TY - JOUR

T1 - On periodic solutions for a reduction of benney chain

AU - Bialy, Misha

PY - 2009/12

Y1 - 2009/12

N2 - 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.

AB - 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.

KW - Benney chain

KW - Genuine nonlinearity

KW - Rich systems

KW - Riemann invariants

UR - http://www.scopus.com/inward/record.url?scp=77951252980&partnerID=8YFLogxK

U2 - 10.1007/s00030-009-0032-y

DO - 10.1007/s00030-009-0032-y

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:77951252980

SN - 1021-9722

VL - 16

SP - 731

EP - 743

JO - Nonlinear Differential Equations and Applications

JF - Nonlinear Differential Equations and Applications

IS - 6

ER -