Integrable equations with Ermakov–Pinney nonlinearities and Chiellini damping

2013/01/31 by Stefan C. Mancas, H. C. Rosu, Haret C. Rosu
Mathematics · Physics and Astronomy · #Dissipation #Dissipative system #Fractional Differential Equations Solutions #Function (biology) #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum mechanics #Type (biology) #math-ph #math.MP

paper · pdf · doi:10.1016/j.amc.2015.02.037

published as Appl. Math. Comp. 259 (2015) 1-11 · 15 pages, 5 figures, 1 appendix, 21 references, published version

openalex publication_date 2015/03/03 · arxiv created 2015/03/05 · arxiv updated 2015/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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)=h₀(v)+cv^-3 and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h₀(v) is a linear function, h₀(v)=λ²v, 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 h₀(v)=Ω₀²(v-v²) and show that it leads to an integrable hyperelliptic case

Citations

• A course of modern analysis an introduction to the general theory of infinite processes and of analytic functions with an account of the principal transcendental functions
• From Feshbach-resonance managed Bose–Einstein condensates to anisotropic universes: Applications of the Ermakov–Pinney equation with time-dependent nonlinearity