A geometric approach to higher-order Riccati chain: Darboux polynomials and constants of the motion

Cariñena, José F.; Guha, Partha; Rañada, Manuel F.

doi:10.1088/1742-6596/175/1/012009

Cited by 22 publications

(36 citation statements)

References 23 publications

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“…There, it is stated that such equations can be used to derive solutions for certain PDEs. In addition, equation (2.25) also appears in the book by Davis [86], and the particular case with f (t) = 0 has recently been treated through geometric methods in [41,66].…”

Section: Painleve-ince Equations and Other Sode Lie Systemsmentioning

confidence: 99%

Lie systems: theory, generalisations, and applications

Cariñena

Lucas

2011

Dissertationes Math.

396

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Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure

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Section: Painleve-ince Equations and Other Sode Lie Systemsmentioning

confidence: 99%

Lie systems: theory, generalisations, and applications

Cariñena

Lucas

2011

Dissertationes Math.

396

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“…We also mention that the second-order Riccati equation has been studied in [13] from a geometric perspective and it has been proved to admit two alternative Lagrangian formulations, both Lagrangians being of a non-natural class (neither potential nor kinetic term). An analysis of the higher-order Riccati equations and all these properties (Lagrangians, symmetries, Darboux polynomials and Jacobi multipliers) is presented in [14].…”

Section: Introductionmentioning

confidence: 99%

Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials

Cariñena¹,

Guha²,

Rañada³

2009

Nonlinearity

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A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and it is proved the existence of two alternative Lagrangian formulations, both Lagrangians being of a non-natural class (neither potential nor kinetic term). These higher-order Abel equations are studied by means of their Darboux polynomials and Jacobi multipliers. In all the cases a family of constants of the motion is explicitly obtained. The general n-dimensional case is also studied.

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“…One can again straightforwardly verify that the contact transformation (25) linearizes the nonlinear ODE (21) toü = 0, which is of the same order as (21). Using the contact transformation (25) we can find the general solution of the second order Abel equation as well.…”

Section: An Example From Abel Chainmentioning

confidence: 83%

“…In the above λ and µ are the infinitesimal symmetries of the given nonlinear ODE, for example (15) or (21). If λ and µ are restricted to the variables t and x only, then they become the Lie point symmetries of the given equation.…”

Section: Algorithmmentioning

confidence: 99%

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Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains

Pradeep¹,

Chandrasekar

Mohanasubha

et al. 2016

Communications in Nonlinear Science and Numerical Simulation

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We identify contact transformations which linearize the given equations in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The identified contact transformations are not of Cole-Hopf type and are new to the literature. The linearization of Abel chain of equations is also demonstrated explicitly for the first time. The contact transformations can be utilized to derive dynamical symmetries of the associated nonlinear ODEs. The wider applicability of identifying this type of contact transformations and the method of deriving dynamical symmetries by using them is illustrated through two dimensional generalizations of the Riccati and Abel chains as well.

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A geometric approach to higher-order Riccati chain: Darboux polynomials and constants of the motion

Cited by 22 publications

References 23 publications

Lie systems: theory, generalisations, and applications

Lie systems: theory, generalisations, and applications

Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials

Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains

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