Symmetry Analysis, Integrability and Completeness of Geodesics of a Double of the Affine Lie Group of the Real Line (original) (raw)

Symmetries, Integrability and Exact Solutions for Nonlinear Systems

2011

The paper intends to offer a general overview on what the concept of integrability means for a nonlinear dynamical system and how the symmetry method can be applied for approaching it. After a general part where key problems as direct and indirect symmetry method or optimal system of solutions are tackled out, in the second part of the lecture two concrete models of nonlinear dynamical systems are effectively studied in order to illustrate how the procedure is working out. The two models are the 2D Ricci flow model coming from the general relativity and the 2D convective-diffusion equation. Part of the results, especially concerning the optimal systems of solutions, are new ones.

Complete Symmetry Groups of Ordinary Differential Equations and Their Integrals: Some Basic Considerations

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. 

Complete Symmetry Groups: A Connection Between Some Ordinary Differential Equations and Partial Differential Equations

The concept of complete symmetry groups has been known for some time in applications to ordinary differential equations. In this Thesis we apply this concept to partial differential equations. For any 1+1 linear evolution equation of Lie’s type (Lie S (1881) U¨ ber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung Archiv f ¨ ur Mathematik og Naturvidenskab 6 328-368 Translation into English by Ibragimov NH in CRC Handbook of Lie Group Analysis of Differential Equations 2 73-508) containing three and five exceptional point symmetries and a nonlinear equation admitting a finite number of Lie point symmetries, the representation of the complete symmetry group has been found to be a six-dimensional algebra isomorphic to sl(2,R) �s A3,1, where the second subalgebra is commonly known as the Heisenberg-Weyl algebra. More generally the number of symmetries required to specify any partial differential equations has been found to equal the numb...

Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

2012

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

Group analysis of differential equations: A new type of Lie symmetries

2018

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...

Application of Lie-group symmetry analysis to an infinite hierarchy of differential equations at the example of first order ODEs

This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function in the hierarchy there exists a corresponding determined equation to which it can be bijectively mapped to. As a direct consequence, its admitted set of symmetry transformations must be identified as a weaker set of indeterminate equivalence transformations. The reason is that no unique general solution can be constructed, not even in principle. Instead, infinitely many disjoint and thus independent general solution manifolds exist. This is in clear contrast to a closed system of differential equations that only allows for a single and thus unique general solution manifold, which, by definition, covers all possible particular solutions this system can admit. Herein, different first order Riccati-ODEs serve as an example, but this analysis is not restricted to them. All conclusions drawn in this study will translate to any first order or higher order ODEs as well as to any PDEs.

Double Bracket Equations and Geodesic Flows on Symmetric Spaces

Communications in Mathematical Physics, 1997

In this paper we consider the geometry of Hamiltonian flows on the cotangent bundle of coadjoint orbits of compact Lie groups and on symmetric spaces. A key idea here is the use of the normal metric to define the kinetic energy. This leads to Hamiltonian flows of the double bracket type. We analyze the integrability of geodesic flows according to the method of Thimm. We obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity. We demonstrate for example integrability of the geodesic flow on the real and complex Grassmannians. We also consider right invariant systems and the generalized rigid body equations in this setting.

Symmetries of linear ordinary differential equations

Journal of Physics A: Mathematical and General, 1997

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.

Solutions of systems of ordinary differential equations using invariants of symmetry groups

AIP Conference Proceedings, 2019

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.