Billiards, Refraction, and Blow-Up for Kirchhoff Equations
Abstract
We start from a simple problem in geometric optics. A particle moves between two homothetic ellipses, with refraction at the inner interface and reflection at the outer one. In a suitable nearly circular regime, the small geometric anisotropy of the ellipses is amplified by the strong refraction, and the corresponding return map develops a transverse heteroclinic connection between the two axial motions. We turn this geometric mechanism into a construction for a two-mode Kirchhoff system while keeping the standard quadratic elastic variable throughout. The singular optical model is first reduced to an explicit kick--drift map, for which the heteroclinic connection is obtained by a contraction argument. The same argument also gives exponential convergence along the two tails and transversality. We then show that the connection persists for the genuine elliptic billiard and, subsequently, through a smooth regularization of the reflecting and refracting interfaces. Finally, a small positive background is added to the Kirchhoff coefficient, making it uniformly positive without destroying the hyperbolicity of the axial modes or the transverse connection. This produces a smooth and uniformly positive Kirchhoff coefficient admitting a transverse heteroclinic orbit between two simple modes. Combined with the road-map theorem developed in our previous work, the construction yields a finite-time blow-up example for a forced abstract Kirchhoff equation with a regular forcing term.
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BibTeX
@misc{ghisi2026billiards,
title = {{Billiards, Refraction, and Blow-Up for Kirchhoff Equations}},
author = {Marina Ghisi and Massimo Gobbino},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14502},
eprint = {2609.14502},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14502},
}
RIS
TY - UNPB
TI - Billiards, Refraction, and Blow-Up for Kirchhoff Equations
AU - Marina Ghisi
AU - Massimo Gobbino
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14502
UR - https://arxiv.org/abs/2609.14502
AB - We start from a simple problem in geometric optics. A particle moves between two homothetic ellipses, with refraction at the inner interface and reflection at the outer one. In a suitable nearly circular regime, the small geometric anisotropy of the ellipses is amplified by the strong refraction, and the corresponding return map develops a transverse heteroclinic connection between the two axial motions. We turn this geometric mechanism into a construction for a two-mode Kirchhoff system while keeping the standard quadratic elastic variable throughout. The singular optical model is first reduced to an explicit kick--drift map, for which the heteroclinic connection is obtained by a contraction argument. The same argument also gives exponential convergence along the two tails and transversality. We then show that the connection persists for the genuine elliptic billiard and, subsequently, through a smooth regularization of the reflecting and refracting interfaces. Finally, a small positive background is added to the Kirchhoff coefficient, making it uniformly positive without destroying the hyperbolicity of the axial modes or the transverse connection. This produces a smooth and uniformly positive Kirchhoff coefficient admitting a transverse heteroclinic orbit between two simple modes. Combined with the road-map theorem developed in our previous work, the construction yields a finite-time blow-up example for a forced abstract Kirchhoff equation with a regular forcing term.
ER -