New solutions for a generalized Benjamin-Bona-Mahony-Burgers equation

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1 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March -6 8 New solutions for a generalized Benjamin-Bona-Mahony-Burgers equation MARIA S. BRUZÓN University of Cádiz Department of Mathematics PO.BOX Puerto Real Cádiz SPAIN matematicas.casem@uca.es MARIA L. GANDARIAS University of Cádiz Department of Mathematics PO.BOX Puerto Real Cádiz SPAIN marialuz.gandarias@uca.es Abstract: In this paper we make a full analysis of the symmetry reductions of a generalized Benjamin-Bona- Mahony- Burgers equation (BBMB) by using the classical Lie method of infinitesimals. The functional forms for which the BBMB equation can be reduced to ordinary differential equations by classical Lie symmetries are obtained. We have used the symmetry reductions as a basis for deriving new exact solutions that are invariant with respect to the symmetries. The exact solutions include compactons solitons kinks and antikinks. Key Words: Symmetries partial differential equation exact solutions Introduction In this paper we solve a group classification problem for equation u t u xxt αu xx βu x (g(u)) x = () where u(x t) represents the fluid velocity in the horizontal direction x α is a positive constant β IR and g(u) is a C -smooth nonlinear function []. We study the functional forms g(u) for which equation () admits the classical symmetry group. When g(u) = uu x with α = and β = equation () is the alternative regularized long-wave equation proposed by Peregrine [] and Benjamin []. Equation () feature a balance between nonlinear and dispersive effects but takes no account of dissipation. In the physical sense equation () with the dissipative term αu xx is proposed if the good predictive power is desired such problem arises in the phenomena for both the bore propagation and the water waves. In [] Khaled-Momani-Alawneh implemented the Adomian s decomposition method for obtaining explicit and numerical solutions of the BBMB equation (). By applying the classical Lie method of infinitesimals Bruzón and Gandarias [3] obtained for a generalization of a family of BBM equations many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic functions. Tari and Ganji [] have applied two methods for solving nonlinear differential equations known as variational iteration and homotopy perturbation methods in order to derive approximate explicit solutions for () with g(u) = u. El Wakil Abdou Hendi [] used the expfunction method with the aid of symbolic computational system to obtain the generalized solitary solutions and periodic solutions for () with g(u) = u. In [6] Fakhari et al. solved the resulting nonlinear differential equation by homotopy analysis method to evaluate the nonlinear equation () with g(u) = u α = and β =. The classical theory of Lie point symmetries for differential equations describes the groups of infinitesimal transformations in a space of dependent and independent variables that leave the manifold associated with the equation unchanged [7 8 9]. The fundamental basis of this method is that when a differential equation is invariant under a Lie group of transformations a reduction transformation exists. For partial differential equations (PDEs) with two independent variables a single group reduction transforms the PDE into a ordinary differential equations (ODEs) which are generally easier to solve. Since the relevant calculations are usually rather laborious they can be conveniently carried out by means of symbolic computations. In our work we used the MACSYMA program symmgrp.max []. Most of the required theory and description of the method can be found in [8 9]. The structure of the work is as follows: In Sec. we study the Lie symmetries of equation () we find the functions g(u) for which we obtain the Lie group of point transformations admitted by the corresponding equation its Lie algebra. We obtain the sym- ISSN: ISBN:

2 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March -6 8 metry reductions similarity variables and the reduced ODEs. In Sec. 3 we derive for some functions g(u) exact solutions which describe solitons kinks antikinks and compactons. Finally in Sec. some conclusions are presented. Classical Symmetries To apply the Lie classical method to equation () we consider the one-parameter Lie group of infinitesimal transformations in (x t u) given by where pr (3) v is the third prolongation of the vector field (). Hence we obtain the following ten determining equations for the infinitesimals: τ u = τ x = ξ u = ξ t = η uu = ατ t η tu = η ux ξ xx = η uxx ξ x = η x g u αη xx βη x η txx η t = αξ xx g u ξ x βξ x g u τ t βτ t ηg uu αη ux η tux =. (7) x = x ɛξ(x t u) O(ɛ ) () t = t ɛτ(x t u) O(ɛ ) (3) u = u ɛη(x t u) O(ɛ ) () where ɛ is the group parameter. We require that this transformation leaves invariant the set of solutions of equation (). This yields to an overdetermined linear system of equations for the infinitesimals ξ(x t u) τ(x t u) and η(x t u). The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form v = ξ(x t u) x τ(x t u) η(x t u) t u. () The functions u = u(x t) which are invariant under the infinitesimal transformations v are in essence solutions to an equation arising as the invariant surface condition : η(x t u) ξ(x t u) u τ(x t u) u =. (6) x t The symmetry variables are found by solving the invariant surface condition. The reduction transforms the PDE into ODEs. We consider the classical Lie group symmetry analysis of equation (). The set of solutions of equation () is invariant under the transformation ()-() provided that pr (3) v( ) = when = From system (7) ξ = ξ(x) τ = τ(t) and η = γ(x t)u δ(x t) where α β ξ τ γ δ and g satisfy γ t α τ t = γ x ξ xx = γ xx ξ x = α γ x γ tx g uu u γ αξ xx g u ξ x β ξ x g u τ t β τ t δ g uu = αu γ xx g u u γ x β u γ x u γ txx u γ t δ x g u α δ xx βδ x δ txx δ t =. From (8) we obtain γ = e x 8 ξ = (k k 3 ) e x 8 ( (k k 3 ) e x (k 8ατ) e x k k 3 ) (k k 3 ) e x 8 (8) k k and α β τ δ and g are related by the following conditions: ((g u β α) k (g u β α) k 3 ) ue x ( ατ t u δ x (g u β) αδ xx δ txx δ t ) e x ((g u β α) k ( g u β α) k 3 ) u = (9) ((g uu k g uu k 3 ) u ( g u β) k (g u β) k 3 ) e x ((g uu k 8αg uu τ) u 8g u τ t 8 β τ t 8 δg uu ) e x ( g uu k 3 g uu k ) u ( g u β) k (g u β) k 3 =. () Solving system (9)-() we obtain that if g is an arbitrary function the only symmetries admitted by () are ξ = k τ = k η = () ISSN: ISBN:

