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We study the dynamics of a rigid body in a central gravitational field modeled as a Hamiltonian system with continuous rotational symmetries following the geometrical framework of Wang et al. Novelties of our work are the use the Reduced…
An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis…
This paper presents elements about the radial orbit instability, which occurs in spherical self-gravitating systems with a strong anisotropy in the radial velocity direction. It contains an overview on the history of radial orbit…
The application of variational principles for analyzing problems in the physical sciences is widespread. Cantilever-like problems, where one end is fixed and the other end is free, have received less attention in terms of their stability…
The main goal of this paper is to present orbital stability results of periodic standing waves for the one-dimensional logarithmic Klein-Gordon equation. To do so, we first use compactness arguments and a non-standard analysis to obtain the…
We consider static solutions of the sine-Gordon theory defined on a cylinder, which can be either periodic or quasi-periodic in space. They are described by the different modes of a simple pendulum moving in an inverted effective potential…
In this document, we deal with the stabilization problem of slow-fast systems (or singularly perturbed Ordinary Differential Equations) at a non-hyperbolic point. The class of systems studied here have the following properties: 1) they have…
It was recently demonstrated that two-dimensional Townes solitons (TSs) in two-component systems with cubic self-focusing, which are normally made unstable by the critical collapse, can be stabilized by linear spin-orbit coupling (SOC), in…
We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its…
We study the Miura map of the KP-II equation on $\mathbb R^2$ and the resulting B\"acklund transform, which adds a line soliton to a given solution. This work aims to develop a complementary approach to T. Mizumachi's method for the…
We present the first concrete evidence for the classical stability of vortons, circular cosmic string loops stabilized by the angular momentum of the charge and current trapped on the string. We begin by summarizing what is known about…
In the paper we have developed a theory of stability preserving structural transformations of systems of second-order ordinary differential equations (ODEs), i.e., the transformations which preserve the property of Lyapunov stability. The…
The permutability of two Backlund transformations is employed to construct a non linear superposition formula and to generate a class of solutions for the N=2 super sine-Gordon model.
We study $\varepsilon$-stability in continuous logic. We first consider stability in a model, where we obtain a definability of types result with a better approximation than that in the literature. We also prove forking symmetry for…
We investigate the stability of the wave equation with spatial dependent coefficients on a bounded multidimensional domain. The system is stabilized via a scattering passive feedback law. We formulate the wave equation in a port-Hamiltonian…
The geometry of an admissible B\"acklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal…
We study 2D discrete integrable equations of order 1 with respect to one independent variable and $m$ with respect to another one. A generalization of the multidimensional consistency property is proposed for this type of equations. The…
In this note we study the generation of attractive oscillations of a class of mechanical systems with underactuation one. The proposed design consists of two terms, i.e., a partial linearizing state feedback, and an immersion and invariance…
We review recent developments in structural stability as applied to key topics in general relativity. For a nonlinear dynamical system arising from the Einstein equations by a symmetry reduction, bifurcation theory fully characterizes the…
We prove the non-linear stability of a large class of spherically symmetric equilibrium solutions of both the collisonless Boltzmann equation and of the Euler equations in MOND. This is the first such stability result that is proven with…