Related papers: Morse Theory and Relative Equilibria in the Planar…
In this paper, we prove a Morse index theorem for the index form of even order linear Hamiltonian systems on the closed interval with reasonable self-adjoint boundary conditions. The highest order term is assumed to be nondegenerate.
We use the Maslov index to study the eigenvalue problem arising from the linearisation about solitons in the fourth-order cubic nonlinear Schr\"odinger equation (NLSE). Our analysis is motivated by recent work by Bandara et al., in which…
We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four body problem with masses 1, m, m, 1, and also in a…
We prove that for some potentials (including the Newtonian one, and the potential of Helmholtz vortices in the plane) relative equilibria consisting of two homothetic regular polygons of arbitrary size can only occur if the masses at each…
We study the resonant prescribed T-curvature problem on a compact 4-dimensional Riemannian manifold with boundary. We derive sharp energy and gradient estimates of the associated Euler-Lagrange functional to characterize the critical points…
The aim of this paper is to present a new, analytical, method for computing the exact number of relative equilibria in the planar, circular, restricted 4-body problem of celestial mechanics. The new approach allows for a very efficient…
An elliptic relative equilibrium (ERE) is a special solution of the planar $N$-body problem generated by a central configuration. Its linear stability depends on the eccentricity $e$ and the masses of the bodies. However, for $e>0$, the…
We study the Robe's restricted three-body problem. Such a motion was firstly studied by A. G. Robe in \cite{Robe}, which is used to model small oscillations of the earth's inner core taking into account the moon attraction. For the linear…
We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium…
In this paper we consider sequences of $p$-harmonic maps, $p>2$, from a closed Riemann surface $\Sigma$ into the $n$-dimensional sphere $\mathbb{S}^n$ with uniform bounded energy. These are critical points of the energy $E_p(u)…
We investigate symmetry properties of solutions to equations of the form $$ -\Delta u = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with…
We give an exact quantitative solution for the motion of three vortices of any strength, which Poincar\'e showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is…
Moran's index is a basic measure of spatial autocorrelation, which has been applied to varied fields of both natural and social sciences. A good measure should have clear boundary values or critical value. However, for Moran's index, both…
Vortices in fluids and superfluids are fundamental to phenomena ranging from Bose-Einstein condensates and superfluid films to neutron stars and hydrodynamic micro-rotors, where background geometry often plays an important role. Curvature…
In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\mathbf{R}^2$. We obtain the symplectic reduction to…
In this paper we study the behaviour of critical points of the Ginzburg-Landau perturbation of the Dirichlet energy into the sphere $E_\varepsilon(u):=\int_\Sigma \frac{1}{2}|du|^2_h\ \,dvol_h…
We show the existence of some infinite families of periodic solutions of the planar Newtonian n-body problem --with positive masses-- which are symmetric with respect to suitable actions of finite groups (under a strong--force assumption,…
Edge-localized stationary states of the focusing nonlinear Schrodinger equation on a general quantum graph are considered in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which…
Central configurations give rise to self-similar solutions to the Newtonian $N$-body problem, and play important roles in understanding its complicated dynamics. Even the simple question of whether or not there are finitely many planar…
The method of Morse theory is used to analyze the distributions of unit charges interacting through a repulsive force and constrained to move on the surface of a sphere -- the Thomson problem. We find that, due to topological reasons, the…