On the two-scale method for one-frequency oscillations
by
Arlen M. Il'in
Institute of Mathematics and Mechanics, Ekaterinburg

A regular series in the degrees of a small parameter \epsilon, representing the solution of the Cauchy problem

d2 u d t2 +\omega2 u = \epsilonf(u,  d u d t ),    u(0) = a,     d u d t (0) = b ,   \epsilon > 0,

(1)

in general is not asymptotic when t >> 1. The known methods ([1], [2]) provide the uniform asymptotic approximation for t=O(\epsilon^-1). After introducing the slow time \tau = \epsilont, the solution of the problem (1) is found in the form u(t, \epsilont, \epsilon), where the function u(t, \tau, \epsilon) is expanded in the asymptotic series

\infty å 0  \epsilonk uk(t, \epsilont).

(2)

We can assume, without loss of generality, that \omega = 1. The requirement of absense of secular terms allows for constructing all u_k(t, \tau), which are periodical in t. The leading term is u₀ (t, \tau) = A(\tau) cos(\omegat +B(\tau)), where A(\tau) is a solution of the problem

2\piA'(\tau) = - 2\pi ó õ 0  f(A(\tau) cosy, - A(\tau) siny) siny dy,    A(0) = Ö   a2+ b2   .

(3)

The following questions arise naturally: does there exist a solution of the problem (1) when t >> \epsilon^-1 and, if it does, what is its asymptotic expansion? Also, the two conjectures are natural: 1) If a solution of (3) exists only in a half-interval [0, L), then the solution of (1) is not extendable onto the whole half-axis [0, \infty]. 2) If a solution of (3) is extendable onto the whole half-axis and, moreover, tends with exponential speed to its limit when \tau --> \infty, then the series (2) is uniformly asymptotic in all the half-axis [0, \infty]. It turns out that, strictly speaking, both conjectures are wrong. Conjecture 1) seems to be valid for almost all initial data of the problem (1). Still, even for equation (3) with f(u, u')=(u')³, the solution of (3) cannot be extended if a²+b² > 0, but there exist such values a and b,     ((a²+ b² > 0) for which the solution of (1) exists in all the halh-axis. (By the way, when f(u, u')=u³, the situation is inverse: the solution of (3) exists for all \tau > 0, while the solution of (1) cannot be extended onto the whole half-axis.) Conjecture 2) is not valid even for the most popular example-the Van-der-Pohl equation (f(u, u')=u'(1-u²)). In this case, the solution of (3) tend exponentially fast to the limit A_\infty = 2 when \tau --> \infty. And the next terms of series (2) contain secular summands, so this series is not uniformly asymptotic for the solution of (1) when t >> \epsilon^-1. For the construction and justification of the asymptotic of problem (1) solution up to any degree of \epsilon^-1 in the time interval t=O(\epsilon^-N), where N is an arbitrary integer, let us pass to usual phase variables. We get the initial-value problem for the equation

dj dt = 1- \epsilonH(\rho, j, \epsilon),

(4)

d\rho dj = \epsilonG(\rho, j, \epsilon),

(5)

j(0, \epsilon)=0,     \rho(0, \epsilon) = Ö   a2+ b2   ,

(6)

where H(\rho, j, \epsilon) = \rho^-1 cos(j-j₀)f(\rhocos(j-j₀), -\rhosin(j-j₀)),       G(\rho, j, \epsilon) = - \rhoH(\rho, j, \epsilon)tan(j-j₀) [1-\epsilonH(\rho, j, \epsilon)]^-1, tanj₀ = b/a Suppose that for some a₀ > 0 the following relations hold:

ó õ 2\pi 0  G(a0, j, 0) dj = 0,

(7)

ó õ 2\pi 0   \partialG(a0, j, 0) \partialr dj = -2\gamma < 0.

(8)

Then let us consider the initial-value problem (5), (6) and search for its solution using the two-scale method.

Theorem. If conditions (7), (8) hold and the value \surd{a²+ b²} is close to a₀, then solution \rho(j, \epsilon) of initial-value problem (5), (6) is expanded in a uniformly asymptotic in the half-line [0, \infty) series:

\rho(j, \epsilon) = \infty å 0  \epsilonk\rhok(j, \epsilonj),

(9)

where \rho_k(j, \theta) in C^\infty are 2\pi-periodical functions of j.

A uniform asymptotics of a solution of problem (5), (6) does not imply a uniform asymptotics of a solution of problem (1), because j and t are connected by the equation (4). Still, for any natural N, we can extract from series (9) and equation (4) the uniform asymptotics in the time interval [0, \epsilon^-N]. To do this, it is sufficient to substitute the partial sum of series (9) in the right hand side of (4). As the result of integration, we will obtain an expression for t in the form of a sum of integrals \int₀^j S(\xi, \epsilon\xi)d\xi, where S(\xi, \theta) are periodic in \xi, while in \theta tend exponentially fast to functions periodical in \xi. Describing the asymptotics of such integrals for \epsilon --> 0, uniform in [0, \epsilon^-N], we can obtain the asymptotics of solution of (1).

[1] Bogolyubov N.N., Mitropol'skii Yu.A. Asymptotic Methods in the Theory of Non-Linear Oscillations. Moscow, 1963 (in Russian).
[2] Grebenikov E.A. Averaging Method in Applied Problems. Moscow: Nauka, 1986 (in Russian).

Date received: March 11, 1998