Integrability and tensor invariants in dynamics

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The project is devoted to the theory and applications of tensor invariants of systems of partial differential equations (pde) and to the problems of integrability of pde's and dynamical systems. I will work on the Courant problems for systems of first order pde's. In papers [*2] and [*3], I have resolved the Courant problems on the decoupling and block-diagonalization of the systems with one spatial variable. I will apply the developed methods to study the Courant problems for systems with several spatial variables and the Courant problem on the invariant characterization of systems of conservation laws. Solution of these problems will use new invariants and identities derived in my papers [*5, 12] and will have applications for their numerical investigations. I will work on the extensions of Courant problems connected with existence of Hamiltonian and Bi-Hamiltonian structures for systems of hydrodynamic type. In collaboration with my PhD students, we will find invariant criteria that will be very useful for examination if a given system of pde's is Hamiltonian. Nowadays, such criteria are known only for the three-component systems due to our paper [7], jointly with P. Reynolds. I will construct integrable Hamiltonian systems generalizing Volterra lattice and systems known in the literature as Bogoyavlenskij's lattices, using the method of descent recently developed in [6,10]. The novelty of the method is that it gives large families of Hamiltonian systems dynamics of which can be described in detail because they possess plenty of first integrals. I will work on the rigorous proof of integrability of these systems in the Liouville's sense and on the integrability of certain special cases in the exact form. I will study also integrability of several integro-differential equations obtained as continual limits of discrete integrable lattices. Jointly with my Master students, we will derive reductions of the ideal magnetohydrodynamics (MHD) equations with helical symmetry and will construct exact equipartition solutions [19] to the viscous MHD equations.