Gr\"{o}bner-Shirshov bases method for Gelfand-Dorfman-Novikov algebras

We establish Gr"{o}bner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand-Dorfman-Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand-Dorfman-Novikov algebra which is not free.