Related papers: Periodic Solutions with Singularities in Two Dimen…
This work is devoted to the study of the existence and periodicity of solutions of initial differential problems, paying special attention to the explicit computation of the period. These problems are also connected with some particular…
We try to prove the existence of choreography solutions for the $n-$body problem on $S^2$. For the three-body problem, we show the existence of the 8-shape orbit on $S^2$.
Simple form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback…
Some properties of the periodic solution of the three-body problem where three particles of equal mass follow the same trajectory are discussed. This trajectory has the shape of a figure-8. The three particles have a constant separation in…
An approximate analytical solution for the two body problem perturbed by a radial, low acceleration is obtained, using a regularized formulation of the orbital motion and the method of multiple scales. Formulating the dynamics with the…
The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our…
In this paper, we investigate the formation of singularity for general two dimensional and radially symmetric solutions for rotating shallow water system from different aspects. First, the formation of singularity is proved via the study…
We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case for when the sphere is 3-dimensional and where we take the group of symmetries to be $SO(4)$. As…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle $\theta \in (0, \pi/7]$ in the equal-mass four-body problem. A…
New travelling wave solutions to the Fornberg-Whitham equation are investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.
A new technique that utilizes surface integrals to find the force, torque and potential energy between two non-spherical, rigid bodies is presented. The method is relatively fast, and allows us to solve the full rigid two-body problem for…
Numerical solutions to Newton's equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible. This is due to the exponential growth of errors introduced by the integration scheme and…
A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the…
We prove the existence of infinitely many classical periodic solutions for a class of semilinear wave equations with periodic boundary conditions. Our argument relies on some new estimates for the linear problem with periodic boundary…
A new efficient approach for searching three-body periodic equal-mass collisionless orbits passing through Eulerian configuration is presented. The approach is based on a symmetry property of the solutions at the half period. Depending on…
We prove existence of infinitely many classical periodic solutions with periodic boundary conditions for a class of monotone semilinear wave equations. Our argument relies on some new estimates for the linear problem with periodic boundary…
We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions…
The necessity and benefit of singular solutions in the study of physical systems is shown. By singular solutions we mean solutions that are not contained in the general solution of the system of equations that describes the dynamic system…
The singularity at a simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than $ C^{8/3}…