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

## TL;DR

This paper investigates smooth periodic solutions of a specific reduction of the Benney chain, classifying solutions in certain regimes and revealing a hyperbolic reduction that challenges existing assumptions.

## Contribution

It provides a complete classification of solutions with one real eigenvalue and introduces a hyperbolic reduction that violates genuine non-linearity.

## Key findings

- Complete classification for single real eigenvalue regime
- Identification of a hyperbolic reduction violating genuine non-linearity
- Insights into the elliptic, hyperbolic, and mixed types of the system

## 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.