, we perform a travelling wave reduction,

u(x, t) = U(Z), v(x, t) = V(Z), Z = x + vt, (3.2)

that converted (3.1) into a system of ODEs,

vU + UU + V + pU =0, vV' + (UV)' + aU'" - pV' = 0,

if we integrate the system (3.3) once, upon setting the constant of integration to zero, we obtain

vU + T + V + PU' = 0, (3.4)

vV + UV + aU'' - pV' = 0.

If we use the first equation in system (3.4) into the second one, we get only one ODE

2v2U + 3vU2 + U3 - 2(ß2 + a)U =0.

Balancing the term U" with the term U3, implies to N = 1. Accordingly, the solution takes on the form

U (Z) = £ at

~G I '

Substituting Eq. (3.6) into Eq. (3.5) and comparing the coefficients of each power of J in both sides, getting an over-determined system of nonlinear algebraic equations with respect to v, j, A and ai; i = 0, 1. Solving the over-determined system of nonlinear algebraic equations using Mathematica, we obtain two sets of constants:

IJ = 0, a0 = —v a! = ±2y/ (a + ß2), and A = —

4 (a + ß2)'

a0 = -V(aK2+ P2V2) - v ai= ±2VW+W>, and A = ^f^ ^

We find the following solutions of Eq. (3.5) For case (a)

(■) ja+W) > 0

(■") a+W] < 0

(iii) v = 0

For case (b)

(a+P2)

(a+P2)

(iii) v = 0

Ui = -v

c cosh

U2 = - v

— c1 sin

1 + 4-

4v2 (a+P2)

4v2 (a+P2)

Z + c2 sinh

Z + c2 cos

(a+P2) j Z/

4v2 (a+P2)

c1 cos

Z + C2 sin

Us = —v ±

(a+P2)

2ci^(a + P2)

(a+P2)j Z /

c1 Z + c2

U4 = —y/av2 + P2v2 — v ± 4v-

c1 sinh

4v2 (a+P2) Z

Z ) + c2 cosh

4v2 . (a+P2) Z

c1 cosh

(0+2) Z> + c2 sinh

U5 = —y/av2 + p2v2 — v ± 4v-

—c1 sin

4v2 (a+P2) '

Z + c2 cos

4v2 Z (a+P2) Z

4v2 (a+P2) '

c1 cos

Z + c2 sin

U6 = —y/av2 + P2v2 — v ±

(a+P2) '

2ciy/(a + P2)

4v2 , (a+P2) Z

c1 Z + c2

Then the solutions of the system (3.1) are:

c1 sinh

u1 = —v

1 + 4-

4v2 (a+P2)

(x + vt) I + c2 cosh

{{^)(X + vt))\

ci cos^yha+2)(x + vM + c2 smh Ja^ (x + vt)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

—c1 si

u2 = —v

4v2 (a+P2)

sin\,l-r^ (x + vt) I + c2 cos | ,/aPj(x + vt)W

4v2 (a+P2)

c1 cos it/ 1,',"o2\ (x + vt) | + c2 sin ( yj (a+p)

(x + vt)

(3.16)

U3 = —v ±

= — V a J2 + ß2j2 — v ± v-

ci slnh

= —V a J2 + ß2J2 — v ± 4v-

— C1 sin

2c,V(a + ß2)

C1X + C2 ,

a+ß2)

(x + vt) I + c2 cosh

a+ß2)

(X + vt)

Ci cosh ( ^ Iy^+pj (X + vtf) + C2 sinh

0+-2) (X + v^ ) + C2 cos

a+ß2)

(X + vt)

a+ßj(x + vt)

Ci co^/(a+%(x + vt) + C2sl^Y(a+ß5)

= — Vaj2 + ß2J2 — v ±

2C^(a + ß2)

Ci X + C2 ,

where v in all above solutions (3.15)-(3.20) given by the relation:

(X + vt)

(3.17)

(3.18)

(3.19)

(3.20)

V = — ( vU + U2+ ßU I .

(3.21)

(a) (b)

Fig. 1 (a) Three-dimensional of the modulus of solitary wave solution u (Eq. 3.15) where a = p = 0.5, j = 1, v = -1, (b) Graphical representation of of the modulus of solitary wave solution u (Eq. 3.15) where a = p = 0.5, j = 1, v = -1.

The soliton solutions for system (3.1) are given in (3.15) and (3.18), while the triangular periodic solutions are given in (3.16) and (3.19). Moreover, for the special values p = c2 = 0, a = 1, we obtain equivalent solutions for the system of the SWW equation that given in [47]. Three-dimensional of the modulus of solitary wave solutions u-i, Eq. (3.15) and u2, Eq. (3.16) are displayed in Figures 1(a) and 2(a) respectively with values of parameters listed in their captions. Moreover, graphical representation for the same solutions are plotted in Figures 1(b) and 2(b) respectively.

