On Periodic solutions for a reduction of Benney chain

TL;DR

This paper classifies all periodic solutions of a specific reduction of the Benney chain, showing they are traveling waves, and demonstrates the nonexistence of nontrivial fourth integrals of motion for the associated Hamiltonian system.

Contribution

It provides a complete classification of periodic solutions for a 3x3 system related to the Benney chain, revealing they are necessarily traveling waves and constraining integrals of motion.

Findings

All periodic solutions are traveling waves.

No nontrivial fourth power integral of motion exists for the Hamiltonian.

New methods using eigenvalue non-linearity and Gibbons-Tsarev compatibility conditions.

Abstract

We study periodic solutions for a quasi-linear system, which is the so called dispersionless Lax reduction of the Benney moments chain. This question naturally arises in search of integrable Hamiltonian systems of the form $ H=p^2/2+u(q,t) $ Our main result classifies completely periodic solutions for 3 by 3 system. We prove that the only periodic solutions have the form of traveling waves, so in particular, the potential $u$ is a function of a linear combination of $t$ and $q$. This result implies that the there are no nontrivial cases of existence of the fourth power integral of motion for $H$: if it exists, then it is equal necessarily to the square of the quadratic one. Our method uses two new general observations. The first is the genuine non-linearity of the maximal and minimal eigenvalues for the system. The second observation uses the compatibility conditions of Gibonns-Tsarev in order to give certain exactness for the system in Riemann invariants. This exactness opens a possibility to apply the Lax analysis of blow up of smooth solutions, which usually does not work for systems of higher order.