Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Symmetry Integrability and Geometry-Methods and Applications

article

Oksana Ye. Hentosh¹  Yarema A. Prykarpatsky²  Denis Blackmore³  Anatolij K. Prykarpatski⁴ 

[1] Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine;Department of Applied Mathematics, University of Agriculture in Krakow;Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights;Department of Physics, Mathematics and Computer Science, Cracow University of Technology

关键词: Lax–Sato equations;    multi-dimensional integrable heavenly equations;    Lax integrability;    Hamiltonian system;    torus diffeomorphisms;    loop Lie algebra;    Lie-algebraic scheme;    Casimir invariants;    R-structure;    Lie–Poisson structure;    conformal structures;    multidimensional heavenly equation;   

DOI  :  10.3842/SIGMA.2019.079

来源: National Academy of Science of Ukraine

【 摘 要 】

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

【 授权许可】

Unknown