On Certain New Exact Solutions of the (2+1)-Dimensional Calogero-Degasperis Equation via Symmetry Approach

Transcription

1 ISSN (print), (online) International Journal of Nonlinear Science Vol.13(01) No.4,pp On Certain New Exact Solutions of the (+1)-Dimensional Calogero-Degasperis Equation via Symmetry Approach Anupma Bansal 1, R.K.Gupta 1 Department of Mathematics D.A.V.College for Women, Ferozepur (Punjab), INDIA School of Mathematics and Computer Applications Thapar University, Patiala (Punjab), INDIA (Received 9 October 011, accepted 15 June 01) Abstract: In this paper, the idea of Lie Group method is used to obtain exact solutions of (+1)-dimensional Calogero-Degasperis (CD) equation. The equation is reduced to (1+1)-dimensional nonlinear equation by applying the Symmetry Group method and again reduced equation is applied by the same method to obtain new solutions. Keywords: Calogero-Degasperis equation; Symmetry group method; Exact solutions 1 Introduction In this paper we consider the (+1)-dimensional CD or breaking soliton equation in the form ψ xt 4ψ x ψ xy ψ y ψ xx + ψ xxxy = 0. (1) Equation (1) was first eastablished by Calogero and Degasperis [1, ] and is used to describe the (+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. The mathematical interest of this equation stems from the fact that it is, in a well defined sense, the generic member of a class of integrable partial differential equations [3], associated with certain infinite-dimensional Lie algebras and groups [4]. Finding special solutions and investigating the corresponding properties of solutions are very important in both practice and theory for understanding these problems. The Hamiltonian structure and the Lax pair of equation (1) have been given by Li [5]. Multi-soliton solutions and algebra-geometric solutions were also found in [6]. In [7], Ma et al. present a general class of Riemann theta function solution to two similar breaking soliton equations. In [8], using computerized symbolic computation, new families of soliton-like solutions are obtained for (+1)-dimensional breaking soliton equations using an ansatz and these solutions contain traveling wave solutions that are of important significance in explaining some physical phenomena. With the help of symbolic computation, sixteen kinds of new special exact soliton-like solutions of (+1)-dimensional breaking soliton equation are obtained by further generalized projective Riccati equation method in [9]. Zhang and Meng [10] derived a general variable separation solution of the (+1)-dimensional breaking soliton system and two classes of novel localized coherent structures like both multipeakon-antipeakon solution and multi compacton-anticompacton solutions are found by selecting appropriate functions. In the study of the (+1)-dimensional nonlinear physical models, much effort has been focused on the single valued localized excitations, such as solitoffs, dromions, rings, lumps, breathers, instantons, peakons, compactons, localized chaotic and fractal patterns and in this direction Zhang et al. [11] obtained folded solitary waves and foldons in the (+1)- dimensional breaking soliton equation. Sheng Zhang [1] obtained many new and more general exact non-traveling wave and coefficient function solutions including soliton-like solutions, trigonometric function solutions, exponential solutions and rational solutions using a generalized auxiliary equation method. Radha and Lakshmanan [13] studied the existence of Corresponding author. address: anupma51@yahoo.co.in Copyright c World Academic Press, World Academic Union IJNS /69

