Integrability cases for the anharmonic oscillator equation

Abstract

Using N. Euler's theorem on the integrability of the general anharmonic oscillator equation [1], we present three distinct classes of general solutions of the highly nonlinear second order ordinary differential equationThe first exact solution is obtained from a particular solution of the point transformed equation d 2 X/dT 2 + X n (T ) = 0, n / ∈ {−3, −1, 0, 1}, which is equivalent to the anharmonic oscillator equation if the coefficients fi(t), i = 1, 2, 3 satisfy an integrability condition. The integrability condition can be formulated as a Riccati equation for f1(t) and 1 f 3 (t) df 3 dt respectively. By reducing the integrability condition to a Bernoulli type equation, two exact classes of solutions of the anharmonic oscillator equation are obtained.

Dynamics of stress propagation in anharmonic crystals: MD simulations

“…Finding exact, general solutions for potentials of that form, and approximate ones, have been discussed by Euler [29], Amore and Fernández [30], Harko et al [31]. They use the idea of performing a non-linear transformation of variables x and t. The general solution of a class of equations of that type can be expressed as an infinite sum of components with the hypergeometric function [31]. That however lacks the simplicity to be analyzed.…”