2015/08/06 by Haiyan Xu, Xu, Haiyan · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1508.01500
openalex publication_date 2015/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the following Hamiltonian equation on the L2 Hardy space on the circle S1 , i∂_ t u = Π(|u|^ 2 u) + α(u|1) , α∈ℝ , where Π is the Szegő projector. The above equation with α= 0 was introduced by Gérard and Grellier as an important mathematical model [5, 7, 3]. In this paper, we continue our studies started in [22], and prove our system is completely integrable in the Liouville sense. We study the motion of the singular values of the related Hankel operators and find a necessary condition of norm explosion. As a consequence, we prove that the trajectories of the solutions will stay in a compact subset, while more initial data will lead to norm explosion in the case α>0.