Behaviour of Solutions
Eqs. (2.7.3) describe how the amplitudes A n develop as functions of time starting from some initial value. We shall particularly be interested in their behaviour in the limit as t → ∞, and in the question whether in that limit the amplitude functions approach finite, stationary values. Inverting the question, we can first construct some stationary solutions of Eqs (2.7.3) and then investigate under what conditions these solutions represent the limit of A n as t → ∞. This is the procedure that we shall ultimately follow, but it should be pointed out at the outset that a complete exact solution of Eqs. (2.7.3), or even of the equations governing stationary solutions, must be considered beyond expectation. We shall hence be concerned with methods of approximation and in a later chapter we shall develop asymptotic methods of solution applicable in certain cases. However, for a better understanding of the phenomena involved some exploratory analysis is useful.
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