TY - UNPB

T1 - Limitations of the Method of Multiple-Time-Scales

AU - Kahn, Peter B.

AU - Zarmi, Yair

PY - 2002/6/1

Y1 - 2002/6/1

N2 - In the Method of Multiple-Time-Scales (MMTS), the introduction of
independent time scales and the elimination of secular terms in the fast
time variable, T0 = t, lead to the well-known solvability conditions.
Starting from first order, free terms (solutions of the unperturbed
equations) emerge in every order in the expansion of the approximate
solution. In orders higher than first, the amplitudes of these free
terms appear in the solvability conditions. Contrary to the common
belief, in the MMTS analysis, these free terms play a role above and
beyond the satisfaction of initial conditions: They make feasible mutual
consistency among solvability conditions that arise in different orders.
In general, this consistency may not be ensured if the free terms are
chosen arbitrarily (e.g., set to zero, as is commonly done in many
applications). If consistency is not ensured, the analysis may lead to
wrong results, or allow only trivial solutions. The solvability
conditions constitute a system of PDE's for the dependence of the
amplitudes that appear in the expansion on the slow time scales.
However, whenever the free amplitudes must be included to ensure a
consistent expansion, these PDE's cannot determine the dependence of the
solution on slow time variables beyond the first one, T1=epsilo*t. The
dependence on slower time scales, Tn=epsilon^n*t, n > 1, must be
imposed either through initial data at, say, T1 = 0, or through
requirements on the structure of the approximate solution (based, for
example, on physical intuition) that are not related to the validity of
the perturbative scheme. These claims are illustrated through several
simple examples, and then discussed in the general case.

AB - In the Method of Multiple-Time-Scales (MMTS), the introduction of
independent time scales and the elimination of secular terms in the fast
time variable, T0 = t, lead to the well-known solvability conditions.
Starting from first order, free terms (solutions of the unperturbed
equations) emerge in every order in the expansion of the approximate
solution. In orders higher than first, the amplitudes of these free
terms appear in the solvability conditions. Contrary to the common
belief, in the MMTS analysis, these free terms play a role above and
beyond the satisfaction of initial conditions: They make feasible mutual
consistency among solvability conditions that arise in different orders.
In general, this consistency may not be ensured if the free terms are
chosen arbitrarily (e.g., set to zero, as is commonly done in many
applications). If consistency is not ensured, the analysis may lead to
wrong results, or allow only trivial solutions. The solvability
conditions constitute a system of PDE's for the dependence of the
amplitudes that appear in the expansion on the slow time scales.
However, whenever the free amplitudes must be included to ensure a
consistent expansion, these PDE's cannot determine the dependence of the
solution on slow time variables beyond the first one, T1=epsilo*t. The
dependence on slower time scales, Tn=epsilon^n*t, n > 1, must be
imposed either through initial data at, say, T1 = 0, or through
requirements on the structure of the approximate solution (based, for
example, on physical intuition) that are not related to the validity of
the perturbative scheme. These claims are illustrated through several
simple examples, and then discussed in the general case.

KW - Exactly Solvable and Integrable Systems

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BT - Limitations of the Method of Multiple-Time-Scales

ER -