2 476 International Journal of NonlinearScience,Vol.13(01),No.4,pp Dromion like structures in the (+1)-dimensional breaking soliton equation. Quan [14] used the new idea of a combination of Lie group method and homoclinic test tecnique to seek non-traveling wave solutions. Nonlinear phenomena [15] play a crucial role in applied mathematics and physics. Calculating exact and numerical solutions of nonlinear equations in mathematical physics play an important role in soliton theory. Mathematical techniques which generate a wide range of solutions and applicable to all types of nonlinear differential equations are few. The group theoretic techniques can be categorized in this class and generally it produces a variety of exact solutions. Lie s method [16-19] is an effective and simplest method among group theoretic techniques and a large number of equations are solved with the aid of this method. Some recent and important contributions are in [0-5]. The paper has been organized as follows. In Section, with the aid of Lie group theory we reduce the CD equation to partial differential equation (PDE) in two variables and the reduced PDE is again operated by Lie classical method to get the ordinary differential equations (ODE) and further solutions of ODEs yields another family of solutions involving upto two arbitrary functions of variable t. Some concluding remarks are given in section 3. Lie Symmetry Analysis The purpose of this article is to study the exact solutions of the CD equation with the aid of symmetry group, i.e., the Lie group of transformations acting on the independent variables (x, y, t) and the dependent variable ψ. Quan [14] and Tian [5] have studied the equation (1) and obtained solutions by assuming particular values for arbitrary constants and functions in symmetries. But, here we are discussing the reductions and solutions of CD equation for the general case and solutions obtained by us are more better and general. Algorithms for calculating the invariance group of an equation or system of equations are well known and for a good summary of this we refer to [16, 19]. The method mainly consists of following steps: 1. Find the Lie group of point transformations ψ = ψ + ɛη(x, y, t, ψ) + O(ɛ ) x = x + ɛξ(x, y, t, ψ) + O(ɛ ) y = y + ɛφ(x, y, t, ψ) + O(ɛ ) t = t + ɛτ(x, y, t, ψ) + O(ɛ ), () which leaves the system (1) invariant. In other words, the transformations are such that if ψ is a solution of equation (1), then ψ is also a solution. The method for determining the symmetry group of (1) consists of finding the infinitesimals η, ξ, φ and τ, which are functions of x, y, t, ψ.. Assuming that the system (1) is invariant under the transformations (), we get the following relations from the coefficients of the first order of ɛ: η xt 4ψ x η xy 4ψ xy η x ψ y η xx η y ψ xx + η xxxy = 0, (3) where η x, η y, η xx, η xy, η xt and η xxxy are extended (prolonged) infinitesimals acting on an enlarged space that includes all derivatives of the dependent variables ψ x, ψ y, ψ xx, ψ xy, ψ xt and ψ xxxy (for more details the readers can refer to [19]). The infinitesimals are determined from invariance condition (3), by setting the coefficients of different differentials equal to zero. We obtain a large number of PDEs in η, ξ, φ and τ that need to be satisfied. The general solution of this large system provides following forms for the infinitesimal elements η, ξ, φ and τ: η = pxy cx ptψ ψ Q (t)y + R(t) ξ = ptx + x + φ = pyt + τ = pt + 4qt, (4) where c, p and q are arbitrary constants,, R(t) are arbitrary functions of t and prime denotes time derivative. We will perform the symmetry reduction using a general form of symmetries. To obtain the symmetry reduction of equation (1), we have to solve the following characterstic equation: dx ξ = dy φ = dt τ = dψ η. (5) IJNS for contribution: editor@nonlinearscience.org.uk

