Symmetry Properties of Autonomous Integrating Factors

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1 Smmetr, Integrabilit and Geometr: Methods and Applications Vol ), Paper 024, 12 pages Smmetr Properties of Autonomous Integrating Factors Sibusiso MOYO and P.G.L. LEACH Department of Mathematics, Durban Institute of Technolog, PO Box 953, Steve Biko Campus, Durban 4000, Republic of South Africa moos@dit.ac.za arxiv:nlin/ v1 [nlin.si] 6 Dec 2005 School of Mathematical Sciences, Howard College, Universit of KwaZulu-Natal, Durban 4041, Republic of South Africa leachp@ukzn.ac.za Received September 27, 2005, in final form November 21, 2005; Published online December 05, 2005 Original article is available at Abstract. We stud the smmetr properties of autonomous integrating factors from an algebraic point of view. The smmetries are delineated for the resulting integrals treated as equations and smmetries of the integrals treated as functions or configurational invariants. The succession of terms pattern) is noted. The general pattern for the solution smmetries for equations in the simplest form of maximal order is given and the properties of the associated integrals resulting from this analsis are given. Ke words: autonomous integrating factors; maximal smmetr 2000 Mathematics Subject Classification: 34A05; 34A30; 34C14; 34C20 1 Introduction It is well-known that, when a smmetr is used to determine a first integral for a differential equation, the smmetr provides an integrating factor for the equation and remains as a smmetr of the first integral. For first-order ordinar differential equations the direct determination of the integrating factor is known [1] and algorithms for finding integrating factors for equations of higher order have been developed. In 1999 Cheb-Terrab and Roche [2] presented a sstematic algorithm for the construction of integrating factors for second-order ordinar differential equations and claimed that their algorithm gave integrating factors for equations which did not possess Lie point smmetries. In 2002 Leach and Bouquet [3] showed that for all equations except one of which Cheb-Terrab and Roche [2] had found integrating factors had smmetries which were not necessaril point smmetries but generalised or nonlocal. In the same ear, 2002, Abraham-Shrauner [4] also wrote a paper to demonstrate the reduction of order of nonlinear ordinar differential equations b a combination of first integrals and Lie group smmetries. The latter and former motivated us hereb to investigate the underling properties of autonomous integrating factors and the associated integrals treated as equations and as functions. Observations are made and inferred in general for an nth-order ordinar differential equation of maximal smmetr. These will also be extended to include other tpes of equations in a separate contribution. Program LIE [5] is used to compute the smmetries for the different cases considered. The knowledge of the smmetries of first integrals of the equation does give rise to some interesting properties of the equation itself. For example, the Ermakov Pinne equation [6, 7] which in its

2 2 S. Moo and P.G.L. Leach simplest form is w + K w 3 = 0, 1) where K is a constant. In theoretical discussions the sign of the constant K is immaterial and in fact it is often rescaled to unit. The general form of 1), videlicet ρ + ω 2 t)ρ = 1 ρ 3 occurs in the stud of the time-dependent linear oscillator, be it the classical or the quantal problem, as the differential equation which determines the time-dependent rescaling of the space variable and the definition of new time. Some of the references for this are [8, 9]. Another origin of 1) of particular interest in this work is as an integral of the third-order equation of maximal smmetr which in its elemental form is = 0. 2 Equations of maximal smmetr Definition 1. We define a first integral I for an equation of maximal smmetr, E = n) = 0, as I = f,,,..., n 1)), where di dx = 0 df E=0 dx = 0. E=0 This means that, if g x,,,,..., n 1)) is an integrating factor, then di dx = ge E=0 x,,,..., n)) E=0 = 0. We start b considering the well-known third-order ordinar differential equation of maximal smmetr = 0 2) which has seven Lie point smmetries. These are G 1 =, G 2 = x, G 3 = x 2, G 4 =, G 5 = x, G 6 = x x +, G 7 = x 2 x + 2x. 3) The algebra is {A 1 s sl2, R)} s 3A 1. The autonomous integrating factors for 2) are and. We list the smmetries and algebra when each of the integrals is treated as an equation and as a function. When we multipl = 0 b the integrating factor, we obtain = 0. Integration of this expression gives 1 2 ) 2 = k, where k is a constant of integration. This gives rise to three

