An Invertible Linearization Map for the Quartic Oscillator

TL;DR

This paper introduces an explicit invertible linearization map that transforms the quartic oscillator's solutions into harmonic oscillator solutions, providing a new method to solve and analyze nonlinear oscillators with even polynomial potentials.

Contribution

The paper presents a novel energy-preserving invertible map that linearizes the quartic oscillator and extends to all even-powered attractive potentials, offering explicit solutions.

Findings

Map explicitly solves Newton's equation for quartic oscillator

Extension to all even-powered attractive potentials

Provides new solutions to initial value problems

Abstract

The set of world lines for the non-relativistic quartic oscillator satisfying Newton's equation of motion for all space and time in 1-1 dimensions with no constraints other than the "spring" restoring force is shown to be equivalent (1-1-onto) to the corresponding set for the harmonic oscillator. This is established via an energy preserving invertible linearization map which consists of an explicit nonlinear algebraic deformation of coordinates and a nonlinear deformation of time coordinates involving a quadrature. In the context stated, the map also explicitly solves Newton's equation for the quartic oscillator for arbitrary initial data on the real line. This map is extended to all attractive potentials given by even powers of the space coordinate. It thus provides classes of new solutions to the initial value problem for all these potentials.