4. (1+1)-dimensional shallow water if we use the transformations

equation u(x,t) = U (Z), Z = x + vt, (4.2)

We next consider the (1+1)-dimensional SWE in the fol- it carries Eq. (4.1) into the ODE lowing form:

(v + 1)U''- vU'''- 6vU'U" = 0, (4.3)

uxt + uxx - uxxxt - 4uxuxt - 2uxxut = 0, (4.1)

(a) (b)

Fig. 2 (a) Three-dimensional of the modulus of solitary wave solution u (Eq. 3.16) where a = p = 0.5, j = 1, v = -1, (b) Graphical representation of of the modulus of solitary wave solution u (Eq. 3.16) where a = p = 0.5, j = 1, v = -1.

where by integrating (4.3) once and taking the constants of integration to be zero we obtain

(v + 1) U'- vU'''- 3v(U')2 = 0. (4.4)

c- slnh Z + c2cosh Z

U- = O0+ 2-\\- )-)v- ; , (4.9)

C- cosh VV- Z + C2Sinh y V+T Z

Considering the homogeneous balance between U and

(U') , then N = 1, and

U (Z) = £ at

Proceeding as in the previous case we obtain

A = "2V ~ V ~ 1 , and a-=2, (4.6)

V = ±V—— , A = 0 and a1=2, (4.7)

A = - , v = 0 and a- =2. (4.8) 4v

For all above cases we find the same three solutions of

Eq. (4.4):

U2 = a0+2

— C1 Sin

+ C2 cos

C1 cos

+ C2 sin

(ill) V = — 1

U3 = a0 +

2c-C-Z + C2 .

Then the solutions of the Eq. (4.1) are: u- = a0

c1 slnh V+1 (x + vt)j + c2 cosh

, (4.1

c1 cosh

u2 = a0

—c1 sin

/t) j + c2 slnh

x + vt) j + c2 cos

(4.10)

(4.11)

■vt))' (4.12)

c- cos^^+vKx + vt)) + C2sln^Jv+1(x + vt))

(4.13)

u3 = a0 +

(4.14)

c- (x — t) + C2

The solution (4.12) is a soliton solution. Moreover the triangular periodic solution is obtained in (4.13). For the special values c2 = 0, we give equivalent solutions for (1+1)-dimensional shallow water equation to that given in [48].

5. (2+1)-dimensional shallow water equation

In this section, we study the (2+1)-dimensional SWE in the following form:

4uxt + uxxxy + 8uxyux + 4uxxuy =0, (5.1)

if we use the following transformations

u(x,t) = U (Z), Z = x + vt, (5.2)

inserting (5.2) into (5.1) yields an ODE.

4vU'' + U'''' + 8U' U'' + 4U' U'' = 0, (5.3)

if we integrate the Eq. (5.3) once, upon setting the constant of integration to zero, we obtain

4vU + U"' +6 (U^2 = 0.

Balancing the term U''' with the term |u'J , we obtain N = 1, and

U (Z) = y at

Proceeding as in the previous case we obtain three sets of constants

(ii) v < 0

U-2 = ao +

-C1 sin((| Z) + C2COs(yq Z) CiCOS (yf Z ) + C2Sin (y/2 Z ) '

(iii) v = 0

U3 = ao+ .

ClZ + C2

Then the solutions of the equation (5.1) are:

(5.10)

(5.11)

U = ao +

C1 sinh (yfv (x + y + vt)) + C2 cosh (^/j(x + y + vt))

C1 cosh (^(x + y + vt)) + C2 sinh (^(x + y + vt)) '

(5.12)

u2 = a0+

-C1 sin (yiv,(x + y + vt)) + C2 cos (yiv,(x + y + vt))

C1 cos (t/2(x + y + vt)) + C2 sin (y^2(x + y + vt)) '

(5.13)

U3 = ao +

Ci (x + y) + C2 .

(5.14)

We conclude that (5.12) is a soliton solution, while (5.13) is a triangular periodic solution. Moreover, we can obtain other soliton solutions that may be derived from the soliton solutions obtained in the last three sections, by setting C =0 or c2 = 0. The same procedure can be applied for triangular periodic solutions.

A = V^+^v and ai=1, (5.6)

V = ±2iy/v, A = 0 and ai=i, (5.7)

A = v, = 0 and ai =1. (5.8)

For all above cases we find the same three solutions of

Eq. (5.4): (i) v > 0

= + Ci Sinh((2 Z)+ c2cosh(Z) 1 0 cicosh 2Z ) + C2 sinh (^ Z ) ' (.)

6. Conclusion

We extend the G-expansion method with symbolic computation to two nonlinear equations of SW and nonlinear system of SW. It is shown that hyperbolic solutions and triangular periodic solutions can be established by the G-expansion method. The travelling wave solutions can be expressed as the hyperbolic functions, trigonometric functions and rational functions, and involved in two arbitrary parameters. As special values of parameters, the solitary waves are also derived from the travelling waves. The rational solutions are computed only by using G-expansion method and they could not be obtained by other methods.

Acknowledgments

The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions which have improved the paper.

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