Deformation of Lie derivative and μ-symmetries

Morando, Paola

doi:10.1088/1751-8113/40/38/007

Cited by 24 publications

(42 citation statements)

References 22 publications

(59 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…note that here the functions ρ i are arbitrary ones. If in addition the matrices Λ i are multiples of the identity, see (37), this reads…”

Section: Simple Twisted Symmetries: µ-Symmetriesmentioning

confidence: 99%

See 1 more Smart Citation

Simple and collective twisted symmetries

Gaeta¹

2021

JNMP

View full text Add to dashboard Cite

After the introduction of λ-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, λ and µ-symmetries), this amounts to a certain kind of gauge transformation.In a previous review paper [23] we have surveyed the first part of the developments of this theory; in the present paper we review recent developments. In particular, we provide a unifying geometrical description of the different types of twisted symmetries; this is based on the classical Frobenius reduction applied to distribution generated by Lie-point (local) symmetries.

show abstract

“…note that here the functions ρ i are arbitrary ones. If in addition the matrices Λ i are multiples of the identity, see (37), this reads…”

Section: Simple Twisted Symmetries: µ-Symmetriesmentioning

confidence: 99%

“…⊙ Remark 19. Here we are not discussing the relation of µ-prolongations and symmetries with µ-related deformed exterior and Lie derivatives; this point of view was advocated by Morando [37].…”

Section: Simple Twisted Symmetries: µ-Symmetriesmentioning

confidence: 99%

Simple and collective twisted symmetries

Gaeta¹

2021

JNMP

View full text Add to dashboard Cite

show abstract

“…It can be shown that (20) has no point symmetries but admits a λ-covering with λ = xe x/u . Hence, if we consider the covering system E ′ defined by (20) together with w 1 = λ, it admits the nonlocal symmetry Y generated by ϕ 1 = e w u 2 /x and ϕ 2 = −e w .…”

Section: Remarkmentioning

confidence: 99%

“…Hence, if we consider the covering system E ′ defined by (20) together with w 1 = λ, it admits the nonlocal symmetry Y generated by ϕ 1 = e w u 2 /x and ϕ 2 = −e w . We will search for solvable structures which extend the nonabelian algebra G =< Y 1 = Y, Y 2 = ∂ w >.…”

Section: Remarkmentioning

confidence: 99%

“…Among these, in our opinion, a special attention is deserved by the notions of λ-symmetry on the one hand and solvable structure on the other hand. Both notions, introduced in [21] and [3], respectively, are suitable for a wide range of applications [4,5,9,11,14,15,20,25,26] and will be the starting point of our forthcoming discussion. In particular, we will provide a more effective reduction method which interconnects both notions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Local and nonlocal solvable structures in the reduction of ODEs

Ferraioli¹,

Morando²

2008

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Abstract.Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure.

show abstract

A relationship between λ‐symmetries and first integrals for ordinary differential equations

Zhang

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we provide some geometric properties of -symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of -symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries.We also investigate the properties of -symmetries in the sense of the deformed Lie derivative and differential operator. We show that -symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases. KEYWORDSdeformed property, first integrals, Frobenius integrable, symmetries, -symmetries MSC CLASSIFICATION 34A05; 34C14 INTRODUCTIONClassical symmetry groups play an important role in analyzing differential equations. They have been widely used to reduce the order of ordinary differential equations and to reduce the number of independent variables of partial differential equations. However, many of the differential equations we study, especially nonlinear differential equations, do not in general possess enough symmetries. It is therefore essential to devise some kinds of proper extensions of classical Lie method in order to handle more situations. In the last few years, various extensions have been done, and some new classes of symmetries have been introduced. Among these extensions, -symmetries have been introduced by Muriel and Romero 1 based on a new method of prolonging vector fields known as -prolongation, which strictly include symmetries.Although -symmetries are not symmetries in the proper sense, as they do not map solutions into solutions, they can be used to perform reduction procedure via exactly the same method used for standard symmetries. Several usual methods of reduction for ODEs that do not come from symmetries are derived from the existence of -symmetries; one can see Muriel and Romero 1 and relevant references for further details. They can also be viewed as a class of twisted symmetries, and the recently developments can be reviewed in Gaeta. 2 Especially, several relations among -symmetries and first integrals for ordinary differential equations are studied in previous studies 3-11 and references therein. For applications of -symmetries to derive first integrals or conservation laws of dynamical systems or Hamiltonian systems, one can see previous studies 12-14 for example. By a geometrical characterization of -prolongation of vector fields, -symmetries have been extended to partial differential equations, leading to -symmetries that can also be used to derive conservation laws of the equations. 15,16 The investigation of -symmetries has practical advantages. First of all, the determining equations for symmetries may not have nontrivial solutions but the corresponding equations for -symmetries may have. In addition, -symmetries can be algorithmically obtained by using the Jacobi last multipliers. 17 Secondly, there exists an unknown function in the determining equations for -symmetries, so comparing with classical Lie method,...

show abstract

Deformation of Lie derivative and μ-symmetries

Cited by 24 publications

References 22 publications

Simple and collective twisted symmetries

Simple and collective twisted symmetries

Local and nonlocal solvable structures in the reduction of ODEs

A relationship between λ‐symmetries and first integrals for ordinary differential equations

Contact Info

Product

Resources

About