Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2601.01923 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909981338501120 |
|---|---|
| author | Lonardoni, Beatriz Signori Natali, Fabio |
| author_facet | Lonardoni, Beatriz Signori Natali, Fabio |
| contents | This paper presents a comprehensive analysis of several aspects of the sinh-Gordon equation within a periodic setting. Our investigation proceeds in three main stages. First we establish the existence of periodic solutions for a fixed wave speed and varying periods by applying the mountain pass theorem. Subsequently, for a fixed period, we construct a family of periodic solutions that depend smoothly on the wave speed; this is achieved via the implicit function theorem. The spectral stability of these waves is then rigorously addressed. We perform a detailed spectral analysis of the linearized operator around the wave of fixed period. A key element in this analysis is the monotonicity of the period map, which, when combined with Morse index theory, enables us to fully characterize the non-positive spectrum of the projected operator in the space of zero-mean periodic functions. Finally, by employing the Hamiltonian-Krein index analysis, we determine the spectral stability and instability of the constructed waves. We additionally discuss qualitative aspects of the Cauchy problem associated with the sinh-Gordon equation, including local well-posedness and blow-up phenomena. The former supports a new linearization of the problem, while the latter predicts the spectral instability of the wave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01923 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Qualitative Aspects of Periodic Traveling Waves for the Sinh-Gordon equation Lonardoni, Beatriz Signori Natali, Fabio Analysis of PDEs This paper presents a comprehensive analysis of several aspects of the sinh-Gordon equation within a periodic setting. Our investigation proceeds in three main stages. First we establish the existence of periodic solutions for a fixed wave speed and varying periods by applying the mountain pass theorem. Subsequently, for a fixed period, we construct a family of periodic solutions that depend smoothly on the wave speed; this is achieved via the implicit function theorem. The spectral stability of these waves is then rigorously addressed. We perform a detailed spectral analysis of the linearized operator around the wave of fixed period. A key element in this analysis is the monotonicity of the period map, which, when combined with Morse index theory, enables us to fully characterize the non-positive spectrum of the projected operator in the space of zero-mean periodic functions. Finally, by employing the Hamiltonian-Krein index analysis, we determine the spectral stability and instability of the constructed waves. We additionally discuss qualitative aspects of the Cauchy problem associated with the sinh-Gordon equation, including local well-posedness and blow-up phenomena. The former supports a new linearization of the problem, while the latter predicts the spectral instability of the wave. |
| title | Qualitative Aspects of Periodic Traveling Waves for the Sinh-Gordon equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.01923 |