On periodic solutions for a reduction of benney chain

Misha Bialy*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study periodic solutions for a quasi-linear system, which naturally arises in search of integrable Hamiltonian systems of the form H = p2/2 + u(q, t). Our main result classifies completely periodic solutions for such a 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 the existence of a fourth power integral of motion for H: if it exists, then it is equal necessarily to the square of a quadratic integral. Our main observation for the quasi-linear system is the genuine non-linearity of the maximal and minimal eigenvalues in the sense of Lax. We use this observation in the hyperbolic region, while the "elliptic" region is treated using the maximum principle.

Original languageEnglish
Pages (from-to)731-743
Number of pages13
JournalNonlinear Differential Equations and Applications
Issue number6
StatePublished - Dec 2009


  • Benney chain
  • Genuine nonlinearity
  • Rich systems
  • Riemann invariants


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