Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

TL;DR

This paper introduces a class of integrable dissipative Ermakov-Pinney equations with Chiellini-based nonlinear damping, providing explicit solutions and extending to higher-order nonlinearities, revealing complex dissipation-gain behaviors.

Contribution

It develops a novel integrable framework for Ermakov-Pinney equations with Chiellini nonlinear damping, including explicit solutions and extensions to higher-order nonlinearities.

Findings

Explicit solutions for linear and nonlinear cases

Identification of dissipation-gain behavior in Chiellini damping

Extension to higher-order Reid nonlinearities

Abstract

We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{\zeta \zeta}+g(v)v_{\zeta}+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h_0(v) is a linear function, h_0(v)=\lambda^2v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipative function is actually a dissipation-gain function because it can be negative on some intervals. We also examine the nonlinear case h_0(v)=\Omega_0^2(v-v^2) and show that it leads to an integrable hyperelliptic case