3 Smmetr Properties of Autonomous Integrating Factors 3 cases which we list as follows: = 0 G 1 =, G 1 =, = k G 2 = x, G 2 = x, ) 1 G 3 =, G 3 = 2 x2 k, G 4 = x, G 4 = x + 2xk, G 5 = x x, G 5 = x x + x 2 k, G 6 = x 2 x + x, G 6 = x 2 x + x + 1 ) 2 x3 k, G 7 = x, G 7 = 3 ) 2 x2 k x x 3 k 2, G 8 = x x + 2, G 8 = x 1 ) 2 x3 k x ) x4 k 2, and, when = k is treated as a function, we have G 1 =, G 2 = x, G 3 = x, G 4 = x x + 2. Remark 1. When = k is treated as an equation, we have two cases, that is, = 0 and = k for which the algebra is sl3, R) : 2A 1 s {sl2, R) A 1 } 2A 1 [11, 12, 13]. If = k is treated as a function, the algebra is A 1 4,9 : A 2 s 2A 1 [11, 12, 13, 14]. If is used an the integrating factor, we obtain = 0. Integration of this equation gives = k which can be written as 1/2) = k/ 1/2) 3 and is the simplest form of the Ermakov Pinne equation [6, 7]. As before we write down the point smmetries corresponding to the three cases of the differential equation u = k/u 3, where u = 1/2. Program LIE [5] gives the following: u = 0 u = k/u 3 u = k/u 3 G 1 = u, G 1 = x, G 1 = x, G 2 = x u, G 2 = 2x x + u u, G 2 = 2x x + u u, G 3 = u u, G 3 = x 2 x + xu u, G 3 = x 2 x + xu u, G 4 = x, G 5 = x x, G 6 = x 2 x + xu u, G 7 = u x, G 8 = xu x + u 2 u. The transformation of = k to u = k/u 3 does not make a difference in terms of the smmetries as we just have a point transformation in this case. The other obvious integrating factors for 2) are 1, x and 1 2 x2 which give 1 = 0 I 3 =, x = 0 I 2 = x, 1 2 x2 = 0 I 1 = 1 2 x2 x +. Note that the numbering of the fundamental first integrals follows the convention given in Flessas et al [15, 16].)

4 4 S. Moo and P.G.L. Leach The integration of equation 2), which is a feature of the calculation of the smmetries of all linear ordinar differential equations of maximal smmetr [10], b means of an integrating factor gives a variet of results depending upon the integrating factor used. The characteristic feature of the Ermakov Pinne equation is that it possesses the threeelement algebra of Lie point smmetries, sl2, R), which in itself is characteristic of all scalar ordinar differential equations of maximal smmetr. The fourth-order ordinar differential equation iv = 0 has autonomous integrating factors and. If we use as an integrating factor in the original equation and integrate, we obtain 1 2 ) 2 = k. 4) Equation 4) is a generalised Kummer Schwartz equation for k = 0 and for k 0 a variation on the Ermakov Pinne equation as it can be written in the form ) 1/2) / = k ) 3/2)). The three cases for the integral in 4) treated as an equation and as a function give the following results: 1 2 ) 2 = ) 2 = k 1 2 ) 2 = k G 1 = x, G 1 = x, G 1 = x, G 2 = x x, G 2 =, G 2 =, G 3 =, G 3 = x x + 2, G 3 = x x + 2, G 4 =. The use of as an integrating factor gives = k. If k = 0, then we just have seven point smmetries as those of equation 2). The two remaining cases give = k = k G 1 = x, G 1 = x, G 2 = x, G 2 =, G 3 = 1 2 x2, G 3 = x, G 4 =, G 4 = x 2, G 5 = x x x3 k G 5 = x x + 3, G 6 = 1 ) 6 x3 k, G 7 = x 2 x + 2x + 1 ) 6 x4 k. Consider the fifth-order equation of maximal smmetr given b v = 0 5) with autonomous integrating factors, and iv. If we multipl 5) b the first integrating factor and integrate, we obtain the integral iv ) 2 = k. 6)

