Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application

Oksana E. Hentosh; Yarema A. Prykarpatski; Denis Blackmore; Anatolij Prykarpatski

doi:10.33187/jmsm.435466

EN

Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application

Abstract

Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were initiated by Plebanski and later analyzed in a series of articles. The AKS-algebraic and related $\mathcal{R}$-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie--Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equations under consideration. Moreover, all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a fascinating Lagrange--d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax--Sato equations is also discussed. We pay a special attention to a generalization of the devised Lie-algebraic scheme to a case of loop Lie superalgebras of superconformal diffeomorphisms of the $1|N$-dimensional supertorus. This scheme is applied to constructing the Lax--Sato integrable supersymmetric analogs of the Liouville and Mikhalev-Pavlov heavenly equation for every $N\in\mathbb{N}\backslash\lbrace 4;5\rbrace.$

Keywords

Lax--Sato equations,Lax--Sato integrability,Hamiltonian system,Lie-algebraic scheme,Casimir invariants,R-structure,Lie-Poisson structure,Heavenly equations,Torus diffeomorphisms,Loop Lie algebra,Lagrange--d'Alembert principle

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Oksana E. Hentosh This is me
Ukraine

Yarema A. Prykarpatski This is me
Poland

Denis Blackmore This is me
United States

Anatolij Prykarpatski
Poland

Publication Date

September 30, 2018

Submission Date

June 21, 2018

Acceptance Date

September 20, 2018

Published in Issue

Year 2018 Volume: 1 Number: 2

DOI

https://doi.org/10.33187/jmsm.435466

IZ

https://izlik.org/JA63MY67EP

APA

Hentosh, O. E., Prykarpatski, Y. A., Blackmore, D., & Prykarpatski, A. (2018). Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application. Journal of Mathematical Sciences and Modelling, 1(2), 105-130. https://doi.org/10.33187/jmsm.435466

AMA

1.Hentosh OE, Prykarpatski YA, Blackmore D, Prykarpatski A. Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application. Journal of Mathematical Sciences and Modelling. 2018;1(2):105-130. doi:10.33187/jmsm.435466

Chicago

Hentosh, Oksana E., Yarema A. Prykarpatski, Denis Blackmore, and Anatolij Prykarpatski. 2018. “Generalized Lie-Algebraic Structures Related to Integrable Dispersionless Dynamical Systems and Their Application”. Journal of Mathematical Sciences and Modelling 1 (2): 105-30. https://doi.org/10.33187/jmsm.435466.

EndNote

Hentosh OE, Prykarpatski YA, Blackmore D, Prykarpatski A (September 1, 2018) Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application. Journal of Mathematical Sciences and Modelling 1 2 105–130.

IEEE

[1]O. E. Hentosh, Y. A. Prykarpatski, D. Blackmore, and A. Prykarpatski, “Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application”, Journal of Mathematical Sciences and Modelling, vol. 1, no. 2, pp. 105–130, Sept. 2018, doi: 10.33187/jmsm.435466.

ISNAD

Hentosh, Oksana E. - Prykarpatski, Yarema A. - Blackmore, Denis - Prykarpatski, Anatolij. “Generalized Lie-Algebraic Structures Related to Integrable Dispersionless Dynamical Systems and Their Application”. Journal of Mathematical Sciences and Modelling 1/2 (September 1, 2018): 105-130. https://doi.org/10.33187/jmsm.435466.

JAMA

1.Hentosh OE, Prykarpatski YA, Blackmore D, Prykarpatski A. Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application. Journal of Mathematical Sciences and Modelling. 2018;1:105–130.

MLA

Hentosh, Oksana E., et al. “Generalized Lie-Algebraic Structures Related to Integrable Dispersionless Dynamical Systems and Their Application”. Journal of Mathematical Sciences and Modelling, vol. 1, no. 2, Sept. 2018, pp. 105-30, doi:10.33187/jmsm.435466.

Vancouver

1.Oksana E. Hentosh, Yarema A. Prykarpatski, Denis Blackmore, Anatolij Prykarpatski. Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application. Journal of Mathematical Sciences and Modelling. 2018 Sep. 1;1(2):105-30. doi:10.33187/jmsm.435466