TY - JOUR

T1 - In search of periodic solutions for a reduction of the Benney chain

AU - Bialy, Misha

AU - Mironov, Andrey E.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - 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.

AB - 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.

KW - QUASI-LINEAR SYSTEM

KW - CONSERVATION-LAWS

KW - POLYNOMIAL INTEGRALS

KW - CLASSIFICATION

KW - EQUATIONS

KW - 2-TORUS

UR - http://www.scopus.com/inward/record.url?scp=85035047680&partnerID=8YFLogxK

U2 - 10.1063/1.4991977

DO - 10.1063/1.4991977

M3 - Article

AN - SCOPUS:85035047680

VL - 58

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 11

M1 - 112701

ER -