5 Smmetr Properties of Autonomous Integrating Factors 5 We consider the three cases for 6) treated as an equation with k = 0, k 0 and as a function. iv ) 2 = 0 iv ) 2 = k iv ) 2 = k G 1 = x, G 1 = x, G 1 = x, G 2 = x x, G 2 = x x + 2, G 2 = x x + 2, G 3 =, G 3 = x 2 x + 4x, G 3 = x 2 x + 4x, G 4 = x 2 x + 4x. Remark 2. For easier closure of the algebra in the first case x x can be written as x x + 2. We also observe that there is no difference when the integral is treated as a function and as an equation. It is important to note that, if is an integrating factor of n) = 0, then the integral obtained using this integrating factor alwas has the sl2, R) subalgebra. We further observe that for the peculiar value of the constant, that is, k = 0, there is the splitting of the self-similarit smmetr into two homogeneit smmetries. The integrating factor with 5) gives the following results iv 1 2 ) 2 = 0 iv 1 2 ) 2 = k iv 1 2 ) 2 = k G 1 = x, G 1 = x, G 1 = x, G 2 = x x, G 2 =, G 2 =, G 3 =, G 3 = x, G 3 = x, G 4 = x, G 4 = x x + 3, G 4 = x x + 3, G 5 =. If we use iv as the integrating factor of 5) and integrate, we obtain iv = k. We delineate the three cases below: iv = 0 iv = k iv = k G 1 =, G 1 = G 1 =, G 2 = x, G 2 = x G 2 = x, G 3 = x 2, G 3 = 1 2 x2 G 3 = x 2, G 4 = x 3, G 4 = 1 6 x3 G 4 = x 3, G 5 =, G 5 = x G 5 = x, G 6 = x, G 6 = 6x x + x 4 k G 6 = x x + 4, G 7 = x x, G 7 = 24 x 3 k ), G 8 = x 2 x + 3x, G 8 = 24x 2 x + 72x + x 5 k ). The differential equation vi = 0 7) has integrating factors, and v. If we use as the integrating factor, we obtain v iv ) 2 = k

6 6 S. Moo and P.G.L. Leach which leads to the cases below. v iv ) 2 = 0 v iv ) 2 = k v iv ) 2 = k G 1 =, G 1 =, G 1 =, G 2 =, G 2 = x, G 2 = x, G 3 = x, G 3 = x x + 3, G 3 = x x + 3, G 4 = x x. The use of as the integrating factor for 7) leads to v 1 2 iv ) 2 = k. The three cases give the following results: v 1 iv ) 2 = 0 2 v 1 iv ) 2 = k 2 v 1 iv ) 2 = k 2 G 1 =, G 1 =, G 1 =, G 2 = x, G 2 = x, G 2 = x, G 3 = x, G 3 = x, G 3 = x, G 4 = x 2, G 4 = x 2, G 4 = x 2, G 5 =, G 5 = x x + 4, G 5 = x x + 4, G 6 = x x. If we use v as an integrating factor, we obtain v = k. We also have the three cases as mentioned above to obtain v = 0 v = k v = k G 1 =, G 1 =, G 1 =, G 2 =, G 2 = x, G 2 = x, G 3 = x, G 3 = x, G 3 = x, G 4 = x, G 4 = x 2, G 4 = x 2, G 5 = x 2, G 5 = x 3, G 5 = x 3, G 6 = x 3, G 6 = x 4, G 6 = x 4, G 7 = x 4, G 7 = x x kx5, G 7 = x x + 5, G 8 = x x, G 8 = 1 ) 120 kx5, G 9 = x 2 x + 4x, G 9 = x 2 x + 4x + 1 ) 120 kx6. For the differential equation vii = 0 we have the integrating factors,, iv and vi. The integrals corresponding to these integrating factors respectivel are vi v + iv 1 2 ) 2 = k, vi v iv ) 2 = k, iv vi 1 2 v ) 2 = k, vi = k. 8)

