Related papers: Global existence and singularity of the Hill's typ…

We consider the planar Hill's lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is bounded,…

In this paper, I prove the fiberwise convexity of the regularized Hill's lunar problem below the critical energy level. This allows us to see Hill's lunar problem of any energy level below the critical value as the Legendre transformation…

Richardson approach provides an exact solution of the pairing Hamiltonian. This Hamiltonian is characterized by the electron-hole pairing symmetry, which is however hidden in Richardson equations. By analyzing this symmetry and using an…

Motivated by the current search for exomoons, this paper considers the stability of tidal equilibrium for hierarchical three-body systems containing a star, a planet, and a moon. In this treatment, the energy and angular momentum budgets…

We investigate the action spectrum of Hill's lunar problem by observing inclusions between the Liouville domains enclosed by the regularized energy hypersurfaces of the rotating Kepler problem and Hill's lunar problem. In this paper, we…

The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn-Hilliard type, is discussed. The existence of a weak solution and its continuous dependence on the data are proved using a…

Consider Hill's operator Q = -D^2 + q(x) in which the potential q(x) is an almost surely continuous and rotation invariant Gaussian process on the circle of perimeter one. Viewing the classical Riccati map as a change of measure, we…

Consider the focusing energy-critical Klein-Gordon equation in dimension d=3,4,5. We describe the global dynamics of real-valued solutions of which the energy is slightly larger than that of the ground states'. We classify the flows of the…

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that…

Domain wall, wormhole, particlelike, and cosmic string general relativistic solutions supported by two interacting phantom or ordinary scalar fields with 4th-, 6th-, and 8th-order potentials are studied. Numerical calculations indicate that…

This paper concerns the existence and related properties of solutions to the Schr\"{o}dinger-Bopp-Podolsky system, which reduces to a nonlinear and nonlocal partial differential equation describing a Schr\"{o}dinger field coupled with its…

Some superlinear fourth order elliptic equations are considered. Ground states are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of nontrivial…

We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions…

In this paper we look at a class of random optimization problems that arise in the forms typically known as Hopfield models. We view two scenarios which we term as the positive Hopfield form and the negative Hopfield form. For both of these…

We study analytically the existence and uniqueness of the ground state of the nonlinear Schr\"{o}dinger equation (NLSE) with a general power nonlinearity described by the power index $\sigma\ge0$. For the NLSE under a box or a harmonic…

An infinite sequence of potential well functions is considered. A trial wavefunction is used with the Schr$\ddot{\text{o}}$dinger equation to obtain an approximate ground state energy for each potential well function. We obtain an…

The energy super-critical Gross--Pitaevskii equation with a harmonic potential is revisited in the particular case of cubic focusing nonlinearity and dimension d > 4. In order to prove the existence of a ground state (a positive, radially…

We consider the nonlinear Schr\"odinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer…

Motivated by a class of orbit problems in astrophysics, this paper considers solutions to Hill's equation with forcing strength parameters that vary from cycle to cycle. The results are generalized to include period variations from cycle to…

Two types of stability boundaries exist for any planetary system consisting of one star and two planets. Lagrange stability requires that the planets remain bound to the star, conserves the ordering of the distance from the star, and limits…