3 Anupma Bansal, R.K. Gupta: On Certain New Exact Solutions of the (+1)-Dimensional Calogero-Degasperis Equation via Symmetry Approach 477 On solving the equations (4) and (5) we have ψ(x, y, t) = χ(x, y, t) + e H(ρ, σ), (6) where χ(x, y, t) = + e (pt +4qt) 4 1 pxyt (pt +4qt) ( R(t) (pt +4qt) 3 4 q cxt (pt +4qt) y (pt +4qt) qye (pt +4qt) 3 4 q )dt). ( )dt) (7) ρ = x e σ = y e (pt +4qt) 4 1 ( ( )dt), )dt). Here ψ(x, y, t) is a solution of the CD equation if and only if H(ρ, σ) satisfies the equation (8) H ρρρσ H σ H ρρ 4H ρ H ρσ ρh ρρ 4qH ρ = 0. (9) Apply Lie Symmetry method to the equation (9) as already applied. In this case we get symmetries as under: η 1 = a 1 H qa σ + a 3 ξ 1 = a 1 ρ + a τ 1 = a 1 σ + a 4, (10) where η 1, ξ 1 and τ 1 are infinitesimals corresponding to H, ρ and σ, respectively and a 1, a, a 3 and a 4 are arbitrary constants. Thus the symmetry algebra admitted by (10) is V 1 = ρ ρ σ σ H H V = ρ qσ H V 3 = σ V 4 = H. In general, there are infinite number of subalgebras of this Lie algebra formed from any linear combination of generators V j ; j = 1,, 3, 4 and to each sublagebra one can get the reduction using characterstic equations: dρ ξ 1 = dσ τ 1 (11) = dh η 1. (1) However, this problem becomes manageable by recognizing that if two algebras are similar, i.e. they are connected to each other by a transformation from the symmetry group, then their corresponding invariant solutions are connected to each other by the same transformation. Therfore, it is sufficient to put all similar subalgebras into one class and select a representative from each class. The set of all these representatives is called an optimal system (For details refer to Olver[19] and Ovsiannikov[16]). The optimal system for (11) consists of following vector fields: (i) V 1, (ii) V, (iii) V + V 3, (iv) V 3, (v) V 4. (13) Because corresponding to vector field V and V 4, PDE (9) is identically satisfied, that s why we confine ourselves to remaining vector fields. In Table 1, we now list the similarity variable, form and the reduced ODEs corresponding to the optimal system. 1. Vector field V 1 The reduced ODE has the following solutions (i) F (ζ) = C 0 (ii) F (ζ) = C 0 qζ (iii) F (ζ) = q coth(c 0 + qζ) (iv) F (ζ) = q tanh(c 0 + qζ), (14) IJNS homepage:

4 478 International Journal of NonlinearScience,Vol.13(01),No.4,pp Table 1: Similarity Reductions of PDE (9) to ODEs Essential Similarity Similarity ReducedODEs f ields variable(ζ) solution(h) V 1 ρσ 1 σ 1 F (ζ) ζf (ζ) + 4F (ζ) F (ζ)f (ζ) 6ζF (ζ)f (ζ) 8(F (ζ)) 4qζF (ζ) 8qF (ζ) = 0 V + V 3 ρ σ σ + F (ζ) F (ζ) + ζf (ζ) 6F (ζ)f (ζ) + 4qF (ζ) = 0 V 3 ρ F (ζ) ζf (ζ) + F (ζ) = 0 where C 0 is arbitrary constant. The solution (14) leads by back substitution to the solution of equation (1) of the form (i) ψ(x, y, t) = χ(x, y, t) + C0e (y e ( )dt)) 1 (ii) ψ(x, y, t) = χ(x, y, t) + e (C 0 q((x e ( ( (iii) ψ(x, y, t) = χ(x, y, t) + e )dt)) 1 )) (y e ( q coth(c 0 + q((x e ( pt dt +4qt )dt)) 1 ))) (iv) ψ(x, y, t) = χ(x, y, t) + e ( ( q tanh(c 0 + q((x e ( )dt)) 1 ))), where χ(x, y, t) is given by equation (7). ( ( pt dt +4qt )dt))(y e (y e (y e ( pt dt +4qt )dt))(y e ( pt dt +4qt )dt))(y e )dt)) 1 )dt)) 1 )dt)) 1 (15) Analysis of Solutions: By the analysis of solutions obtained corresponding to vector field V 1, we conclude that solutions (15) depend upon four constants p, q, c, C 0 and two arbitrary functions, R(t) of time. Depending upon these constants and functions of time, we obtain