The generalized centrally extended Lie algebraic structures and related integrable heavenly type equations

There are studied Lie-algebraic structures of a wide class of heavenly type non-linear integrable equations, related with coadjoint flows on the adjoint space to a loop vector field Lie algebra on the torus. These flows are generated by the loop Lie algebras of vector fields on a torus and their coadjoint orbits and give rise to the compatible Lax-Sato type vector field relationships. The related infinite hierarchy of conservations laws is analysed and its analytical structure, connected with the Casimir invariants, is discussed. We present the typical examples of such equations and demonstrate in details their integrability within the scheme developed. As examples, we found and described new multidimensional generalizations of the Mikhalev-Pavlov and Alonso-Shabat type integrable dispersionless equation, whose seed elements possess a special factorized structure, allowing to extend them to the multidimensional case of arbitrary dimension

The Electromagnetic Lorentz Condition Problem and Symplectic Properties of Maxwell and Yang-Mills Type Dynamical Systems

Symplectic structures associated to connection forms on certain types of principal fiber bundles are constructed via analysis of reduced geometric structures on fibered manifolds invariant under naturally related symmetry groups. This approach is then applied to nonstandard Hamiltonian analysis of of dynamical systems of Maxwell and Yang-Mills type. A symplectic reduction theory of the classical Maxwell equations is formulated so as to naturally include the Lorentz condition (ensuring the existence of electromagnetic waves), thereby solving the well known Dirac -Fock - Podolsky problem. Symplectically reduced Poissonian structures and the related classical minimal interaction principle for the Yang-Mills equations are also considered. 1

Classical R-matrix theory for bi-Hamiltonian field systems

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The theory is developed for (1+1)-dimensional case where the space variable belongs either to R or to various discrete sets. Then, the extension onto (2+1)-dimensional case is made, when the second space variable belongs to R. The formalism presented contains many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.Comment: review article, 39 page