22 research outputs found
Linear Stability Analysis of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem
We apply the symmetry reduction method of Roberts to numerically analyze the
linear stability of a one-parameter family of symmetric periodic orbits with
regularizable simultaneous binary collisions in the planar pairwise symmetric
four-body problem with a mass as the parameter. This reduces the
linear stability analysis to the computation of two eigenvalues of a matrix for each obtained from numerical integration of the
linearized regularized equations along only the first one-eighth of each
regularized periodic orbit. The results are that the family of symmetric
periodic orbits with regularizable simultaneous binary collisions changes its
linear stability type several times as varies over , with linear
instability for close or equal to 0.01, and linear stability for close
or equal to 1.Comment: 13 pages, 1 figur
Existence of hyperbolic motions to a class of Hamiltonians and generalized -body system via a geometric approach
For the classical -body problem in with ,
Maderna-Venturelli in their remarkable paper [Ann. Math. 2020] proved the
existence of hyperbolic motions with any positive energy constant, starting
from any configuration and along any non-collision configuration. Their
original proof relies on the long time behavior of solutions by Chazy 1922 and
Marchal-Saari 1976, on the H\"{o}lder estimate for Ma\~{n}\'{e}'s potential by
Maderna 2012, and on the weak KAM theory.
We give a new and completely different proof for the above existence of
hyperbolic motions. The central idea is that, via some geometric observation,
we build up uniform estimates for Euclidean length and angle of geodesics of
Ma\~{n}\'{e}'s potential starting from a given configuration and ending at the
ray along a given non-collision configuration. Note that we do not need any of
the above previous studies used in Maderna-Venturelli's proof.
Moreover, our geometric approach works for Hamiltonians
, where is lower semicontinuous and decreases
very slowly to faraway from collisions. We therefore obtain the existence
of hyperbolic motions to such Hamiltonians with any positive energy constant,
starting from any admissible configuration and along any non-collision
configuration. Consequently, for several important potentials , we get similar existence of hyperbolic motions to the
generalized -body system , which is an extension
of Maderna-Venturelli [Ann. Math. 2020].Comment: 37 pages, 6 figure
Convergence of least energy sign-changing solutions for logarithmic Schr\"{o}dinger equations on locally finite graphs
In this paper, we study the following logarithmic Schr\"{o}dinger equation
-\Delta u+\lambda a(x)u=u\log u^2\ \ \ \ \mbox{ in }V on a connected locally
finite graph , where denotes the graph Laplacian, is a constant, and represents the potential. Using variational
techniques in combination with the Nehari manifold method based on directional
derivative, we can prove that, there exists a constant such that
for all , the above problem admits a least energy
sign-changing solution . Moreover, as , we
prove that the solution converges to a least energy sign-changing
solution of the following Dirichlet problem \begin{cases} -\Delta u=u\log
u^2~~~&\mbox{ in }\Omega,\\ u(x)=0~~~&\mbox{ on }\partial\Omega, \end{cases}
where is the potential well.Comment: Submitted to CNSN
Existence and Stability of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem
We extend our previous analytic existence of a symmetric periodic
simultaneous binary collision orbit in a regularized fully symmetric equal mass
four-body problem to the analytic existence of a symmetric periodic
simultaneous binary collision orbit in a regularized planar pairwise symmetric
equal mass four-body problem. We then use a continuation method to numerically
find symmetric periodic simultaneous binary collision orbits in a regularized
planar pairwise symmetric 1, m, 1, m four-body problem for between 0 and 1.
Numerical estimates of the the characteristic multipliers show that these
periodic orbits are linearly stability when , and are
linearly unstable when .Comment: 6 figure