3 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March -6 8 which are defined by the group of space and time translations v = λ x v = µ t. Substituting () in the invariant surface condition (6) we obtain the similarity variable and the similarity solution z = µx λt () u(x t) = h(z). Substituting () into () we obtain λµ h αµh (βµ λ)h µh g h (h) =. (3) Integrating (3) once we get λµ h αµh (βµ λ)h µg(h) k =. () In the following cases equation () have extra symmetries: (i) If α = g(u) = βu a k a(n) (au b)n ξ = k τ = k t k 3 η = k (au b). an Besides v and v we obtain the infinitesimal generator v 3 = t t au b an u. v 3 : We obtain the reduction where h(t) satisfies z = x u = t n h(x) b a h kna n h n h h =. () Equation () does not admit Lie symmetries. making the change of variables y(s) = h (z) equation () becomes s = h(z) y y kna n s n y s =. v v 3 : The reduction is By z = x ln t u = t n h(z) b a. (6) The reduced ODE is nh h nh nka n h n h h =. (7) Equation (7) does not admit Lie symmetries. We can observe that for the reduction (6) we have that u(x t) = t b n h(x ln t ) a. This solution describes a travelling wave with decaying velocity v = t and decaying amplitude t n if n >. 3 Travelling wave solutions (ii) If α β and g(u) = au b where δ satisfy ξ = k τ = k η = δ(x t) αδ xx g u δ x βδ x δ txx δ t =. We do not considerer case (ii) because in this case equation () is a linear PDE. In order to determine solutions of PDE () that are not equivalent by the action of the group we must calculate the one-dimensional optimal system [8]. The generators of the nontrivial one-dimensional optimal system are the set µv λv v 3 v v 3. Since equation () has additional symmetries and the reductions that correspond to v and v have already been derived we must determine the similarity variables and similarity solutions corresponding to the generators v 3 and v v 3. If g is an arbitrary function the similarity variables are given by z = µx λt u = h so that u(x t) = h(z) = h(µx λt). Consequently the corresponding solutions of () are travelling-wave solutions. As the derivative of trigonometric hyperbolic and exponential functions can be expressed in terms of themselves we can choose g as an algebraic function of h so that the equation () admits the trigonometric functions (p sin q z p cos q z p tan q z p sinh q z p cosh q z p tanh q z) hyperbolic functions (p sn q (z m) p cn q (z m) p dn q (z m)) and exponential function (exp(z)) as solutions. In the following we present same exact solutions of equation () for k =. h(z) = p sin q (z) is solution of equation () for µ pq q λ h q α h µ q λ µ p q q λ h λ h q µ p q h q q h q β h. (8) ISSN: ISBN:

4 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March -6 8 Consequently an exact solution of equation () where g(u) is obtained substituting h by u in (8) is u(x t) = p sin q (µx λt). (9) For µ = λ = k k = p = q = the solution { sin (µx λt) x t π u(x t) = k x t > π k is a sine-type double compacton (that is solution which has two peaks see Fig. ) Figure : Solution () for µ = λ = k k = p = and q =.... Figure : Solution (9) for µ = λ = k k = p = and q =. µ pq q λ h q α a solution of () is p q h q q h q h µ q λ µ p q q λ h λ h q µ β h h(z) = p cos q (z). So an exact solution of equation () is... () u(x t) = p cos q (µx λt) () where g(u) is obtained substituting h by u in (). For µ = λ = k k = p = and q = the solution u(x t) = { cos (µx λt) x t π k x t > π k is a compacton solution with a single peak (see Fig.). µ pq qλ( q) h q h λ µ α pq q h q α h h q µ qλ(q) p q q q p q β h h µ q λ () a solution of () is h(z) = p tan q (z). So an exact solution of equation () where g(u) is obtained substituting h by u in () is u(x t) = p tan q (µx λt). (3) µ pq qλ( q) α h q p q h q q h q a solution of equation () is h µ q λ h λ µ β h h(z) = p sinh q (z) () So an exact solution of equation () where g(u) is obtained substituting h by u in () is u(x t) = p sinh q (µx λt). () µ pq qλ(q ) α h q p q h q q a solution of equation () is h µ q λ h λ µ β h h q (6) h(z) = p cosh q (z). Consequently an exact solution of equation () where g(u) is obtained substituting h by u in (6) is u(x t) = p cosh q (µx λt). (7) For λ = µ = p = and q = the solution u(x t) = sech (x t) ISSN: ISBN:

