2017/12/05 by Moudhaffar Bouthelja, Bouthelja, Moudhaffar
Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1712.01597
openalex publication_date 2017/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider the nonlinear wave equation on the circle:u_tt - u_xx + m u = g(x,u), t ∈ ℝ, x ∈ \mathbbS1,where m ∈ [1,2] is a mass and g(x,u)=4u3+ O(u4). This equation will be treated as a perturbation of the integrable Hamiltonian: u_t= v, v_t = - u_xx + m u.Near the origin and for generic m, we prove the existence of small amplitude quasi-periodic solutions close to the solution of the linear equation\eqreffirst equation. For the proof we use an abstract KAM theorem in infinite dimension and a Birkhoff normal form result.