Quasi-linearization and stability analysis of some self-dual, dark equations and a new dynamical system † In memoriam of Boris Kupershmidt (†2010), a mathematical light in the mysterious world of ‘dark’ equations.

Denis Blackmore, Mykola M. Prytula, Anatolij K. Prykarpatski

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We describe a class of self-dual dark nonlinear dynamical systems a priori allowing their quasi-linearization, whose integrability can be effectively studied by means of a geometrically based gradient-holonomic approach. A special case of the self-dual dynamical system, parametrically dependent on a functional variable is considered, and the related integrability condition is formulated. Using this integrability scheme, we study a new self-dual, dark nonlinear dynamical system on a smooth functional manifold, which models the interaction of atmospheric magneto-sonic Alfvén plasma waves. We prove that this dynamical system possesses a Lax representation that allows its full direct linearization and compatible Poisson structures. Moreover, for this self-dual nonlinear dynamical system we construct an infinite hierarchy of mutually commuting conservation laws and prove its complete integrability.

Original languageEnglish (US)
Article number105007
JournalCommunications in Theoretical Physics
Volume74
Issue number10
DOIs
StatePublished - Oct 1 2022

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy (miscellaneous)

Keywords

  • Hamiltonian system
  • Poisson structure
  • asymptotic analysis
  • complete integrability
  • conservation laws
  • dark evolution system

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