Symmetry Reductions of (2+1) dimensional Equal Width. Wave Equation

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1 Authors: Symmetry Reductions of (2+1) dimensional Equal Width 1. Dr. S. Padmasekaran Wave Equation Asst. Professor, Department of Mathematics Periyar University, Salem 2. M.G. RANI Periyar University, Salem 3. T. Shanmugapriya, Madurai Abstract: We discuss the symmetries and reductions of the two dimensional Equal Width Wave equation. The (2 + 1) dimensional Equal width Wave equation is u t + u 2 u x μ (u xxt + u yyt ) = 0 is subjected to the Lie s Classical Method. Classification of its Symmetry algebra into one and two dimensional subalgebra s are carried out inorder to facilitate its reduction systematically to (1 + 1) dimensional PDE and then to ODEs. 1. INTRODUCTION In Mathematical Physics the simple model equation is the KdV equation [1], u t + 6uu x + δu xxx = 0 (1) stands for the effect of nonlinear convection and the linear dispersion. Equation (1) is integrable in the sense that it possesses solitons, Backlund transformations, Lax Pair, infinite number of conservation laws and Painleve Property. Whitham [2] has given a representation of a periodic wave as a sum of solitons for (1). Miura [3] established a relation between (1) and the modified KdV equation [4] u t + au 2 u x + u xxx = 0 (2) Liu and Yang [5] studied the bifurcation problems for a generalized KdV equation u t + au n u x + u xxx = 0, n 1, a R (3) 724

2 The Equal Width (EW) Wave equation was suggested by Morrison and Meiss [6] to be used as a model partial differential equation for the simulation of one dimensional wave propagation in a non-linear medium with a dispersion process. The EW wave equation is an alternative description of the nonlinear dispersive waves to the more usual KdV equation. In this paper, we consider the Equal Width Wave equation in two dimensions as, u t + u 2 u x μ (u xxt + u yyt ) = 0 (4) We shall show that the equation (4) admits a four dimensional symmetry group and determine the corresponding Lie algebra, classify the one and two dimensional subalgebras of the symmetry algebra of (4) in order to reduce eqn (4) to ( 1 + 1) dimensional partial differential equations and then to ODEs. We shall establish that the symmetry generators form a closed Lie algebra and this allowed us to use the recent method due to Ahmed, Bokhari, Kara and Zaman [7] to successively reduce (4) to (1+1) dimensional PDEs and then to ODEs with the help of two dimensional Abelian and solvable non-abelian subalgebras. 2. The Symmetry group and Lie Algebra of u t + u 2 u x μ (u xxt + u yyt ) = 0 [8], Olver [9]). If (4) is invariant under a one Parameter Lie group of point transformations (Bluman and Kumei x = x+ ξ 1 (x, y, t; u) + o( 2 ) (5) y = y+ ξ 2 (x, y, t; u) + o( 2 ) (6) t = t+ ξ 3 (x, y, t; u) + o( 2 ) (7) u = u+ φ 1 (x, y, t; u) + o( 2 ) (8) with infinitesimal generator, ξ 11 = ξ, ξ 12 = η, ξ 13 = τ and φ 1 = φ then the fourth prolongation Pr (4) V of the corresponding Vector field, which satisfies, V = ξ (x, y, t; u)x + η (x, y, t; u)y + τ(x, y, t; u)t + φ(x, y, t; u)u (9) We introduced the following quantities: Pr (4) VΩ(x, y, t; u) Ω(x,y,t;u)=0 = 0 (10) 725

3 φ x 1 = Dφ 1 x u Dξ 2 y u Dξ 3 t φ y 1 = Dφ 1 x u Dξ 2 y u Dξ 3 t (11) (12) φ t 1 = Dφ 1 Dt x u Dξ 2 Dt y u Dξ 3 Dt t Dt φ 1 xx = Dφ 1 x xx u Dξ 2 xy u Dξ 3 xt φ yy 1 = Dφ y 1 xy u Dξ 2 yy u Dξ 3 yt (13) (14) (15) where D = x + u x u + u xx u x + u xy u y + u xt u t (16) D = + u y y u + u xy u x + u yy u y + u yt u t (17) D Dt = t + u t u + u xt u x + u yt u y + u tt u t (18) The determining equations are obtained from (10) are as follows: ξ 1,u = 0 (19) ξ 2,u = 0 (20) ξ 3,u = 0 (21) φ 1,uu = 0 (22) ξ 1,t = 0 (23) φ 1,tu = 0 (24) ξ 3y = 0 (25) ξ 3x = 0 (26) φ 1t + u 2 φ 1x μφ 1xxt μφ 1yyt = 0 (27) 2uφ 1 u 2 ξ 1x + u 2 ξ 3t 2μφ 1xtu + μu 2 φ 1xxu + μu 2 φ 1yyu = 0 (28) 726

