SynopsisModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems wuh no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.As an example for the dIssipative case. we find that the Ginzburg-Landau equation is the modulation equation for the Swift-Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equatton is shown to describe the modulation of wave packets in the Sine-Gordon equation
A completeness result for Lamb modes in homogeneous waveguides is proved. The problem is formulated as a linear eigenvalue problem in an appropriate Hilbert space of functions. Orthogonality and biorthogonality relations are given. A detailed spectral analysis which is necessary for the application of a general completeness theorem is presented.
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