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On periodic solutions for a reduction of Benney chain. (English) Zbl 1182.35014

Summary: We study periodic solutions for a quasilinear system, which 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 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 quasilinear system is the genuine nonlinearity 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.

MSC:

35B10 Periodic solutions to PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
35L60 First-order nonlinear hyperbolic equations
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