arXiv subjects

Miguel Ayala

Publications and source records attributed to Miguel Ayala.

4 recordsLinked to original sources

arXiv2026

Lagrange points of the restricted three-body problem in spaces of constant curvature

We continue the study initiated by Kilin ({\em Reg. Chaot. Dyn.} 4, (1999)) and by Martínez and Simó ({\em Celest. Mech. Dynam. Astronom.} 128, (2017)) on the classification and stability of the relative equilibria of the restricted three-body problem in two-dimensional spaces of constant curvature, which generalize the classical Lagrange points of the planar problem. After formulating the problem as an autonomous Lagrangian system with two degrees of freedom, whose only parameters are the curvature $κ$ and the mass ratio $μ$ of the primaries, we establish several classification results for the case $κ>0$ by combining analytical methods with computer-assisted proofs. These results provide rigorous confirmation of phenomena for which previously only numerical evidence was available. We also provide topological explanations for the qualitative differences between the behavior of relative equilibria in positive curvature and that observed in the planar and negative-curvature cases. Our analysis focuses on the regime of small $μ$ and indicates that positive curvature has a stabilizing effect on the triangular equilibria $\Ll_4$ and $\Ll_5$, whereas negative curvature appears to have the opposite effect.

math-ph↗

arXiv2026

Computer-Assisted Proofs for Geometric Optimization: From Crystallization to Carbon Nanotubes

We present a framework based on computer-assisted proofs that turns geometry optimization simulations for atomistic structures into mathematical proofs. Starting from a numerically computed approximation of a local minimizer or saddle point, we use validated numerical computations to prove the existence of a critical point of the potential energy close to this approximation. We demonstrate this framework in two settings. In the first, we study capped carbon nanotubes modeled as minimizers of carbon interatomic potentials (harmonic, Tersoff, and a Huber potential) and obtain proven bounds on tube diameter, bond lengths, and bond angles. In particular, we show that caps induce diameter oscillations along the tube. As a second application, we consider a finite Lennard-Jones crystal in a face-centered cubic (fcc) lattice and provide computer-assisted proofs of a local minimizer representing the perfect crystal, a local minimizer with a single vacancy defect, and a saddle point that connects two single-vacancy configurations on the energy landscape.

physics.comp-ph↗

arXiv2026

Computer-Assisted Proofs of Gap Solitons in Bose-Einstein Condensates

We provide a framework for turning a numerical simulation of a gap soliton in the one-dimensional Gross-Pitaevskii equation into a rigorous mathematical proof of its existence. These nonlinear localized solutions play a central role in the study of Bose-Einstein condensates (BECs). We reformulate the problem of proving their existence as the search for homoclinic orbits in a dynamical system. We then apply computer-assisted proof techniques to obtain verifiable conditions under which a numerically approximated trajectory corresponds to a true homoclinic orbit. This work also presents the first examples of computer-assisted proofs of gap solitons in the Gross-Pitaevskii equation on non-perturbative parameter regimes.

math.DS↗

arXiv2024

$\mathfrak{gl}(3)$ Polynomial Integrable System: Different Faces of the 3-Body/${\mathcal A}_2$ Elliptic Calogero Model

It is shown that the $\mathfrak{gl}(3)$ polynomial integrable system, introduced by Sokolov-Turbiner in [arXiv:1409.7439], is equivalent to the $\mathfrak{gl}(3)$ quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian as well as their third-order integral can be rewritten in terms of $\mathfrak{gl}(3)$ algebra generators. In turn, all these $\mathfrak{gl}(3)$ generators can be represented by the non-linear elements of the universal enveloping algebra of the 5-dimensional Heisenberg algebra $\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)$, thus, the Hamiltonian and integral are two elements of the universal enveloping algebra $U_{\mathfrak{h}_5}$. In this paper, four different representations of the $\mathfrak{h}_5$ Heisenberg algebra are used: (I) by differential operators in two real (complex) variables, (II) by finite-difference operators on uniform or exponential lattices. We discovered the existence of two 2-parametric bilinear and trilinear elements (denoted $H$ and $I$, respectively) of the universal enveloping algebra $U(\mathfrak{gl}(3))$ such that their Lie bracket (commutator) can be written as a linear superposition of nine so-called artifacts - the special bilinear elements of $U(\mathfrak{gl}(3))$, which vanish once the representation of the $\mathfrak{gl}(3)$-algebra generators is written in terms of the $\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2},I)$-algebra generators. In this representation all nine artifacts vanish, two of the above-mentioned elements of $U(\mathfrak{gl}(3))$ (called the Hamiltonian $H$ and the integral $I$) commute(!); in particular, they become the Hamiltonian and the integral of the 3-body elliptic Calogero model, if $(\hat{p},\hat{q})$ are written in the standard coordinate-momentum representation.

math-ph↗