Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations

TL;DR

This paper investigates how nonlocal transformations influence the linearizability and exact solvability of generalized modified Emden equations, revealing new explicit solutions for damped harmonic oscillators and illustrating their phase-space dynamics.

Contribution

It demonstrates the role of nonlocal transformations in linearizing GMEE into harmonic oscillators and provides the first explicit solutions for the damped harmonic oscillator case.

Findings

Exact solutions for GMEE transformed into HO and DHO

Novel explicit solution for DHO with comprehensive damping cases

Illustrative phase-space trajectories for all solutions

Abstract

The nonlinear generalized modified Emden type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of the nonlocal transformation and the linearizability into HO or DHO determine the nature/structure of the dynamical forces involved (hence, determine the structure of the dynamical equation). Yet, a reverse engineering strategy is used so that the exact solutions of the emerging GMEE are nonlocally transformed to find the exact solutions of the HO and DHO dynamical equations. Consequently, whilst the exact solution for the HO remains a textbook one, the exact solution for the DHO (never reported elsewhere, to the best of our knowledge) turns out to be manifestly the most explicit and general solution that offers consistency and comprehensive coverage for the associated under-damping, critical-damping, and over-damping cases (i.e., no complex settings for the coordinates and/or the velocities are eminent/feasible). Moreover, for all emerging dynamical system, we report illustrative figures for each solution as well as the corresponding phase-space trajectories as they evolve in time.