C~∞-symmetry And Integrability Of Differential Equations

Differential equations and dynamical systems are usually used to describe natural phenomena for which local processes are known. Generally speaking, these elementary processes are nonlinear, their evolution is governed by nonlinear differential equations. These nonlinear effects give rise to complex structures to differential equations, hence, it is extremely difficult to completely describe these differential equations. Once the equations in description of the natural phenomena are formulated, the next problem is to "solve" these equations.The first attempt to solve differential equations goes back to Newton, Leibniz and Euler, they did some celebrated works in the context of solving differential equations. In the late 18th and 19th centuries, the theory of integration for the equations of motion was developed by the work of Lagrange, Poisson, Hamilton and Liouville. The basic idea of these works is that the solution can always be represented by the combination of known functions. The notion of "integrability" was then introduced to describe the property of differential equations.For hundred years, by the efforts of several eras of mathematicians, several methods are developed to determine the integrability.·In the early 19th century. Liouville's theorem showed that the exis- tence of n first integrals (including the Hamiltonian) in involution allows one to reduce the system to quadratures, that is, it is integrable.·In 1870, S. Lie introduced the concept of Lie group, and Lie group method is an important method to determine the integrability. In general, a Hamiltonian system with n degrees of freedom which admits n-parameter abelian symmetry group is considered to be completely integrable.·In the late 19th century. Frobenius presented Frobenius theorem to determine the integrability of a system of Pfaff equations.·In the end of 19th century, Painleve singularity analysis method showed that the system of differential equations which exhibits Painleve property may be considered "integrable".·In 1918, E. Noether proposed Noethers theorem, it is the first time to determine the general principle relating symmetry groups and conservation laws.·In 1932, T. Carleman presented Carleman embedding method to obtain first integrals of nonlinear differential equations.·In 1963, Arnold generalized Liouville's theorem by giving it a geometric interpretation, that is the celebrated Arnold-Liouville's theorem.·In 1968, P. Lax defined Lax pair in the context of evolution equation (and more particularly for the KdV equation). The importance of Lax pair for the theory of integrability is that one can obtain first integrals from Lax pair.·In 1988, J.M. Strelcyn and S. Wojciechowski developed linear consistent analysis method for the differential equations in R³ according to Frobenius theorem to obtain first integrals of nonlinear differential equations.Lie symmetry group theory provides a powerful tool for analyzing dif- ferential equations. Its infinitesimal analysis technique, that is, symmetry technique has been applied to the problems in mathematics, physics and mechanics. However, not every technique can be based on symmetry analysis, and this requires generalizations of classical Lie methods. In 2001, Muriel and Romero introduced a new class of symmetry based on a new method of prolonging vector fields known as the C^∞-prolongation, leading to the notion of C^∞-symmetry, that strictly includes Lie symmetry. And C^∞-symmetry provided with some excellence properties similar to symmetry. Our aim here is to find first integrals of differential equations and to obtain the integrability of differential equations using C^∞-symmetry. Consider the following system of differential equations(?). (1) In general, the knowledge of an r-parameter solvable group of symmetries allows one to reduce the order of differential equations by r. However, there is no general conclusion in verifying the integrability of a system of differential equations from the knowledge of a single symmetry. Until 1999, G. Unal showed that the existence of a certain Lie symmetry determines the algebraic integrability. G. (?)nal introduced Liouville vector field and obtained n - 1 first integrals of (1). We can expect that a generalization of the concept of Liouville vector field, based on C^∞-symmetry, will generate a new method for obtaining first integrals. We establish this generalization and introduce the concept of C^∞-Liouville vector field. Some essential properties of the C^∞-Liouville vector field are presented. We also provide an algorithmic procedure to obtain first integrals of any system of ordinary differential equations that admits a C^∞-Liouville vector field.Theorem 1 Let L beαλ-Liouville vector field andγsatisfy (?) =λγ. IfγL is divergence free, then we can obtain n - 1 independent first integrals of (1) from the n-1 independent invariants of the vector field L.Hence, for a system of autonomous differential equations, we can obtain the integrability from the above theorem. We also include several examples to illustrate how this new method works in practice, including the Whittaker's differential equation, the Lorenz system and the Pikovski-Rabinovich-Trakhtengerts system. Consider the first order differential equation(?). (2) For the first order differential equation, the correspondence between symmetry and integrating factor is well known. That is, we can obtain an integrating factor from a symmetry. Common wisdom has it that the technique can not be used except in this case. In 1992, J. Sherring and G. Prince extended the traditional idea of an integrating factor for a first order differential equation with symmetry using the differential geometry of vector fields and forms. It is our aim here to provide the generalization of the integrating factor approach using C^∞-symmetry, that is, to obtain integrating factors from C^∞-symmetries. Furthermore, we can obtain the integrability of differential equations. In the case of a first order equation, we obtain the following result.Theorem 2 Let X beαλ-symmetry of (2), and X_ωis nowhere zero. Thenis an integrating factor of (2), whereγsatisfiesΓ(γ) =λ_γ. Consider the second order Newton motion equation We can also obtain the relationship between C^∞-syinmetries and integrating factors.Theorem 3 Let X beαλ₁-symmetry of (3), Y beαλ₂-symmetry of (3), satisfy [γ₁X,γ₂ Y] = 0, and X, Y span withΓis three dimensional. Then the two forms (?) and (?)are closed and locally provided a complete set of two functionally independent first integrals, whereΓ(γ_i) =λ_iγ_i, i = 1,2.Theorem 4 Let X and Y be vector fields satisfy X isαλ₁-symmetry of (3) and Y, and Y isαλ₂-symmetry of (3) and X, whose span withΓis three dimensional. Then the two forms(?) and (?) are closed.Theorem 5 Let X beαλ₁-symmetry of (3), Y beαλ₂-symmetry of span{X,Γ}, and X, Y andΓare linearly independent everywhere. Then is closed, whileis closed moduloω¹. That is, there exist two functionally independent first integrals I₁ and I₂, satisfyω¹ = dI₁,ω² = dI₂ - Y(I₂)ω¹. According to the above theorems, if we find two C^∞-symmetries of (3), then we can obtain two functionally independent first integrals of (3), hence, we can obtain the integrability of (3). Consider the nth order differential equation(?). (4) We obtain the following result on the relationship between C^∞-symmetries and integrating factors.Theorem 6 Let X_i beαλ_i-symmetry of (4), i = 1,…,n, which satisfy [γ_iX_i,γ_jX_j] = 0, i,j =1,…,n, and X₁,…, X_n andΓare linearly independent. PutThenare closed, and locally we can obtain n independent first integrals, where satisfyΓ(γ_i) =λ_iγ_i, i = 1,…, n. Theorem 7 Let X_n beαλ_n-symmetry of (4), X_i beαλ_i-symmetry of span{X_i+1,…,X_n,Γ}, i = 1,…,n, and X₁,…, X_n and T are linearly independent. PutThen dω¹ = 0, dω² = 0 modω¹, dωⁱ= 0 modω¹,…,ω^i-1, i = 1,…,n. So that locallyFrom the above theorems, we can obtain n first integrals of (4), that is. we obtain the integrability of (4).Thus, we investigate the integrability of differential equations using C^∞-symmetry. That is, the existence of C^∞-symmetries which satisfy some special conditions can determine the integrability of differential equations.