5 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March Figure 3: Solution (7) for λ = µ = p = and q =. Figure : Solution (9) for µ = λ = p = and q = 3. describes a soliton moving along a line with constant velocity (see Fig.3). describes an anti-kink solution (see Fig.). µ p q qλ( q) h q µqλ 3hµqλ pq hλ µ h µ q λ α pq q α h q β h h(z) = p exp(qz). Consequently an exact solution of equation () where g(u) is obtained substituting h by u in (3) is h(z) = p tanh q (z). Consequently an exact solution of equation () where g(u) is obtained substituting h by u in (8) is q For µ = λ = p = u(x t) = p exp[q(µx λt)]. (9) u(x t) = p tanh (µx λt). t u(x t) = tanh x µ p q qλ( q) h m describes a kink solution (see Fig.). µ q λ mµ qλ(m)(q) h m µ q λ hµλ pq h µ q α h h m µ q λ q λ βq p q h q a solution of equation () is. p q h q m q pq (3) h(z) = p snq (z m).. (3) and q = the solution (3) a solution of () is (8) a solution of equation () is h µ q λ λ α µ q β µ µ µ p q qλ(q ) -. Consequently an exact solution of equation () where g(u) is obtained substituting h by u in (3) is u(x t) = p snq (µx λt m). Figure : Solution (9) for µ = λ = p = q =. and For µ = λ = p = q = and m =.996 the solution u(x t) = sn(x t.996) For µ = λ = p = and q = 3 the solution u(x t) = tanh ISSN: t x (33) shows a stable nonlinear nonharmonic oscillatory periodic wave (see Fig.6). 63 ISBN:

6 Figure 6: Solution (33) for µ = λ = p = q = and m =.996. AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8) Harvard Massachusetts USA March -6 8 (m ) µ p q q λ h q h q mµ q λ(m)(q) p q h µ q λ ( m ) µ p q q λ β h h q h m µ q λ( q) h m µ q λ h λ µ α p q h q a solution of equation () is m p q p q h q m q h q p q h(z) = p cn q (z m). (3) Consequently an exact solution of equation () where g(u) is obtained substituting h by u in (3) is Conclusion u(x t) = p cn q (µx λt m). In this paper we have seen a classification of symmetry reductions of a generalized Benjamin Bona Mahony Burgers equation depending on the values of the constants α and β and the function g(u) by making use of the theory of symmetry reductions in differential equations. We have found the functions g(u) for which we have obtained the Lie group of point transformations. We have constructed all the invariant solutions with regard to the one-dimensional system of subalgebras. Besides the travelling wave solutions we have found new similarity reductions for this equation. We have constructed all the ODEs to which () is reduced. We have obtained for some functions many exact solutions which are solitons kinks anti-kinks and compactons. Acknowledgements: The support of DGICYT project MTM6-3 Junta de Andalucía group FQM and project P6-FQM-8 are gratefully acknowledged. References: [] K. Al Khaled S. Momani and A. Alawneh Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations Appl. Math. Comput. 7 pp [] T. B. Benjamin J. L. Bona and J. J. Mahony Model Equations for Long Waves in Nonlinear Dispersive Systems Phil. Trans. R. Soc. A 7 97 pp [3] M. S. Bruzón and M. L. Gandarias Symmetry for a Family of BBM Equations Preprint. [] B. Champagne W. Hereman and P. Winternitz The computer calculations of Lie pont symmetries of large systems of differential equations Comp. Phys. Comm pp [] S. A. El-Wakil M. A. Abdou and A. Hendi New periodic wave solutions via Exp-function method Phys. Lett. A Preprint. [6] A. Fakhari G. Domairry and Ebrahimpour Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution Phys. Lett. A pp [7] N. Kh. Ibragimov Transformation Groups Applied to Mathematical Physics Reidel Dordrecht 98. [8] P. J. Olver Applications of Lie Groups to Differential Equations Springer Verlag Berlin Heidelberg New York Tokyo 993. [9] L. V. Ovsyannikov Group Analysis of Differential Equations Academic New York 98. [] D. H. Peregrine Calculations of the development of an undular bore J. Fluid Mech. 996 pp [] H. Tari and D. D. Ganji Approximate explicit solutions of nonlinear BBMB equations by Hes methods and comparison with the exact solution Phys. Lett. A pp. 9-. ISSN: ISBN: