Existence proofs of traveling wave solutions on an infinite strip for the suspension bridge equation and proof of orbital stability
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arXiv
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| Format: | Preprint |
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2025
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| author | van der Aalst, Lindsey Cadiot, Matthieu |
| author_facet | van der Aalst, Lindsey Cadiot, Matthieu |
| contents | In this paper, we present a computer-assisted approach for constructively proving the existence of traveling wave solutions of the suspension bridge equation on the infinite strip $Ω= \mathbb{R} \times (-d_2,d_2)$. Using a meticulous Fourier analysis, we derive a quantifiable approximate inverse $\mathbb{A}$ for the Jacobian $D\mathbb{F}(\bar{u})$ of the PDE at an approximate traveling wave solution $\bar{u}$. Such approximate objects are obtained thanks to Fourier coefficients sequences and operators, arising from Fourier series expansions on a rectangle $Ω_0 = (-d_1,d_1) \times (-d_2,d_2)$. In particular, the challenging exponential nonlinearity of the equation is tackled using a rigorous control of the aliasing error when computing related Fourier coefficients. This allows to establish a Newton-Kantorovich approach, from which the existence of a true traveling wave solution of the PDE can be proven in a vicinity of $\bar{u}$. We successfully apply such a methodology in the case of the suspension bridge equation and prove the existence of multiple traveling wave solutions on $Ω$. Finally, given a proven solution $\tilde{u}$, a Fourier series approximation on $Ω_0$ allows us to accurately enclose the spectrum of $D\mathbb{F}(\tilde{u})$. Such a tight control provides the number of negative eigenvalues, which in turns, allows to conclude about the orbital (in)stability of $\tilde{u}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16693 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence proofs of traveling wave solutions on an infinite strip for the suspension bridge equation and proof of orbital stability van der Aalst, Lindsey Cadiot, Matthieu Analysis of PDEs Dynamical Systems In this paper, we present a computer-assisted approach for constructively proving the existence of traveling wave solutions of the suspension bridge equation on the infinite strip $Ω= \mathbb{R} \times (-d_2,d_2)$. Using a meticulous Fourier analysis, we derive a quantifiable approximate inverse $\mathbb{A}$ for the Jacobian $D\mathbb{F}(\bar{u})$ of the PDE at an approximate traveling wave solution $\bar{u}$. Such approximate objects are obtained thanks to Fourier coefficients sequences and operators, arising from Fourier series expansions on a rectangle $Ω_0 = (-d_1,d_1) \times (-d_2,d_2)$. In particular, the challenging exponential nonlinearity of the equation is tackled using a rigorous control of the aliasing error when computing related Fourier coefficients. This allows to establish a Newton-Kantorovich approach, from which the existence of a true traveling wave solution of the PDE can be proven in a vicinity of $\bar{u}$. We successfully apply such a methodology in the case of the suspension bridge equation and prove the existence of multiple traveling wave solutions on $Ω$. Finally, given a proven solution $\tilde{u}$, a Fourier series approximation on $Ω_0$ allows us to accurately enclose the spectrum of $D\mathbb{F}(\tilde{u})$. Such a tight control provides the number of negative eigenvalues, which in turns, allows to conclude about the orbital (in)stability of $\tilde{u}$. |
| title | Existence proofs of traveling wave solutions on an infinite strip for the suspension bridge equation and proof of orbital stability |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2509.16693 |