4 2ξ 1x + μφ 1xxu + μφ 1yyu = 0 (29) 2ξ 2y + μφ 1xxu + μφ 1yyu = 0 (30) ξ 2xx ξ 2yy + 2φ 1yu = 0 (31) ξ 1xx + ξ 1yy 2φ 1xu = 0 (32) u 2 ξ 2x 2mφ 1ytu = 0 (33) ξ 1y + ξ 2x = 0 (34) The above determining equations are solved and the infinitesimals ξ, η, τ and φ are as follows: ξ = K 1, η = K 2, τ = K 3 + K 4 t & φ = K 4 u 2 (35) Let us write down the four symmetry generators corresponding to each of the constants k i, where i = 1 to 4 involved in the infinitesimals Viz., V 1 = x ; V 2 = y ; V 3 = t ; V 4 = 2tt udu (36) table. These symmetry generators form a closed (Lie) algebra as is seen from the following commutator Table: 1 [vi, vj] V1 V2 V3 V4 V V V V3 V V3 0 for this four dimensional Lie algebra the commutator table for Vi is a (4 4) table whose (i, j)th entry expresses the Lie bracket [vi, vj] given by the above Lie algebra L. The table is skew symmetric and the diagonal elements all vanish. The coefficient Cijk is the coefficient of Vi of the (i, j)th entry of the 727

5 commutator table and the related structure constants can be easily calculated from the commutator table. Hence the Lie algebra L is solvable. The radical of G is R = <V1, V2> <V3, V4>. In the next section, we derive the reductions of (4) to PDEs with two independent variable and then to ODEs. There are four one-dimensional Lie subalgebras, namely Ls,1 = {V1}; Ls,2 = {V2}; Ls,3 = {V3} and Ls,4 = {V4}. And also for the corresponding one dimensional subalgebras we reduce eqn (4) to a PDE. Further reductions to ODEs are associated with the two-dimensional subalgebras. It is evident from the commutator table that there are five two-dimensional solvable abelian subalgebras namely L A,1 = {V 1, V 2 };L A,2 = {V 1, V 3 }; L A,3 = {V 1, V 4 }; L A,4 = {V 2, V 3 }; L A,5 = {V 2, V 4 } and only one nonabelian subalgebra (ie)., L na,1 = {V 3, V 4 } 3. Reduction of u t + u 2 u x μ(u xxt + u yyt ) = 0 by one dimensional subalgebras We consider the reductions of (4) under each generator separately. Case: a Sub algebra L S,1 = {V 1 } The characteristic equation associated with the generator V1 is dx 1 = dy 0 = dt 0 = du 0 (37) Integration of (37) yields the similarity transformation y = p ; t = q and u = w(p,q) (38) using (38) in (4) we arrive at w q + w 2 (o) μ(wppq) = 0 (i.e)., w q + w 2 (o) μ(wppq) = 0 (39) Case: b Subalgebra L S,2 = {V 2 } The characteristic equation and its solutions are dx 0 = dy 1 = dt 0 = du 0 (40) and 728

6 In view of (41) (4) transforms into Case: c Subalgebra L S,3 = {V 3 } x = p; t = q; u = w(p, q) (41) w q + w 2 w p μ(wppq) = 0 (42) The characteristic equation associated with V3 is dx 0 = dy 0 = dt 1 = du 0 (43) Integration of (43) gives rise to x = p; y = q; u = w(p, q) (44) In view of (44) in (4) we find that w 2 w p μ(0) = 0 Case: d Subalgebra L S,4 = {V 4 } w 2 w p = 0 (45) The characteristic equation associated with V4 is dx 0 = dy 0 = dt 2t = du u (46) Integrating (46) we get x = p; t = q; u = w(p, q)t 2 (47) By inserting (47) in (4) we get w q + w 2 w p q 2 2q 1 w μ(wppq) 2μq 1 w pp = 0 (48) 4. Reductions of two dimensional subalgebras for u t + u 2 u x μ(u xxt + u yyt ) = 0 Here we consider the abelian subalgebras and non-abelian subalgebras separately. 4.1 Two dimensional Abelian subalgebras: Case : 729

