一类强非线性振子的解析解

A. Salas, S. El-Tantawy
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引用次数: 4

摘要

振子无处不在;它们中的大多数本质上是非线性的。虽然非线性方程本身大多不能产生精确的解析解,但存在大量的基本而实用的技术来提取有关方程解的重要信息。本章的目的是为读者介绍一些新的技术,这些技术主要是用Duffing振荡器的例子来详细说明的。利用三次Duffing和三次五次Duffing振子的精确解析解,我们描述了其他保守和一些非保守阻尼非线性振子可以用本文描述的解析技术来研究的方法。我们不使用摄动技术。然而,与这些方法进行了一些比较。我们考虑具有x¨+fx=0和x¨+2ε +fx=Ft形式的振子,其中x=xt和f=fx和Ft是连续函数。在本章中,有时我们将使用f−x=−fx并取近似fx≈∑j=1Npjxj,其中j=1,3,5,⋯N仅为奇数整数值且x∈−AA。此外,我们将取近似fx≈∑j=0Npjxj,其中j=1,2,3,⋯N,并且x∈- AA。考虑任意初始条件。主要思想是用合适的三次或五次多项式逼近函数f=fx。解析解用Jacobian和weerstrass椭圆函数表示。应用于等离子体物理,电子电路,孤子理论和工程提供。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Analytical Solutions of Some Strong Nonlinear Oscillators
Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing’s oscillator. Using the exact analytical solution to cubic Duffing and cubic-quinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form x¨+fx=0 as well as x¨+2εẋ+fx=Ft, where x=xt and f=fx and Ft are continuous functions. In the present chapter, sometimes we will use f−x=−fx and take the approximation fx≈∑j=1Npjxj, where j=1,3,5,⋯N only odd integer values and x∈−AA. Moreover, we will take the approximation fx≈∑j=0Npjxj, where j=1,2,3,⋯N, and x∈−AA. Arbitrary initial conditions are considered. The main idea is to approximate the function f=fx by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma physics, electronic circuits, soliton theory, and engineering are provided.
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