certain periodic and bell profile solutions. For p = q = (t +4t) 5 4 1, c = C 0 = 0 and = e ( 1 4 log(t+) 1 4 log(t)), R(t) = 0, with y = sin(x) solution (15 (iv)) takes the form of periodic solutions as shown in Fig. 1. By taking the same considerations for constants and arbitrary functions, for y = cos(x) solution (15 (iii)) takes the form of bell profile solutions and with = 0, y = x sin(x) solution (15 (iv)) behaves as kink wave as shown in Fig. and Fig. 3 respectively.. Vector field V + V 3 For q 0, the reduced ODE corresponding to vector field V + V 3 has trivial solutions. For q = 0, integrating IJNS for contribution: editor@nonlinearscience.org.uk

5 Anupma Bansal, R.K. Gupta: On Certain New Exact Solutions of the (+1)-Dimensional Calogero-Degasperis Equation via Symmetry Approach 479 Figure 1: The periodic solution (15)(iv) for p = q = 1, c = C 0 = 0 and = (t +4t) 5 4 e ( 1 4 log(t+) 1 4 y = sin(x) log(t)), R(t) = 0, Figure : The bell profile solution (15)(iii) for p = q = 1, c = C 0 = 0 and = (t +4t) 5 4 e ( 1 4 log(t+) 1 4 y = cos(x) log(t)), R(t) = 0, Figure 3: The kink wave solution (15)(iv) for p = q = 1, c = C 0 = 0 and = 0, R(t) = 0, y = x sin(x) equation once, multiplying integrated equation with F (ζ), again integrating and on substituting F (ζ) = J(ζ), we get J (ζ) J(ζ) 3 + K 1 J(ζ) + K = 0, (16) where K 1, K are arbitrary constants. Solving equation (16), we get J (ζ) = (1/ /3 ζ + C 1, K ) 3 3 1, K, (17) where C 1 is arbitrary constant and denotes the WeirstrassP function. 3. Vector field V 3 The reduced ODE has the following solution: F (ζ) = C 1 + C ζ, (18) where C 1, C are arbitrary constants and solution (18) corresponds to the following solution of Eq. (1): ψ(x, y, t) = χ(x, y, t) + e where χ(x, y, t) is given by (7). C 1 + C x e (pt 1 ( +4qt) 4 (pt 5 +4qt) 4 )dt), (19) 3 Conclusions In this paper, we have presented the similarity reductions of Eq. (1) by classical Lie group method and some new solutions are given, including the periodic solutions, bell profile solutions etc. The discussions on our results are as follows: Eq. (1) has been reduced to (1+1)-dimensional nonlinear Eq. (9) by means of the classical Lie group method. The (1+1)-dimensional reduced PDE is further reduced to ODEs using several transformations and reduced ODEs are studied to get several exact general solutions which corresponds to solutions of Eq. (1). IJNS homepage:

6 480 International Journal of NonlinearScience,Vol.13(01),No.4,pp The solutions obtained by us are such that one can choose the arbitrary functions, R(t) along with various other parameters, in a suitable manner, to simulate physical situations governed by Eq. (1) to obtain particular solutions having desired features (as shown in Figures 1-3). To understand the solutions well, we plot the graphs of the solution surfaces with some special parameters. As shown in Figures 1, and 3 the periodic solutions, bell profile solutions and kinky wave solutions can be obtained through solutions (15). Some solutions of the CD equation which are already in literature can also be recovered from our general solutions and general solutions obtained by us have not been reported earlier. The availability of mathematical computer software like Maple facilitates the tedious algebraic calculations. It is worth to mention here that the correctness of the solutions has been checked with the aid of software Maple. References [1] F. Calogero and A. Degasperis, Nonlinear Evolution Equation Solvable by the Inverse Spectral Transform. Nuovo Cimento B, 31 (1976):01. [] F. Calogero and A. Degasperis, Nonlinear Evolution Equation Solvable by the Inverse Spectral