7 Subalgebra L A,1 = {V 1, V 2 } From table 1, we find that the generators V1 & V2 commute (ie) [V1, V2] = 0. We can initiate the reduction procedure by taking V1 or V2. If we begin with V1 then (4) is reduced to the PDE (39). We now write V 2 which is V2 but expressed interms of the similarity variables given in (38). Hence the associated characteristic equation is V 2 = p (49) dp 1 = dq 0 = dw 0 (50) Whose solution is q = ξ & w = R(ξ) (51) Consequently (39) is replaced by an ODE R 1 = 0 (52) Case: b Subalgebra L A,2 = {V 1, V 3 } It follows from the table: 1, that [V1, V3] = 0, we begin with V1 to transform eqn (4) to (39). Then V3 changes to V 3 = q by (38). Integration of the characteristic equation associated with V 3 gives p = ξ & w = R(ξ) (53) and its solution is satisfied by eqn (39) Case: c Subalgebra L A,3 = {V 1, V 4 } It follows from table: 1 that [V1, V4] = 0. We shall begin with V4 to transform (4) to (48). Then V1 changes to V 1 = p by (47). Integration of the characteristic eqn associated with V 1 gives q = ς & w = R(ς) (54) and its solution by eqn (48) is R 1-2ς 1 R = 0 (55) Case: d Subalgebra L A,4 = {V 2, V 3 } 730

8 From table: 2 we find that the generators V2 & V3 commute we can initiate the reduction procedure by taking V2 or V3. If we begin with V2 then (4) is reduced to the PDE (42). We now write V 3 in terms of eqn (41) as Hence the associated characteristic equation is V 3 = q (56) dp 0 = dq 1 = dw 0 (50) Whose solution is p = ξ & w = R(ξ) (58) Consequently (42) is replaced by an ODE R 2 R 1 = 0 (59) Case: e Subalgebra L A,5 = {V 2, V 4 } Similarly following the above procedure we begin with V2. Then V 4 in terms of (41) is V 4 = 2qq ww. Hence by solving the associated characteristic eqn for V 4 is p = ξ & w = R(ξ) (60) Hence (42) reduces to R 2 R 1 = 0 (61) 4.2 Two=dimensional solvable non-abelian subalgebra Case: 1 Subalgebra L na,1 = {V 3, V 4 } From commutator table, we find that [V3, V4] = 2V. we begin with V3 to transform eqn (4) to (45). Then V4 changes tov 4. Integration of the characteristic eqn associated with V 4 and solved for the solution. That solution is satisfied by (44) & (45). 5. CONCLUSION: 1. A (2+1) dimensional Equal Width Wave equation u t + u 2 u x μ (u xxt + u yyt ) = 0 is subjected to Lie s classical method. 731

9 2. Eqn. (4) admits a four dimensional symmetry group. 3. It is established that the symmetry generators form a closed Lie algebra. 4. Classification of symmetry algebra of (4) into one and two dimensional subalgebras is carried out. 5. Systematic reduction to (1+1) dimensional PDE and then to first or second order ODEs are performed using one-dimensional and two-dimensional Abelian and solvable non-abelian subalgebras. 6. REFERENCES: 1. Bluman. G.W and Kumei S, Symmetries and Differential Equations, Springer Verlag, New York, (1989). 2. Whitham G.B, Linear and Non Linear Waves, Wiley, New York, (1974). 3. Miura R.M, Korteweg-de Vries equations and generalizations: A remarkable explicit nonlinear transformation, I. Math phys., 9, , (1968). 4. Olver P.J. Applications of Lie Groups to differential equations, Graduate Texts in Mathematics, 107, Springer Verlag, New York, (1986). 5. Liu. Z. and Yang. C., The application of bifurcation method to a higher order KdV equation, J. Math, Anal. Appl, 275, 1-12, (2002). 6. Morrison PJ and Meiss JD, 1984 Physica 11D, 30, Ahmed A., Ashfaque. H Bokhari, Kara A.H. and Zaman F.D, Symmetry Classifications and reductions of some classes of (2+1) nonlinear heat equation, J. Math. Anal Appl, 339, , (2008). 732