Mathematics

We will consider systems of PDEs endowed with a pair of  compatible Hamiltonian operators of differential-geometric  (Dubrovin-Novikov) type of first and third order respectively. We will  describe the structure of the third order operator in flat coordinates  of the first order operator. We use this description on a simple  example, the WDVV equation in three components (Ferapontov et al.),  where we are able to find a Lagrangian representation according with  Nutku and Pavlov (arxiv:0108214). With the same description we tackle  a 6-component system recently introduced by Sergyeyev and Pavlov  (arXiv:1204.2514, oriented WDVV associativity equation) which has only  one known first-order Hamiltonian structure and find evidences of a  second third order Hamiltonian structure of the above type.