7 Smmetr Properties of Autonomous Integrating Factors 7 If is used as the integrating factor, we have the integral vi v + iv 1 2 ) 2 = k which is treated as an equation for k = 0, k 0 and as a function. This gives the following results: G 1 = x, G 1 = x, G 1 =, G 2 = x x, G 2 = x x + 3, G 2 = x x + 3, G 3 =, G 3 = x 2 x + 6x, G 3 = x 2 x + 6x, G 4 = x 2 x + 6x. The integral corresponding to the integrating factor leads to the following cases: vi v + 1 iv ) 2 = 0 2 vi v + 1 iv ) 2 = k 2 vi v + 1 iv ) 2 = k 2 G 1 =, G 1 =, G 1 =, G 2 = x, G 2 = x, G 2 = x, G 3 = x, G 3 = x, G 3 = x, G 4 = x x, G 4 = x x + 4, G 4 = x x + 4, G 5 =. For the integrating factor iv we have the cases: iv vi 1 v ) 2 = 0 2 iv vi 1 v ) 2 = k 2 iv vi 1 v ) 2 = k 2 G 1 =, G 1 =, G 1 =, G 2 = x, G 2 = x, G 2 = x, G 3 =, G 3 = x, G 3 = x, G 4 = x, G 4 = x 2, G 4 = x 2, G 5 = x 2, G 5 = x 3, G 5 = x 3, G 6 = x 3, G 6 = x x + 5, G 6 = x x + 5, G 7 = x x. The last of the four integrating factors vi leads to vi = k. We have for the three cases the following results: vi = 0 vi = k vi = k G 1 =, G 1 =, G 1 =, G 2 = x, G 2 = x, G 2 = x, G 3 =, G 3 = x, G 3 = x, G 4 = x, G 4 = 1 2 x2, G 4 = x 2, G 5 = x 2, G 5 = 1 6 x3, G 5 = x 3, G 6 = x 3, G 6 = 1 24 x4, G 6 = x 4, G 7 = x 4, G 7 = x5, G 7 = x 5, G 8 = x 5, G 8 = x x kx6, G 8 = x x + 6, G 9 = x x, G 9 = 1 ) 720 kx6, G 10 = x 2 + 5x, G 10 = x 2 x + 5x + 1 ) 3600 k5x7.

8 8 S. Moo and P.G.L. Leach The differential equation viii = 0 9) has integrating factors,, v and vii. If we use in 9) and integrate the resulting equation, we obtain the integral vii vi + v 1 2 iv ) 2 = k. 10) The three cases of the integral in 10) being treated as an equation with k = 0 and k 0 and as a function are given respectivel below: G 1 = G 1 = G 1 =, G 2 =, G 2 = x, G 2 = x, G 3 = x, G 3 = x x + 4, G 3 = x x + 4, G 4 = x x. If is used as an integrating factor, we obtain vii iv vi v ) 2 = k with the following respective cases: G 1 =, G 1 =, G 1 =, G 2 =, G 2 = x, G 2 = x, G 3 = x, G 3 = x, G 3 = x, G 4 = x, G 4 = x 2, G 4 = x 2, G 5 = x 2. The use of v as an integrating factor gives v vii 1 2 vi )2 = k. 11) Equation 11) is of the Ermakov Pinne tpe. The three cases can be delineated as follows: v vii 1 2 vi ) 2 = 0 v vii 2 1 vi ) 2 = k v vii 1 2 vi ) 2 = k G 1 =, G 1 =, G 1 =, G 2 =, G 2 = x, G 2 = x, G 3 = x, G 3 = x, G 3 = x, G 4 = x, G 4 = x 2, G 4 = x 2, G 5 = x 2, G 5 = x 3, G 5 = x 3, G 6 = x 3, G 6 = x 4, G 6 = x 4, G 7 = x 4, G 7 = x x + 6, G 7 = x x + 6, G 8 = x x.

