Dynamics of the Modified Emden and pseudo-Modified Emden Equations: position-dependent mass, invariance and exact solvability

TL;DR

This paper explores the dynamics of the modified Emden equation and its position-dependent mass variant, providing exact solutions and phase-space analysis through nonlocal transformations, advancing understanding of nonlinear oscillatory systems.

Contribution

It introduces a general solution for the modified Emden equation and transforms it into a position-dependent mass framework, offering exact solutions and illustrative examples.

Findings

Exact solutions for PDM-MEE-type particles

Phase-space trajectories for specific parameters

Transformation linking MEE to PDM classical particles

Abstract

We consider the modified Emden equation (MEE) and introduce its most general solution, using the most general solution for the simple harmonic oscillator's linear dynamical equation (i.e., the initial conditions shall be identified by the PDM-MEE problem at hand) . We use a general nonlocal point transformation and show that modified Emden dynamical equation is transformed to describe position-dependent mass (PDM) classical particles. Two PDM-MEE-type classical particles are used as illustrative examples, and their exact solutions are reported. Under specific parametric considerations, the phase-space trajectories are reported for the MEE-type and for PDM-MEE-type classical particles.