Existence of generalized solitary waves for a diatomic Fermi-Pasta-Ulam-Tsingou lattice
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| Format: | Preprint |
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2024
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| _version_ | 1866917740990693376 |
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| author | Deng, Shengfu Sun, Shu-Ming |
| author_facet | Deng, Shengfu Sun, Shu-Ming |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02017 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of generalized solitary waves for a diatomic Fermi-Pasta-Ulam-Tsingou lattice Deng, Shengfu Sun, Shu-Ming Dynamical Systems 37L60, 74J35, 34C37, 34D10 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. |
| title | Existence of generalized solitary waves for a diatomic Fermi-Pasta-Ulam-Tsingou lattice |
| topic | Dynamical Systems 37L60, 74J35, 34C37, 34D10 |
| url | https://arxiv.org/abs/2408.02017 |