9 Smmetr Properties of Autonomous Integrating Factors 9 If vii is used as an integrating factor in 9), we obtain vii = k with the following smmetries for each of the three cases: vii = 0 vii = k vii = k G 1 =, G 1 =, G 1 =, G 2 =, G 2 = x, G 2 = x, G 3 = x, G 3 = x, G 3 = x, G 4 = x, G 4 = 1 2 x2, G 4 = x 2, G 5 = x 2, G 5 = 1 6 x3, G 5 = x 3, G 6 = x 3, G 6 = 1 24 x4, G 6 = x 4, G 7 = x 4, G 7 = x5, G 7 = x 5, G 8 = x 5, G 8 = x6, G 8 = x 6, G 9 = x 6, G 9 = x x + 1 G 10 = x x, G 10 = 1 G 11 = x 2 x + 6x, G 11 = x 2 x kx7, G 9 = x x + 7, ), 5040 kx7 6x + x k ). 3 Relationship between fundamental integrals and integrals obtained from integrating factors Consider the example of the third-order ordinar differential equation = 0 with the three fundamental integrals together with the appropriate associated point smmetries from the subalgebra sl2, R): G 7 = x 2 x + 2x, I 1 = 1 2 x2 x +, G 6 = x x +, I 2 = x, G 5 = x, I 3 =. The numbering of the smmetries follows that of the listing of Lie point smmetries in 3) and the ordering of the integrals is in terms of their solution smmetries. Then the autonomous integral associated with the integrating factor comes from the combination J = I 1 I I2 2 = Proposition 1. All the integrals obtained using as an integrating factor alwas have the sl2, R) subalgebra whereas the fundamental integrals onl have one of the sl2, R) elements. Proof. To prove the first proposition we consider the sl2, R) subalgebra Λ 1 = x, Λ 2 = x x + and Λ 3 = x 2 x +2x and the fundamental integrals I 1, I 2 and I 3 respectivel. Then we have the following: Λ 1 I 1 = I 2, Λ 2 I 1 = I 1, Λ 3 I 1 = 0, Λ 1 I 2 = I 3, Λ 2 I 2 = 0, Λ 3 I 2 = 2I 1, Λ 1 I 3 = 0, Λ 2 I 3 = I 3, Λ 3 I 3 = 2I 2.

10 10 S. Moo and P.G.L. Leach We also observe that Λ i J = 0 for i = 1,2,3. In fact it is eas to show that Λ i J = ǫ ijk I j I k. This is shown below as follows: Λ 1 J = I 2 I 3 I 2 I 3 = 0, Λ 2 J = I 1 I 3 I 1 I 3 = 0, Λ 3 J = 2I 2 I 1 + 2I 1 I 2 = 0. In general we have I ni = n i 1 j=0 1) j x n j i 1) n j i 1)! n j 1), i = 0,1,...,n 1, 12) so that for n = 3, I 30 = I 1, I 31 = I 2 and I 32 = I 3. The smmetries Λ 1,Λ 2 and Λ 3 operating on the fundamental integrals then ield in general Λ 1 I ni = I n,i+1, Λ 2 I ni = 1 i)i ni, Λ 3 I ni = n + i 3)n i)i n,i 1, I nn = 0. If we take for example Λ 3 I 3i = i3 i)i 3,i 1 with n = 3 and i = 0,1,2, we obtain Λ 3 I 30 = 0, Λ 3 I 31 = 2I 30, Λ 3 I 32 = 2I 31, where as above I 30 = I 1, I 31 = I 2 and I 32 = I 3. Proposition 2 [17]). If we take the equation of maximal smmetr n) = 0, the sl2, R) subalgebra maps back to itself and is preserved. Proposition 3. For the fifth-order equation v = 0 the autonomous integral emanating from the integrating factor can be obtained from J = I 0 I 4 I 1 I I2 2, where I 0 = 1 24 x4 iv 1 6 x x2 x +, I 1 = 1 6 x3 iv 1 2 x2 + x, I 2 = 1 2 x2 iv x +, I 3 = x iv, I 4 = iv. Proposition 4. The fourth-order equation also has an autonomous integral J defined as J = I 1 I I2 2, where I 1 = 1 2 x2 x +, I 2 = x, I 3 =. Proposition 5. It can be shown that the differential equation vi = 0 also has the autonomous integral J which is defined as J = I 0 I 6 I 1 I 5 + I 2 I I2 3 with the I i i = 0,6) being redefined appropriatel.

