Global dynamics near solitary waves for the generalized Boussinesq equation under even-odd perturbations
Abstract
We study the global dynamics of the generalized Boussinesq equation near standing solitary waves. For even-odd perturbations, Maulen [J. Math. Pures Appl. 177 (2023)] constructed an asymptotically stable center-stable manifold M near the solitary wave Q. We complete this local description by proving that any sufficiently small even-odd perturbation of Q lying outside M either scatters in the energy space as t -> +infinity or blows up in finite time. The proof relies on two main ingredients. First, we classify even-odd solutions with energy below the ground-state energy by identifying two invariant regions, associated with scattering and blow-up, respectively. Second, we establish a one-pass theorem in one dimension: a non-scattering solution cannot re-enter a suitably chosen neighborhood of the solitary wave once it has left that neighborhood. Consequently, the solution remains confined to one of the two invariant regions. Together with the classification above, this yields a dichotomy between scattering and blow-up outside M.
Related
- The asymptotic stability of the stable solitons of the generalized good Boussinesq equation
- Exact Elliptic and Solitary Waves in a Fourth-Order Generalized Burgers–Huxley-Type Equation with Truncated M-Fractional Derivatives
- Long-time validity of the Korteweg-de Vries approximation for the Boussinesq equation
- On the long wave limit of the good Boussinesq equation
- Initial and initial-boundary value problems for a cubic sixth-order Boussinesq equation
BibTeX
@misc{li2026global,
title = {{Global dynamics near solitary waves for the generalized Boussinesq equation under even-odd perturbations}},
author = {Xiaoguang Li and Jun Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39356},
eprint = {2609.39356},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39356},
}
RIS
TY - UNPB
TI - Global dynamics near solitary waves for the generalized Boussinesq equation under even-odd perturbations
AU - Xiaoguang Li
AU - Jun Wang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39356
UR - https://arxiv.org/abs/2609.39356
AB - We study the global dynamics of the generalized Boussinesq equation near standing solitary waves. For even-odd perturbations, Maulen [J. Math. Pures Appl. 177 (2023)] constructed an asymptotically stable center-stable manifold M near the solitary wave Q. We complete this local description by proving that any sufficiently small even-odd perturbation of Q lying outside M either scatters in the energy space as t -> +infinity or blows up in finite time. The proof relies on two main ingredients. First, we classify even-odd solutions with energy below the ground-state energy by identifying two invariant regions, associated with scattering and blow-up, respectively. Second, we establish a one-pass theorem in one dimension: a non-scattering solution cannot re-enter a suitably chosen neighborhood of the solitary wave once it has left that neighborhood. Consequently, the solution remains confined to one of the two invariant regions. Together with the classification above, this yields a dichotomy between scattering and blow-up outside M.
ER -