Travelling wave solutions for a CBS equation in dimensions

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1 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, 8 Travelling wave solutions for a CBS equation in + dimensions MARIA LUZ GANDARIAS University of Cádiz Department of Mathematics PO.BOX, Puerto Real, Cádiz SPAIN marialuz.gandarias@uca.es MARIA SANTOS. BRUZÓN University of Cádiz Department of Mathematics PO.BOX, Puerto Real, Cádiz SPAIN matematicas.casem@uca.es Abstract: One of the more interesting solutions of the ( + )-dimensional integrable Calogero-Bogoyavlenskii- Schiff (CBS) equation are the soliton solutions. We previously derived a complete group classification for the CBS in ( + )-dimensions written as a system of partial differential equations. We now consider the classical Lie symmetries of the CBS equation written in a potential form. We obtain travelling-wave reductions variable velocity depending on the form of an arbitrary function. The corresponding solutions of the ( + )-dimensional equation involve arbitrary smooth functions. Consequently the solutions exhibit a rich variety of qualitative behaviours. Indeed by making adequate choices for the arbitrary functions, we exhibit periodic and solitary waves. Key Words: Symmetries, partial differential equation, exact solutions Introduction The study of higher-dimensional integrable systems is one of the main themes in integrability theory. Several models in the context of ( + )-dimensional equations, i.e. equations two spatial and one temporal variables, which are integrable have been developed by Toda and Yu []. These equations has been recently derived by using a method proposed by Calogero. That is, by modifying one of the operators of the Lax pair for ( + )-dimension. In this way from the Calogero-Bogoyavlenskii-Schiff CBS equation they obtain W t + W W z + W x x W z + W xxz = () where f = fdx. Although this ( + )-dimensional CBS equation () arises in a non-local form, it can be written as the system of differential equations: v x u z = u x v + u xxz + u t + uu z = () In a previous paper we applied the Lie group method of infinitesimals transformations to system (). By using this method we bring out the unexplored invariance properties and similarity reduced systems of ( + ) partial differential equations (PDE s) of the above system (). First we obtain a point transformation group which leaves the system () invariant. In order to find all invariant solutions respect to s- dimensional subalgebras, it is sufficient to construct invariant solutions for the optimal system of order s. The set of invariant solutions obtained in this way is called an optimal system of invariant solutions. We only consider one-parameter subgroups. For further details [9]. It was shown in [] that the ( + )-dimensional CBS equation which can be written in the potential form u tx + u x u xz + u xx u z + u xxxz =, () W = u x, admits a Lax representation and is integrable by the one-dimensional inverse scattering transform. In [] Bogoyavlenskii proved that an equation equivalent to Eq. () has an overturning soliton By using the classical Lie method, we derive exact solutions for the ( + )-dimensional integrable potential breaking soliton equation (BS) (), as well as for the ( + )-dimensional integrable generalization of the CBS equation (). Some of these solutions are soliton solutions, localized on a curve and that decays exponentially apart from the curve. Lie symmetries. In this section we perform Lie symmetry analysis for the (+)-dimensional CBS in potential form (). Let us consider a one-parameter Lie group of infinitesimal ISSN: 797 ISBN:

