arXiv subjects

Carlos García-Azpeitia

Publications and source records attributed to Carlos García-Azpeitia.

31 records · Page 2Linked to original sources

arXiv2019

Standing waves of fixed period for $n+1$ vortex filaments

The $n+1$ vortex filament problem has explicit solutions consisting of $n$ parallel filaments of equal circulation in the form of nested polygons uniformly rotating around a central filament which has circulation of opposite sign. We show that when the relation between temporal and spatial periods is fixed at certain rational numbers, these configurations have an infinite number of homographic time dependent standing wave patterns that bifurcate from these uniformly rotating central configurations.

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arXiv2018

Choreographies in the $n$-vortex problem

We consider the equations of motion of $n$ vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, corresponds to choreographies of the $n$ vortices. We include numerical results for all cases, for various values of $n$, and we provide key details on the computational approach.

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arXiv2018

Nonlinear Vibrations in the Fullerene Molecule $C_{60}$

In this paper we analyze nonlinear dynamics of the fullerene molecule. We prove the existence of global branches of periodic solutions emerging from an icosahedral equilibrium (nonlinear normal modes). We also determine the symmetric properties of the branches of nonlinear normal modes for maximal orbit types. We find several solutions which are standing and rotating waves that propagate along the molecule with icosahedral, tetrahedral, pentagonal and triangular symmetries. We complement our theoretical results with numerical computations using Newton's method.

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arXiv2018

Standing waves in a counter-rotating vortex filament pair

The distance among two counter-rotating vortex filaments satisfies a beam-type of equation according to the model derived in [15]. This equation has an explicit solution where two straight filaments travel with constant speed at a constant distance. The boundary condition of the filaments is 2$π$-periodic. Using the distance of the filaments as bifurcating parameter, an infinite number of branches of periodic standing waves bifurcate from this initial configuration with constant rational frequency along each branch.

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arXiv2018

Solutions of Fixed Period in the Nonlinear Wave Equation on Networks

The wave equation on network is defined by $\partial_{tt}u=Δ_{G}u+g(u)$, where $u\in\mathbb{R}^{n}$ and the graph Laplacian $Δ_{G}$ is an operator on functions on $n$ vertices. We suppose that $g:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is an odd continuous function that satisfies $g(0)=g^{\prime }(0)=0$ and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations $Γ$, using a $Γ$-equivariant topological invariant we prove the existence of multiple non-constant $p$-periodic solutions characterized by their symmetries.

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arXiv2018

Symmetries of Nonlinear Vibrations in Tetrahedral Molecular Configurations

We study nonlinear vibrational modes of oscillations for tetrahedral configurations of particles. In the case of tetraphosphorus, the interaction of atoms is given by bond stretching and van der Waals forces. Using equivariant gradient degree, we present a full topological classification of the spatio-temporal symmetries of the periodic solutions. This procedure gives all the symmetries of the nonlinear vibrations for general force fields.

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arXiv2018

Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of n bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three equilibrium Zn-orbits from which families of Lyapunov orbits emerge. Numerical continuation of these families using a boundary value formulation is used to construct the bifurcation diagram for the case n=7, also including some secondary and tertiary bifurcating families. We observe symmetry-breaking bifurcations in this system, as well as certain period-doubling bifurcations.

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arXiv2018

Choreographies in the Discrete Nonlinear Schrödinger Equations

We study periodic solutions of the discrete nonlinear Schrödinger equation (DNLSE) that bifurcate from a symmetric polygonal relative equilibrium containing $n$ sites. With specialized numerical continuation techniques and a varying physically relevant parameter we can locate interesting orbits, including infinitely many choreographies. Many of the orbits that correspond to choreographies are stable, as indicated by Floquet multipliers that are extracted as part of the numerical continuation scheme, and as verified \textit{a posteriori} by simple numerical integration. We discuss the physical relevance and the implications of our results.

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arXiv2017

Symmetries and choreographies in families bifurcating from the polygonal relative equilibrium of the $n$-body problem

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits. These arise from the polygonal system of $n$ bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases $n=4,~6,~7,~8$, and $9$. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link: http://mym.iimas.unam.mx/renato/choreographies/index.html

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arXiv2016

Molecular chains interacting by Lennard-Jones and Coulomb forces

We study equations for the mechanical movement of chains of identical particles in the plane interacting with their nearest-neighbors by bond stretching and by van der Waals and Coulomb forces. We find collinear and circular equilibria as minimizers of the energy potential for chains with Neumann and periodic boundary conditions. We prove global bifurcation of periodic brake orbits from these equilibria applying the global Rabinowitz alternative. These results are complemented with numeric computations for ranges of parameters that include carbon atoms among other molecules.

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arXiv2016

Traveling and standing waves in coupled pendula and Newton's cradle

The existence of traveling and standing waves is investigated for chains of coupled pendula with periodic boundary conditions. The results are proven by applying topological methods to subspaces of symmetric solutions. The main advantage of this approach comes from the fact that only properties of the linearized forces are required. This allows to cover a wide range of models such as Newton's cradle, the Fermi-Pasta-Ulam lattice and the Toda lattice.

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arXiv2016

Global bifurcation of traveling waves in discrete nonlinear Schrödinger equations

We consider discrete nonlinear Schrödinger equations of n sites with periodic boundary conditions. These equations have n branches of standing waves that bifurcate from zero. Traveling waves appear as a symmetry-breaking from the standing waves for different amplitudes. The bifurcation is proved using the global Rabinowitz alternative in subspaces of symmetric functions. As examples, we present applications to the Schrödinger and Saturable lattices.

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arXiv2016

Global bifurcation of vortex and dipole solutions in Bose-Einstein condensates

The Gross-Pitaevskii equation for a Bose-Einstein condensate (BEC) with symmetric harmonic trap is given in (1). Periodic solutions of (1) play an important role in the understanding of the long term behavior of its solutions. In this note we prove the existence of several global branches of solutions to (1) among which there are vortex solutions and dipole solutions.

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