On a countable sequence of homoclinic orbits arising near a saddle-center point
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909412212342784 |
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| author | Baldomá, Inmaculada Guardia, Marcel Pelinovsky, Dmitry E. |
| author_facet | Baldomá, Inmaculada Guardia, Marcel Pelinovsky, Dmitry E. |
| contents | Exponential small splitting of separatrices in the singular perturbation theory leads generally to nonvanishing oscillations near a saddle--center point and to nonexistence of a true homoclinic orbit. It was conjectured long ago that the oscillations may vanish at a countable set of small parameter values if there exist a quadruplet of singularities in the complex analytic extension of the limiting homoclinic orbit. The present paper gives a rigorous proof of this conjecture for a particular fourth-order equation relevant to the traveling wave reduction of the modified Korteweg--de Vries equation with the fifth-order dispersion term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a countable sequence of homoclinic orbits arising near a saddle-center point Baldomá, Inmaculada Guardia, Marcel Pelinovsky, Dmitry E. Dynamical Systems Mathematical Physics Analysis of PDEs Pattern Formation and Solitons Exponential small splitting of separatrices in the singular perturbation theory leads generally to nonvanishing oscillations near a saddle--center point and to nonexistence of a true homoclinic orbit. It was conjectured long ago that the oscillations may vanish at a countable set of small parameter values if there exist a quadruplet of singularities in the complex analytic extension of the limiting homoclinic orbit. The present paper gives a rigorous proof of this conjecture for a particular fourth-order equation relevant to the traveling wave reduction of the modified Korteweg--de Vries equation with the fifth-order dispersion term. |
| title | On a countable sequence of homoclinic orbits arising near a saddle-center point |
| topic | Dynamical Systems Mathematical Physics Analysis of PDEs Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2412.01563 |