Small-speed solitary-wave stability and global well-posedness for a Bessel-regularized Airy equation
Abstract
We prove orbital stability of minimizing solitary waves at sufficiently small positive speeds for an Airy equation with a quadratic interaction filtered by a Bessel potential of order one. The stability hypotheses are verified: constrained minimization gives one negative direction, the small-speed KdV limit and operator-norm perturbation of the profile resolvent give a one-dimensional translation kernel, and a differentiable rescaled branch with uniformly bounded derivative gives a positive slope of the quadratic invariant. A conserved Lyapunov functional then proves orbital stability in the three-halves-derivative energy space and the perturbation also gives uniqueness of centered minimizers at small speed. The dynamical framework is supplied by a direct Bourgain-space bilinear estimate, local well-posedness in the square-integrable class, unconditional uniqueness from one-quarter derivative onward, and global well-posedness in the half-derivative energy space. For every prescribed positive speed, constrained minimization constructs smooth positive profiles that are symmetric, strictly monotone away from their center, and exponentially localized. At general positive speeds the stability criterion remains conditional on nondegeneracy and the slope condition.
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BibTeX
@misc{lopez2026smallspeed,
title = {{Small-speed solitary-wave stability and global well-posedness for a Bessel-regularized Airy equation}},
author = {Germán Preciado López},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15751},
eprint = {2610.15751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15751},
}
RIS
TY - UNPB
TI - Small-speed solitary-wave stability and global well-posedness for a Bessel-regularized Airy equation
AU - Germán Preciado López
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15751
UR - https://arxiv.org/abs/2610.15751
AB - We prove orbital stability of minimizing solitary waves at sufficiently small positive speeds for an Airy equation with a quadratic interaction filtered by a Bessel potential of order one. The stability hypotheses are verified: constrained minimization gives one negative direction, the small-speed KdV limit and operator-norm perturbation of the profile resolvent give a one-dimensional translation kernel, and a differentiable rescaled branch with uniformly bounded derivative gives a positive slope of the quadratic invariant. A conserved Lyapunov functional then proves orbital stability in the three-halves-derivative energy space and the perturbation also gives uniqueness of centered minimizers at small speed. The dynamical framework is supplied by a direct Bourgain-space bilinear estimate, local well-posedness in the square-integrable class, unconditional uniqueness from one-quarter derivative onward, and global well-posedness in the half-derivative energy space. For every prescribed positive speed, constrained minimization constructs smooth positive profiles that are symmetric, strictly monotone away from their center, and exponentially localized. At general positive speeds the stability criterion remains conditional on nondegeneracy and the slope condition.
ER -