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.