arXiv 1301.35672013-01math-phmath.MP

具有Ermakov-Pinney非线性和Chiellini阻尼的可积方程

Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

Stefan C. Mancas, Haret C. Rosu

备注：: 15 pages, 5 figures, 1 appendix, 21 references, published version
发表信息：: Appl. Math. Comp. 259 (2015) 1-11
DOI：: 10.1016/j.amc.2015.02.037

摘要：

引入一类特殊的耗散Ermakov-Pinney方程，其形式为v_{ζζ} +g（v）v_v +h（v）=0，其中h（v）=h_0（v）+cv^{-3}，非线性耗散g（v）是基于相应的Chiellini可积Abel方程.当h_0（v）为线性函数时，h_0（v）=λ^2v，则按Abel方程的路线得到了通解.在特解的基础上，我们还给出了含有Milne型相位因子的一般解。此外，对于高阶Reid非线性项，也构造了同样的通解.基耶利尼耗散函数实际上是一个耗散增益函数，因为它在某些区间上可以是负的。我们还研究了非线性情形h_0（v）=Ω_0^2（v-v^2），并证明了它是一个可积的超椭圆情形

英文摘要：

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

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