Asymptotic Solutions of Initial Value Problems for Nonlinear Partial Differential Equations
Asymptotic Solutions of Initial Value Problems for Nonlinear Partial Differential Equations
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非线性偏微分方程初值问题的渐近解
DOI：
10.1137/0118067
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发表时间：
1970
期刊：
影响因子：
--
通讯作者：
S. Kogelman
中科院分区：
文献类型：
--
作者：
J. Keller;S. Kogelman
1.导论.考虑一类非线性偏微分方程初值问题的解u（x，t，e），其中非线性度与参数e成正比。我们希望确定的渐近行为的解决方案时，e是小的，是大的。通常的微扰法，其中u是作为一个幂级数在e中寻找，是不适用的，因为它会导致长期的条款，因此是无用的大t。相反，我们将使用两次方法，该方法已被应用于常微分方程的科尔和Kevorkian和其他人（见科尔[1]）和统计力学的弗里曼，桑德里和其他人（见弗里曼[2]）。另一种密切相关的方法，Bogolyubov和Mitropol'skii的平均法，可能被用来代替。在分析常微分方程的周期解时，Lindstedt和Poincar6表明，可以通过扩展解的周期来消除长期项，以及解本身，以e的幂。Keller和Ting [3]、Millman和Keller [4]等将其方法用于偏微分方程。我们考虑的问题是一个非线性波动方程的初边值问题。我们的详细结果得到了一个特定的方程，可以被看作是一个广义的货车der Pol方程。该方程的边值问题具有无穷多个周期解或极限环，Millman和Keller [4]找到了这些解的渐近展开式。我们将看到，一般情况下，初值问题的解趋向于这些周期解之一，对于给定的初始数据，我们将说明是哪一个。然而，对于某些特殊的初始条件，解往往是有限个周期解的组合。这些特解是不稳定的，只有个别周期解是稳定的。在最后一节中，我们展示了如何将两次方法应用于Hilbert空间中向量的非线性初值问题。
1. Introduction. Let us consider the solution u (x, t, e) of an initial value problem for a nonlinear partial differential equation in which the nonlinearity is proportional to a parameter e. We wish to determine the asymptotic behavior of the solution when e is small and is large. The usual perturbation method, in which u is sought as a power series in e, is not applicable because it leads to secular terms and is therefore useless for large t. Instead we shall use the twotime method, which has been applied to ordinary differential equations by Cole and Kevorkian and others (see Cole [1]) and to statistical mechanics by Frieman, Sandri and others (see Frieman [2]). An alternative closely related procedure, the averaging method of Bogolyubov and Mitropol’skii, might have been used instead.In the analysis of periodic solutions of ordinary differential equations, Lindstedt and Poincar6 showed that secular terms could be eliminated by ex-panding the period of the solution, as well as the solutionitself, in powers of e. Their method has been adopted to partial differential equations by Keller and Ting [3], Millman and Keller [4] and others. The problem we consider is an initial boundary value problem for a nonlinear wave equation. Our detailed results are obtained for a particular equation which may be viewed as a generalized van der Pol equation. The boundary value problem for this equation possesses infinitely many periodic solutions or limit cycles, asymptotic expansions of which were found by Millman and Keller [4]. We shall see that in general the solution of the initial value problem tends to one of these periodic solutions, and for given initial data we shall show which one. However, for certain special initial conditions, the solution tends toa combinationof a finite number ofperiodic solutions. Thesespecial solutions are unstable, and only the individual periodic solutionsare stable. In a final sectionwe show how the two-time method can beapplied to a nonlinear initial value problem for a vector in a Hilbert space.