Non-integrability of Gross–Neveu systems

We investigate a class of Hamiltonian systems with n degrees of freedom, the so-called Gross–Neveu systems. They appear in the classical and quantum field theories and statistical physics. Till now, it has been proved that these systems are non-integrable only for small n. In this paper, we prove that these systems are not meromorphically integrable in the Liouville sense for an arbitrary large n. In our proof, we apply the Morales–Ramis theory. We also propose a certain n-parametric generalisation of the classical Gross–Neveu systems and analyse their integrability. Some comments concerning the proved non-integrability and the presence of chaos are also given.