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The method of derivative-expansion with multiple scales is applied in a generalized from to the analysis of weak nonlinear dispersive waves. It is shown that in a certain case a nonlinear Schrödinger equation, which describes a nonlinear slow amplitude modulation of wave trains, can be derived from the condition that the perturbation expansion be free from secular terms. A reduction to the Korteweg-de Vries equation for long waves is also discussed.
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Takuji Kawahara (1973) studied this question.
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