Transformation Groups Applied to Mathematical Physics

Ibragimov, Nail H.

doi:10.1007/978-94-009-5243-0

Cited by 629 publications

(831 citation statements)

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“…All the equations (41) can be linearized by the Cole-Hopf transformation. The obtained recursion operator R (40), which generates the hierarchy of equations (41), is well known as a recursion operator of the Burgers equation in the sense of higher symmetries (or generalized symmetries, or Lie-Bäcklund algebras) (see, e.g., [10]). …”

Section: Burgers Equation's Recursion Operatormentioning

confidence: 99%

Cyclic Bases of Zero-Curvature Representations: Five Illustrations to One Concept

Sakovich

2004

Acta Applicandae Mathematicae

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Section: Burgers Equation's Recursion Operatormentioning

confidence: 99%

Cyclic Bases of Zero-Curvature Representations: Five Illustrations to One Concept

Sakovich

2004

Acta Applicandae Mathematicae

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“…We remark here that the strong and weak accessibility Lie rank conditions possess an easy and nice interpretation within the framework of the differential geometry of jets and prolongations of infinite order, which is becoming a mainstay in some parts of mathematics and physics (see, e.g., [3,4,32,38,41,49,66,67,69] and the references therein). Motivated by the study of differential flatness [25], i.e., by dynamic feedback linearizabilty, this infinite-dimensional geometry is now being developed in nonlinear control (cf.…”

Section: Introductionmentioning

confidence: 96%

A remark on nonlinear accessibility conditions and infinite prolongations

Fliesś

Lévine

Martin

et al. 1997

Systems & Control Letters

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The Lie rank condition for strong nonlinear accessibility is interpreted via the differential geometry of jets and prolongations of infinite order. It yields to a new Lie algebraic criterion and to the consideration of first integrals, which apply to nonlinear systems in quite general form; the latter characterization in particular is valid without distinguishing between the system variables and is independent of time-scalings. Weak accessibility is also considered.

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“…The mathematical tools necessary for the expression of the above-mentioned concept are provided by Symmetry Analysis [3]. We begin with a preliminary discussion and basic notions.…”

Section: Introductionmentioning

confidence: 99%

Invariant Manifolds in Parametric turbulent Models

Grebenev

Черных

2004

RMTA

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The article is devoted to examining the so-called local-equilibrium approximations used while modeling turbulent flows. The dynamics of a far plane turbulent wake are investigated as an example. In this article, we analyze these approximations by using the method of differential constraints. We show that some algebraic models based on using the local-equilibrium approximation can be interpreted as equations of invariant manifolds generated by the models under consideration. Reduction of the models on the corresponding invariant manifolds made it possible to find self-similar solutions and to separate explicit solutions. Moreover, some empirical constants may be calculated and their obtained values are close to the recommended quantities.Keywords: local-equilibrium approximation, turbulent wake, method of differential constrains, invariant manifold, self-similar solutions. ResumenEl artículo trata de examinar el llamado equilibrio local de aproximaciones usado cuando se modelan fluidos turbulentos. La dinámica de una estela turbulenta plana es estudiada como un ejemplo. Analizamos estas aproximaciones usando el método de restricciones diferenciales. Mostramos que algunos modelos algebraicos basados en el uso de la aproximación de equilibrio local pueden ser interpretadas como ecuaciones de variedades invariantes generadas por los modelos bajo consideración. La reducción de los modelos a las variedades correspondientes hicieron posible encontrar soluciones auto-similares y separar soluciones explícitas. Es más, algunas constantes empíricas pueden ser calculadas y los valores obtenidos son cercanos a las cantidades recomendadas.

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Transformation Groups Applied to Mathematical Physics

Cited by 629 publications

References 0 publications

Cyclic Bases of Zero-Curvature Representations: Five Illustrations to One Concept

Cyclic Bases of Zero-Curvature Representations: Five Illustrations to One Concept

A remark on nonlinear accessibility conditions and infinite prolongations

Invariant Manifolds in Parametric turbulent Models

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