λ −Symmetries and unification of techniques for ordinary differential equations
λ −Symmetries and unification of techniques for ordinary differential equations Generalized Sundman transformations admits a generalized symmetry Different methods to search solutions of ODEs are unified by using the λ -symmetry approach. These techniques include reduction in order through some types of nonlocal symmetries, the search of first integrals with the Prelle-Singer method and linearization through generalized Sundman transformations. Motivation: exponential vector fields In the book  by P. Olver appears the concept of exponential vector field in order to show that not every method of integrating an ODE comes from the classical Lie method. Exponential vector fields are of the form R ∂ ∂ ∗ P(x,u)dx (1) v =e ξ (x, u) + η(x, u) ∂x ∂u v = ξ (x, u, ux , w)∂x + η(x, u, ux , w)∂u + ψ(x, u, ux , w)∂w (6) that lets reduce the order of (4). In terms of the original variables, the coefficients of (6) involve nonlocal terms. It can be checked that if Hux = 0, such nonlocal symmetries are exponential vector fields introduced by P. Olver. In the general case, the nonlocal symmetries that let reduce the order of the equation are connected with the following λ -symmetries of the equation: Nonlocal symmetries and λ −symmetries where λ , µ are functions and Dx is the total derivative operator. Given a vector field V defined on M and a function λ ∈ C∞ (M k) ), formula (2) can be used to prolong V. Such prolongation has been called the λ -prolongation of V and denoted by V [λ ,(n)] : n V [λ ,(n)] = ξ ∂x + ∑ η [λ ,(i)] ∂ui i=0 where η [λ ,(0)] = η and η [λ ,(i)] = (Dx + λ )(η [λ ,(i−1)] ) − (Dx + λ )(ξ )ui . A pair (V, λ ) such that V [λ ,(n)] leaves invariant a given nth order ODE is called a λ symmetry of the ODE. In this case [V [λ ,(n)] , A] = λV [λ ,(n)] + µA, Extended Prelle-Singer method Prelle and Singer  introduced a method to construct integrating factors of first-order ODEs. This method has been adapted and applied to second-order ODEs in . The extended Prelle-Singer method tries to find a function S such that the sum of the differential forms The associated λ -symmetry is defined by V = ∂u and λ = α(x, u)ux + β (x, u) where −1 −1 Fx Fu Fu Fu , β =− (13) α =− G u G G u G φ dx − dux and (φ + Sux ) dx − [Sdu + dux ] uxx = φ (x, u, ux ) (7) be proportional to the differential form dI = Ix dx + Iu du + Iux dux , for some function I(x, u, ux ). In this case I is a first integral of (4) and µ = −Iux is an integrating factor. For second-order ODEs the following relations between λ -symmetries and integrating factors hold (): First integrals and λ −symmetries If I is a first integral of A then V = ∂u is a λ −symmetry of (4) for λ = −Iu /Iux . If V a λ −symmetry then there exists a first integral I such (4) is based on the concept of H-coverings (). Briefly, the method tries to find a function H such the system (H-covering): uxx = φ (x, u, ux ), (5) wx = H(x, u, ux ), Such λ −symmetries and the associated first integrals can be calculated by systematic methods, which allow us construct the nonlocal linearizing transformations (11). Generalized Sundman transformations and λ −symmetries If the GST (11) linearizes the equation (4) then V = ∂x is a λ -symmetry for the function λ = α ux + β where α and β are defined by (13). Conclusion: The λ -symmetry approach can be helpful in understanding and unifying apparently no related paths for solving ODEs. that V [λ ,(1)] (I) = 0. Integrating factors and λ −symmetries µ is an integrating factor if and only if µ satisfies µu + [A(µ) + µφux ]ux = 0 and V = ∂u is a λ −symmetry for λ = A(µ)/µ + φux If V = ∂u is a λ −symmetry of equation (4) then an associated integrating factor is a solution of the first-order linear system of PDEs References  O LVER P J, 1986 Applications of Lie Groups to Differential Equations (New-York: Springer)  M URIEL C AND ROMERO J L 2002 Theo. Math. Phys. 133 (2) 1565–1575  M URIEL C AND ROMERO J L 2001 IMA J. Appl. Math. 66 (2) 111–125  M URIEL C, ROMERO J L AND O LVER P J 2006 J. Diff. Eqs. 222:164–184 µx + ux µu + φ µux + (φux − λ )µ = 0, µu + λ µux + λux µ = 0. (8) Iu = −λ µ, Iux = µ  G AETA G 2009 J. Nonlinear Math. Phys. 16 (2) 107–136  G ANDARIAS M L 2009 Theoretical and Mathematical Physics 159:779–786 The corresponding first integral G can be calculated through a line-integral from  C ATALANO D 2007 J. Phys. A: Math. Theor. 40 5479–5489 (9) This reveals the connection between the extended Prelle-Singer method and the λ symmetry approach: A method to obtain some nonlocal symmetries of a given second order ODE () (11) Due to the nonlocal nature of (11), only I1 = UX provides first integrals of the nonlinear ODE, that are of the form Fu (x, u)ux + Fx (x, u) . (12) I= G(x, u) Ix = µ(λ ux − φ ), Nonlocal symmetries and the λ -covering method are characterized by the Lie linearization test. The two independent integrals of UT T = 0, I1 = UX and I2 = U − XUX , provide two independent first integrals of the nonlinear equation. Since there are many ODEs that do not pass the Lie test of linearization, many recent studies consider transformations involving nonlocal terms (, , ). The most simple type of such transformations are the generalized Sundman transformations (GST): U = F(x, u), dX = G(x, u)dx. (3) where A denotes the vector field associated to the ODE. By the ID property, the order of such ODE can be reduced by the method of the differential invariants. The λ −symmetry concept, as introduced in , has been generalized for systems, PDEs, variational problems () and difference equations (see  for details). Apart from order reductions, there are different methods to find solutions of ODEs, such as the search of first integrals, integrating factors or linearization processes. The second-order ODEs that can be transformed into UXX = 0 through a local change of variables U = R(x, u), X = S(x, u), (10) Case ξ = 6 0 : V = ∂x + η/ξ ∂u λ = (ξx + ξu ux + ξux φ + ξw H)/ξ Case ξ = 0 : V = ∂u , λ = (ηx + ηu ux + ηux φ + ηw H)/η R where (x, u) ∈ M and P(x, u)dx is, formally, the integralR of the function P(x, u), once one has chosen a function u = f (x). He noticed that v∗ (n) = e P(x,u)dxV, where V is an ordinary vector field on M (n) , and that an exponential vector field can be used to reduce the order of the ODE by the method of the differential invariants. It should be noted that V is not a standard prolongation of a vector field on M. This situation motivated us to investigate the prolongations of vector fields that admit invariants constructed by derivation of lower order invariants. The unique prolongations with the invariant-by-derivation (ID) property () are the vector fields V ∗ on M (n) that satisfy the relation [V ∗ , Dx ] = λV ∗ + µDx (2) C. Muriel and J L Romero Dept. de Matemáticas Universidad de Cádiz, Spain Extended Prelle-Singer method and λ −symmetries If S is the function of the extended Prelle-Singer method then V = ∂x is a λ -symmetry for λ = −S. 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