Complete Symmetry Groups: A Connection Between Some Ordinary Differential Equations and Partial Differential Equations (original) (raw)
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Journal of Mathematical Analysis and Applications, 2001
The concept of the complete symmetry group of a differential equation introduced by J. Krause (1994, J. Math. Phys. 35, 5734-5748) is extended to integrals of such equations. This paper is devoted to some aspects characterising complete symmetry groups. The algebras of the symmetries of both differential equations and integrals are studied in the context of equations for which the elements are represented by point or contact symmetries so that there is no ambiguity about the group. Both algebras and groups are found to be nonunique.
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We discuss the Lie symmetry approach to homogeneous, linear, ordinary di erential equations in an attempt to connect it with the algebraic theory of such equations. In particular we pay attention to the elds of functions over which the symmetry vector elds are de ned and, by de ning a noncharacteristic Lie subalgebra of the symmetry algebra, are able to establish a general description of all continuous symmetries. We use this description to rederive a classical result on di erential extensions for second order equations.
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We investigate the use of invariants of the admitted Lie groups of transformation in finding solutions of the systems of ordinary differential equations (ODEs). Bluman's theorem (1990) of invariant solutions of ODEs is extended for systems of ODEs. Differential invariants of a Lie group are used in reducing order of the given system. Examples are given to illustrate the methods.
Group analysis of differential equations: A new type of Lie symmetries
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We construct a new type of symmetries using the regular Lie symmetries as the basis, which we call Modified symmetries. The contrast is that while Lie symmetries arise from point transformations, the Modified symmetries result from the transformations of the neighborhood of that point. The similarity is that as the neighborhood contracts to the central point, the two sets of symmetries become indistinguishable from one another, meaning the Modified symmetries will cease to exist if there were no Lie symmetries in the first place. The advantage is that the group invariant solutions are not affected by all these, because they result from ratios of the symmetries, and will therefore exist in the absence of Lie symmetries, i.e,. zero symmetries. Zero symmetries lead to 0/0, and no further. With the Modified symmetries we get f (x, ω)/g(x, ω) = 0/0 as ω goes to zero, and there are numerous mathematical techniques through which this can be resolved. We develop this concept using tensors a...
1993
We give a method for using explicitly known Lie symmetries of a system of differential equations to help find more symmetries of the system. A Lie (or infinitesimal) symmetry of a system of differential equations is a transformation of its dependent and independent variables, depending on continuous parameters, which maps any solution of the system to another solution of the same systkun. Infinitesimal Lie symmetries of a system of differential equations arise as solutions of a related system of linear homogeneous partial differential equations called infinitesimal determining equations. The importance of symmetries in applications has prompted the development of many software packages to derive and attempt to integrate infinitesimal determining equations. For a. given system of differential equations we usually have a priori explicit knowledge of many symmetries of the system because of their simple form or the physical origin of the system. Current methods for finding symmetries d...
On the Virasoro Structure of Symmetry Algebras of Nonlinear Partial Differential Equations
Symmetry Integrability and Geometry-methods and Applications, 2006
We discuss Lie algebras of the Lie symmetry groups of two generically nonintegrable equations in one temporal and two space dimensions arising in different contexts. The first is a generalization of the KP equation and contains 9 arbitrary functions of one and two arguments. The second one is a system of PDEs that depend on some physical parameters. We require that these PDEs are invariant under a Kac-Moody-Virasoro algebra. This leads to several limitations on the coefficients (either functions or parameters) under which equations are prime candidates for being integrable.
The Lie symmetry group of the general Liénard-type equation
Journal of Nonlinear Mathematical Physics
We consider the general Liénard-type equationü = ∑ n k=0 f ku k for n ≥ 4. This equation naturally admits the Lie symmetry ∂ ∂ t. We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions f 0 ,. .. , f n. Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases n = 1 or n = 2. That is, this paper handles all remaining cases except for n = 3.