Transform-II. Nuovo Cimento B, 39 (1977):54. [3] T. Alagesan, Y. Chung and K. Nakkeeran, Painlevé Test for the Certain (+1)-Dimensional Nonlinear Evolution Equations. Chaos, Solitons and Fractals, 6 (005): [4] V. Kac, Infinite Dimensional Lie Algebras. Birkhäuser, Boston [5] Y.S. Li, Differential Geometric Methods in Theoretical Physics. Proceeding of the XXI International Conference, Tianjin, China [6] X. Geng and C. Cao, Explicit Solutions of the (+1)-Dimensional Breaking Soliton Equation. Chaos, Solitons and Fractals, (004): [7] W. Ma, R. Zhou and L. Gao, Exact One-Periodic and Two-Periodic Wave Solutions to Hirota Bilinear Equations in +1 Dimensions. Modern Physics Letters A, 4 (009): [8] Z. Yan and H. Zhang, Constructing families of soliton-like Solutions to a (+1)-Dimensional Breaking Soliton Equation Using Symbolic Computation. Computers and Mathematics With Applications, 44 (00): [9] Z. Xie and H. Zhang, Symbolic Computation and Construction of Soliton-like Solutions for a (+1)-Dimensional Breaking Soliton Equation. Applied Mathematics and Computation, 16 (005): [10] J. Zhang and J. Meng, New Localized Coherent Structures to the (+1)-Dimensional Breaking Soliton Equation. Physics Letters A, 31 (004): [11] J. Zhang, J. Meng, C. Zheng and W. Huang, Folded Solitary Waves and Foldons in the (+1)-Dimensional Breaking Soliton Equation. Chaos, Solitons and Fractals, 0 (004): [1] S. Zhang, New Exact Non-traveling Wave and Coefficient Function Solutions of the (+1)-Dimensional Breaking Soliton Equations. Physics Letters A, 368 (007): [13] R. Radha and M. Lakshmanan, Dromion Like Structures in the (+1)-Dimensional Breaking Soliton Equation. Physics Letters A, 197 (1995):7-1. [14] X. Da-Quan, Symmetry Reduction and New Non-traveling Wave Solutions of (+1)-Dimensional Breaking Soliton equation. Communications in Nonlinear Science and Numerical Simulation, 15 (010): [15] W.F. Ames, Nonlinear Partial Differential Equations in Engineering. Academic, New York [16] L.V. Ovsiannikov, Group Analysis of Differential Equations. Academic, New York [17] G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations. Springer Verlag, New York [18] R.L. Anderson and N.H. Ibragimov, Lie-Bäcklund Transformations in Applications. SIAM, Philadelphia [19] P.J. Olver, Applications of Lie Groups to Differential Equations. Graduate Texts Math., Vol. 107, Springer Verlag, New York [0] M.L. Gandarias and M.S. Bruzon, Nonclassical Symmetry Reduction for an Inhomogeneous Nonlinear Diffusion Equation. Communications in Nonlinear Science and Numerical Simulation, 13 (008): [1] O. Bogoyavlenskij, Restricted Lie Point Symmetries and Reductions for ideal Magnetohydrodynamics Equilibria. Journal of Engineering Mathematics, 66 (010): [] K. Singh and R.K. Gupta, Lie Symmetries and Exact Solutions of a New Generalized Hirota-Satsuma Coupled KdV System with Variable Coefficients. International Journal of Engineering Science, 44 (006): IJNS for contribution: editor@nonlinearscience.org.uk

7 Anupma Bansal, R.K. Gupta: On Certain New Exact Solutions of the (+1)-Dimensional Calogero-Degasperis Equation via Symmetry Approach 481 [3] G. Bluman, P. Broadbridge, J.R. King and M.J. Ward, Similarity: Generalizations, Applications and Open Problems. Journal of Engineering Mathematics, 66 (010):1-9. [4] R.K. Gupta, Anupma, The Dullin-Gottwald-Holm Equation: Classical Lie Approach and Exact Solutions. International Journal of Nonlinear Science, 10 (010): [5] T. Ying-Hui, C. Han-Lin and L. Xi-Qiang, Reduction and New Explicit Solutions of (+1)-Dimensional Breaking Soliton Equation. Communications in Theoretical Physics, 45 (006): IJNS homepage: