2024/08/04 by Shengfu Deng, Deng, Shengfu, Shu-Ming Sun +1
Mathematics · Physics and Astronomy · #34C37 #34D10 #37L60 #74J35 #Advanced Mathematical Physics Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2408.02017
openalex publication_date 2024/08/04 · openalex created_date 2025/01/17 · openalex updated_date 2026/07/28
This paper concerns the existence of generalized solitary waves (solitary waves with small ripples at infinity) for a diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice. It is proved that the FPUT lattice problem has a generalized solitary-wave solution with the amplitude of those ripples algebraically small using dynamical system approach. The problem is first formulated as a dynamical system problem and then the center manifold reduction theorem with Laurent series expansion is applied to show that this system can be reduced to a system of ordinary differential equations with dimension five. Its dominant system has a homoclinic solution. By applying a perturbation method and adjusting some appropriate constants, it is shown that this homoclinic solution persists for the original dynamical system, which connects to a periodic solution of algebraically small amplitude at infinity (called generalized homoclinic solution), which yields the existence of a generalized solitary wave for the FPUT lattice. The result presented here with the algebraic smallness of those ripples will be needed to show the existence of generalized multi-hump waves for the FPUT lattice later.