2 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, 8 transformations in (x, t, z, u) given by τ, α and β must satisfy the following equations x = x + εξ(x, z, t, u) + O(ε ), z = z + εη(x, z, t, u) + O(ε ), t = t + ετ(x, z, t, u) + O(ε ), u = u + εφ(x, z, t, u) + O(ε ), () where ε is the group parameter. Then one requires that this transformation leaves invariant the set of solutions of the equation (). This yields to the overdetermined, linear system of eighteen equations for the infinitesimals ξ(x, z, t, u),η(x, z, t, u), τ(x, z, t, u) and φ(x, z, t, u, v). The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form v = ξ x + η z + τ t + φ u. () Having determined the infinitesimals, the symmetry variables are found by solving the invariant surface condition Φ ξ u x + η u z + τ u φ =. (6) t Applying the classical method to the () yields a system of eighteen equations ξ z = ξ u = η x = η u = τ x = τ z = τ u = φ ux = φ uz = (7) φ uu = ξ xx = φ xx = φ tx = φ z ξ t = φ x τ t = η t τ z ξ x = φ xz + φ tu ξ tx = φ u + τ t η z ξ x = By solving this system we get that ξ = ξ(x, t), η = η(z, t), τ = τ(t) φ = α(t)u + β(x, z, t) where ξ, η, β tx =, β xx =, β z ξ t =, ξ tx + β xz + α t =, (8) ξ xx =, β x τ t =, τ t η z ξ x =, τ t η z ξ x + α =. (9) Solving this system lead to a six-parameter Lie group. Associated this Lie group we have a Lie algebra which can be represented by the generators, these generators are : v = t, v = z, v = t z + x u, v = z z + t t v = tx x + tz z + t + (xz tu) t u v 6 = x x z z u u, and the infinite-dimensional v α = α(t) x + α (t)z u, v β = β(t) u. Optimal systems and reductions In order to construct the one-dimensional optimal system, following Olver, we construct the commutator table and the adjoint table which shows the separate adjoint actions of each element in v i, i =... 6, as it acts on all other elements. This construction is done easily by summing the Lie series. The corresponding generators of the optimal system ISSN: 797 ISBN:

3 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, 8 of subalgebras are. Equation E, admits the following symmetries av + bv 6, v, av + bv 6, av + bv, av + bv. where a R, and b R are arbitrary. Our aim in this paper is to use the theory of symmetry reductions to find traveling-wave solutions for the (+ )-dimensional CBS equation. In order to obtain these solutions, we consider the following reductions arising from translations and the infinite-dimensional vector field, i.e., v, v, v α and v β. Reduction By using the generator v + λv +v α + v β we obtain the similarity variables and similarity solution z = x α(t)dt, z = z λt, () u = z α(t) + λ (tα (t) β(t))dt + h(z, z ) () and the PDE E h z z h z + h z z z z + h z h z z λh z z = () Reduction By using the generator v +v α +v β we obtain the similarity variables and similarity solution z = x z α(t)dt, z = t, () u = α(t)z + β(t)zdt + h(z z ) () and the PDE E h z z β(z ) + h z z =. () v = z, v = h, v = z z + (λz h) h, v δ = δ(z ) z, (6) where γ(z ) is an arbitrary function of z. By using v + v γ we obtain the similarity variable and similarity solutions w = z k δ(z ), (7) h = k δ(z ) + g(w), where δ(z ) = dz and the autonomous γ(z ) ODE k (g + (g ) λg ) k g k =. (8) Setting g = y, can be reduced to the following second order autonomous ODE k (y + y λy) k y k =. (9) By multiplying by y and integrating once we get (y ) + y ( k k + λ)y k k y =. () Setting y = θ n we get (θ ) = n θn+ + ( k k + λ n )θ k k n θ n. For n = we get () Symmetry reductions to ordinary differential equations (ODE s) The reduced PDE s in ( + ) variables admit symmetries which lead to further reductions to ODE s, we shall use again the techniques of Lie group theory. (θ ) = θ + ( k k λ)θ + k k. For n = we get (θ ) = θ(k k λ)θ + k k θ. () () ISSN: 797 ISBN:

4 Some travelling wave solutions In the following we present some explicit solutions of the second order ODE s as well as the corresponding travelling solution of the ( + ) BS equation (),as well as for the ( + ) CBS equation (). Equations () and () can be integrated in terms of elliptic functions. Consequently, after making the change of variables y = θ n, some exact solutions for (9) and () are y = sn (kw p) () AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, 8 k p + =, k = k k, k = k ( k λ) () y = sn (kw p) (6) k + =, k = k pk, k = k ( + p λ) (7) y = cn (kw p) (8) k (p + p) =, k = k (p ) p, + p k = k [(8λ 9)p + (λ )p + 6] p. + p (9) y = cn (kw p) () k (p ) + p =, k = k (p ) (p, ) k = k [p + (λ )p p + 6 λ)] p. () Clearly any of the rational, hyperbolic or trigonometric degenerations of the Jacobi sn functions also give solutions. In particular, solitary waves result: Setting in () k =, and p = we get p y = tan ( w ), h = z tan ( w ) + w (6) By considering (6), and the corresponding symmetry reductions () and (7) we obtain that a solution for the breaking soliton equation and for the CBS equation in ( + ) dimensions can be written as W = -sec ( w ) + d + (7) u = tan( w ) + d + w + g(t) (8) w = (x k δ(z )), z = z λt,. (9) In figure we can see solution (7). λ =, δ(z λt) = (z t ) for t =. k (p ) =, k = k p(p ) p, k = k [(λ )p + p λ + )] p. () y = dn (kw p) () k (p + ) p =, k = k (p ), p + k = k [p + p + (λ 6)p + (λ 6)]. p + () y = dn (kw p) () Figure : Solution (7) δ = (z t ), t =. Setting in (6) k = and p = we get y = cosech ( w ), h = coth ( w ). () Setting in (6) k = and p = we get - ISSN: ISBN:

5 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, Figure : Solution (8) δ = (z t ), t =, g(t) =. Figure : Soliton solution () t= δ = z t, t =. y = cot ( w ), h = cot ( w ). () Setting in (8) k = p and p = we get + p y = sech ( w ), h = tanh ( w ). () By considering (), and the corresponding symmetry reductions () and (7) we obtain that solutions for the breaking soliton equation BS and for the CBS equation in ( + ) dimensions can be written as W = sech ( w ) () u = tanh( w ) () w = (x k δ(z ), δ(z ) = z, z = z λt,. () In figure we can see solution () and in figure we can see () λ =, δ(z λt) = (z t ) for t =. In figure we can see solution () λ =, δ(z λt) = (z t ) for t = which is a curve soliton. In figure 6 we can see solution () and in figure 7 we can see () λ =, δ(z λt) = sin(z t ) for t =. In figure 8 we can see solution () k = p =, w = x δ,λ =, δ(z λt) = (z t ) for t = which is a periodic solution. In figure 9 we can see solution () k =, p =, w = x δ, λ =, δ(z λt) = sin(z t ) for t = which is a periodic solution. Figure : Kink solution () δ = z t, t =, g(t) =. 6 Conclusions In this work we have discussed symmetry reductions and similarity variables and similarity solutions for the (+)-dimensional integrable BS equation as well as for the ( + )-dimensional integrable generalization of the CBS equation. By using the classical Lie method, we obtained solutions which have a rich variety of qualitative behaviours. This is due to the freedom in the choice of the arbitrary function δ(z λt). We have obtained soliton solutions. Because of these arbitrary functions are included in the single soliton solution () the solution is localized on a curve and the curve may have quite a free form. References: [] Abramowitz Handbook of Mathematical Functions (Dover, New York 97). [] Bogoyavlenskii O I 99 Math. USSR Izvestiya [] Bogoyavlenskii O I 99 Math. USSR Izvestiya ISSN: ISBN:

6 AMERICAN CONFERENCE ON APPLIED MATHEMATICS (MATH '8), Harvard, Massachusetts, USA, March -6, Figure : Curve soliton δ = (z t ), t = Figure 7: Kink solution () δ = sin(z t ), t =, g(t) = Figure 6: Soliton solution () t= δ = sin(z t ), t =. [] Bogoyavlenskii O I 99 Math. USSR Izvestiya 6 9 [] F. Calogero Lett. Nuovo Cimento.,,, 97. [6] B. Champagne, W. Hereman and P. Winternitz, Comp. Phys. Comm., 66, 9, Figure 8: Solution () w = x δ, δ = z t, f = t, k =, p =, t =.. [7] E.L. Ince Ordinary Differential Equations (Dover, New York, 96). [8] S. Lou J. Math Phys.,9,, 998. [9] P.J. Olver Applications of Lie Groups to Differential Equations (Springer, Berlin, 986). [] L.V. Ovsiannikov Group Analysis of Differential Equations, (Academic Press, New York, 98). [] K. Toda S. Yu J. Math. Phys,, 77,. [] J. Weiss J.M. Tabor and G. Carnevale, J. Math Phys.,,, Figure 9: Solution () w = x δ, δ = sin(z t ), f = t, k =, p =, t =. - - ISSN: ISBN: