Related papers: A remark on rational isochronous potentials
We give a short proof of Urabe's criteria for the isochronicity of periodical solutions of the equation $\ddot{x}+g(x)=0$. We show that apart from the harmonic oscillator there exists a large family of isochronous potentials which must all…
We are interested at first in the study of the monotonicity for the period function of the conservative equation \ $(1)\quad \ddot x + g(x) = 0.$\quad Some refinements of known criteria are brought. Moreover, we give necessary and…
It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level $k^2>0$. However, numerical experiments show that two quite different…
In this work the existence of periodic solutions is studied for the Hamiltonian functions (Formula presented.) where the first term consist of a harmonic oscillator and the second term are homogeneous polynomials of degree 5 defined by two…
This paper considers a family of second-order periodic parabolic equations with highly oscillating potentials, which have been considered many times for the time-varying potentials in stochastic homogenization. Following a standard…
The motion in a simple, time independent rational galactic potential is studied. The potential is a generalization of a two dimensional harmonic oscillator potential and can be considered to describe plane motion in the central parts of a…
Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the…
By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain…
We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. $\cal PT$ symmetry may…
An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context,…
A complex potential is a holomorphic function $\Omega:\mathbb{C} \to \mathbb{C}$ whose real and imaginary parts generate a pair of orthogonal foliations, representing the equipotential lines and the streamlines of $\dot{z} =…
We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1)…
The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials $V_{A,B,{\rm ext}}(x)$, obtained from a conventional Morse…
Combining recent results on rational solutions of the Riccati-Schr\"odinger equations for shape invariant potentials to the finite difference B\"acklund algorithm and specific symmetries of the isotonic potential, we show that it is…
In this paper, the quantum spectrum of isochronous potentials is investigated. Given that the frequency of the classical motion in such potentials is energy-independent, it is natural to expect their quantum spectra to be equispaced.…
In this paper we consider several families of potential non-isochronous systems and study their associated period functions. Firstly, we prove some properties of these functions, like their local behavior near the critical point or…
In this paper, we investigate the revivals of the one-dimensional periodic Schr\"odinger equation with a piecewise $C^2$ potential function. As has been observed through numerical simulations of the equation with various initial data and…
We introduce the relativistic version of the well-known Henon's isochrone spherical models: static spherically symmetrical spacetimes in which all bounded trajectories are isochrone in Henon's sense, i.e., their radial periods do not depend…
We study stationary rotating topological solitons in (2+1)-dimensional ${\mathbb C}P^2$ non-linear sigma model with a stabilizing potential term. We find families of $U(1)\times U(1)$ symmetric solutions with topological degrees larger than…
We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on $E^2$…