2008/04/14 by Michael, Bialy
Mathematics · #35L65 #35L67 #70H06 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematics and Applications #Rings, Modules, and Algebras #Symplectic Geometry (math.SG) #math.AP #math.SG #msc:35L65 #msc:35L67 #msc:70H06
paper · pdf · doi:10.48550/arxiv.0804.2187
arxiv created 2008/04/14 · openalex publication_date 2008/04/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study periodic solutions for a quasi-linear system, which is the so called dispersionless Lax reduction of the Benney moments chain. This question 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 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 existence of the fourth power integral of motion for H: if it exists, then it is equal necessarily to the square of the quadratic one. Our method uses two new general observations. The first is the genuine non-linearity of the maximal and minimal eigenvalues for the system. The second observation uses the compatibility conditions of Gibonns-Tsarev in order to give certain exactness for the system in Riemann invariants. This exactness opens a possibility to apply the Lax analysis of blow up of smooth solutions, which usually does not work for systems of higher order.