In search of periodic solutions for a reduction of the Benney chain

Abstract: We search for smooth periodic solutions for the system of quasi-linear PDEs known as the Lax dispersionless reduction of the Benney moments chain. It is naturally related to the existence of a polynomial in momenta integral for a Classical Hamiltonian system with 1,5 degrees of freedom. For the solution in question it is not known a-priori if the system is elliptic or hyperbolic or of mixed type. We consider two possible regimes for the solution. The first is the case of only one real eigenvalue, where we can completely classify the solutions. The second case of strict Hyperbolicity is really a challenge. We find a remarkable 2 by 2 reduction which is strictly Hyperbolic but violates the condition of genuine non-linearity.