Cent Eur J Phys • 11(4) • 2013 • 518-525 DOI: 10.2478/s11534-013-0197-1 Central European Journal of Physics Computational study of some nonlinear shallow water equations Research Article Ali H Bhrawy1,2 , Mohamed A Abdelkawy2∗ Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt Received 09 December 2012; accepted 27 February 2013 Abstract: 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 periodic solutions can be obtained from the G G G G -expansion method Hyperbolic and triangular -expansion method PACS (2008): 02.30.Gp, 02.30.Jr, 02.70.-c, 02.70.Wz Keywords: G G -expansion method • nonlinear physical phenomena • nonlinear shallow water equations • time dependent nonlinear system of shallow water © Versita sp z o.o 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 518 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 Ali H Bhrawy, Mohamed A Abdelkawy The GG -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, discontinuous Galerkin 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], Bäcklund 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: 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, ut + uux + vx + βuxx = 0, vt + (uv)x + αuxxx − βvxx = 0, where u is the velocity and v the total depth, which can be used as a model for water waves When α = and β = 0, the CWBK equations are reduced to the approximate long-wave (ALW) equations in SW When α = and β = 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 − 4ux uxt − 2uxx ut = (1.2) The third one, we will study in this paper is the (2+1)dimensional SWE in the following form: 4uxt + uxxxy + 8uxy ux + 4uxx uy = In case of the Painlevè test of integrability fails, G -expansion method can be used as a powerful G technique to integrate the NEEs, (1.1) (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], function methods [26]– [32], exp-function method [33]– [36], Jacobi and Weierstrass elliptic function method [37]– [39] etc In this article we find exact solutions of three SWEs using the GG 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 The rational solutions are computed by using GG expansion method and they could not be obtained by other methods G G expansion method This section is devoted to the study of implementing the GG expansion method for a given partial differential equation G(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 ζ = x + y + νt, u(x, y, t) = U(ζ), where ν 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 , = (2.2) The ODE is integrated as long as all terms contain derivatives in ζ, 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(ζ) can be expressed as a finite series of GG , i N u(x, y, t) = U(ζ) = i=0 G , G G + µG + λG = (2.3) 519 Computational study of some nonlinear shallow water equations The parameter N can be determined by balancing the linear term(s) of highest order with the nonlinear one(s), where N is a positive integer, so that an analytic solution in closed form may be investigated Expressing the ODE (2.2) in terms of Eq (2.3) and comparing the coefficients of each power of GG in both sides, to get an over-determined system of nonlinear algebraic equations in the parameters ν, µ, λ and , i = 1, · · · , N "We next solve the overdetermined system of nonlinear algebraic equations by using Mathematica program Nonlinear system of shallow water In this section, we firstly implement the proposed method for solving the nonlinear system of SW: ut + uux + vx + βuxx = 0, (3.1) vt + (uv)x + αuxxx − βvxx = 0, as described in the introductory section, we perform a travelling wave reduction, u(x, t) = U(ζ), v(x, t) = V (ζ), ζ = x + νt, (3.2) that converted (3.1) into a system of ODEs, νU + UU + V + βU = 0, (3.3) νV + (UV ) + αU − βV = 0, if we integrate the system (3.3) once, upon setting the constant of integration to zero, we obtain U2 + V + βU = 0, νV + UV + αU − βV = νU + (3.4) If we use the first equation in system (3.4) into the second one, we get only one ODE 2ν U + 3νU + U − 2(β + α)U = (3.5) Balancing the term U with the term U , implies to N = Accordingly, the solution takes on the form U (ζ) = i=0 G G i (3.6) Substituting Eq (3.6) into Eq (3.5) and comparing the coefficients of each power of GG in both sides, getting an overdetermined system of nonlinear algebraic equations with respect to ν, µ, λ and ; i = 0, Solving the over-determined system of nonlinear algebraic equations using Mathematica, we obtain two sets of constants: a) µ = 0, 520 a0 = −ν a1 = ±2 (α + β ), and λ=− ν2 , (α + β ) (3.7) Ali H Bhrawy, Mohamed A Abdelkawy b) (αµ + β µ ) − ν a0 = − a1 = ±2 (α + β ), and λ= αµ + β µ − ν (α + β ) (3.8) We find the following solutions of Eq (3.5) For case (a) (i) ν2 (α+β ) >0 U1 = −ν 1 ∓ (ii) ν2 c1 sinh 4ν (α+β ) ζ + c2 cosh 4ν (α+β ) ζ c1 cosh 4ν ζ + c2 sinh 4ν ζ ( α+β ) ( α+β , (3.9) ) 0 c1 sinh U4 = − αµ + β µ − ν ± 4ν c1 cosh (ii) ν2 (α+β ) ζ + c2 cosh 4ν ζ (α+β ) + c2 sinh 4ν (α+β ) ζ 4ν (α+β ) , (3.12) , (3.13) 4ν ζ (α+β ) 0 (4.4) (ii) and ν+1 ν U2 = a0 +2 U (ζ) = i=0 G G ν+1 ζ 2ν + c2 cosh ν+1 ζ 2ν c1 cosh ν+1 ζ 2ν + c2 sinh ν+1 ζ 2ν −c1 sin ν+1 ζ 2ν + c2 cos i (4.5) , (4.9) U1 = a0 + 524 c1 sinh c1 cosh ν ζ ν ζ + c2 cosh + c2 sinh Conclusion (5.6) ν ζ ν ζ , (5.9) We extend the GG -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 GG 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 GG -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 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