Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

Abstract

We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{ζζ}+g(v)v_ζ+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)=λ^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)=Ω_0^2(v-v^2) and show that it leads to an integrable hyperelliptic case

15 pages, 5 figures, 1 appendix, 21 references, published version

References in corpus (6)

• Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in General Relativity
• Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations
• A nonlinear superposition rule for solutions of the Milne--Pinney equation
• Applications of Lie systems in dissipative Milne--Pinney equations
• From Feshbach-Resonance Managed Bose-Einstein Condensates to Anisotropic Universes: Some Applications of the Ermakov-Pinney equation with Time-Dependent Nonlinearity
• Nonlinear superpositions and Ermakov systems

Cited by in corpus (4)

• Noncommutative quantum mechanics in a time-dependent background
• Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach
• Generalized damped Milne-Pinney equation and Chiellini method
• Exact travelling wave solutions of non-linear reaction-convection-diffusion equations -- an Abel equation based approach