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

Misha Bialy, Andrey E. Mironov

Research output: Contribution to journalArticlepeer-review

Abstract

We search for smooth periodic solutions for the system of quasi-linear equations 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 × 2 reduction which is strictly hyperbolic with one umbilic point but violates the condition of genuine non-linearity.

Original languageEnglish
Article number112701
Number of pages9
JournalJournal of Mathematical Physics
Volume58
Issue number11
DOIs
Publication statusPublished - 1 Nov 2017

Keywords

  • QUASI-LINEAR SYSTEM
  • CONSERVATION-LAWS
  • POLYNOMIAL INTEGRALS
  • CLASSIFICATION
  • EQUATIONS
  • 2-TORUS

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