11 Smmetr Properties of Autonomous Integrating Factors 11 4 Conclusion If n) = f x,,,..., n 1) is an nth-order ordinar differential equation and g x,,,..., n 1)) = k is an integral, the integral obtained b multipling the equation b the integrating factor and integrating once possesses certain smmetries when treated as a function, an equation for the general constant and a configurational invariant k=0). It is important to note that, if is an integrating factor of n) = 0, then the integral obtained using this integrating factor alwas has the sl2, R) subalgebra whereas the fundamental integrals onl have one of the sl2, R) elements. We further observe that for the peculiar value of the constant, k = 0, there is the splitting of the self-similarit smmetr into two homogeneit smmetries. The thirdorder ordinar differential equation is actuall special and leads to the Ermakov Pinne tpe equation. The fourth-order ordinar differential equation iv = 0 has as one of its autonomous integrating factors which leads together with the the original equation upon integration to the generalised Kummer Schwartz equation. An extension to other tpes of equations will be completed in a separate contribution. The question of what Lie point smmetries of an ordinar differential equation are also shared b all its first integrals will form the basis for the next contribution. Acknowledgments SM thanks the National Research Foundation of South Africa and the Durban Institute of Technolog for their support. PGLL thanks the Universit of KwaZulu-Natal for its continuing support. [1] Bluman G.W., Kumei S., Smmetries and differential equations, New York, Springer-Verlag, [2] Cheb-Terrab E.S., Roche A.D., Integrating factors for second order ordinar differential equations, J. Smbolic Comput., 1999, V.27, ; math-ph/ [3] Leach P.G.L., Bouquet S.É., Smmetries and integrating factors, J. Nonlinear Math. Phs., 2002, V.9, suppl. 2, [4] Abraham-Shrauner B., Hidden smmetries, first integrals and reduction of order of nonlinear ordinar differential equations, J. Nonlinear Math. Phs., 2002, V.9, suppl. 2, 1 9. [5] Head A., LIE, a PC program for Lie analsis of differential equations, Comput. Phs. Comm., 1993, V.77, [6] Ermakov V., Second-order differential equations. Conditions of complete integrabilit, Universitetskie Izvestia, Kiev, 1880, N 9, 1 25 translated b A.O. Harin). [7] Pinne E., The nonlinear differential equation x)+px)+c 3 = 0, Proc. Amer. Math. Soc., 1950, V.1, 681. [8] Lewis H.R., Classical and quantum sstems with time-dependent harmonic oscillator-tpe Hamiltonians, Phs. Rev. Lett., 1967, V.18, [9] Lewis H.R.Jr., Motion of a time-dependent harmonic oscillator and of a charged particle in a time-dependent, axiall smmetric, electromagnetic field, Phs. Rev., 1968, V.172, [10] Mahomed F.M., Leach P.G.L., Smmetr Lie algebras of nth order ordinar differential equations, J. Math. Anal. Appl., 1990, V.151, [11] Mubarakzanov G.M., On solvable Lie algebras, Izv. Vs. Uchebn. Zaved. Matematika, 1963, N 1 32), [12] Mubarakzanov G.M., Classification of real structures of five-dimensional Lie algebras, Izv. Vs. Uchebn. Zaved. Matematika, 1963, N 3 34), [13] Mubarakzanov G.M., Classification of solvable Lie algebras of sixth order with a non-nilpotent basis element, Izv. Vs. Uchebn. Zaved. Matematika, 1963, N 4 35), [14] Patera, J., Sharp R.T., Winternitz P., Invariants of real low dimension Lie algebras, J. Math. Phs., 1976, V.17,

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