A non-linear Oscillator with quasi-Harmonic behaviour: two- and nn-dimensional Oscillators

A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the nonlinear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of $n$ degrees of freedom. Finally, the relation of this nonlinear system with the harmonic oscillator on spaces of constant curvature, two-dimensional sphere $S^2$ and hyperbolic plane $H^2$, is discussed.Comment: 30 pages, 4 figures, submitted to Nonlinearit

Dynamical behaviour of solitons in a -invariant nonlocal nonlinear Schrödinger equation with distributed coefficients

We present one-, two- and three-soliton solutions of a parity-time ()-invariant nonlocal nonlinear Schrödinger (NNLS) equation with distributed coefficients, namely dispersion, nonlinearity and loss/gain parameters. We map the considered equation into constant coefficient -invariant NNLS equation with a constraint. We prove that the considered system is -invariant only when the distributed coefficients are even functions. To investigate the dynamical behaviour of the constructed one- and two-soliton solutions, we consider three different forms of dispersion parameters, namely (i) constant, (ii) periodically distributed, and (iii) exponentially distributed one. We report how the intensity profiles of solitons get modified in the background by considering the aforementioned dispersion parameters. By performing asymptotic analysis, we also explain how the dispersion parameters influence the interactions of nonlocal solitons