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Abstract The nonlinear partial differential equations for dispersive waves have special solutions representing uniform wavetrains. An expansion procedure is developed for slowly varying wavetrains, in which full nonlinearity is retained but the scale of the nonuniformity introduces a small parameter. The first-order results agree with those obtained previously by averaging techniques (Whitham 1965 a, b). The perturbation method provides a detailed description and deeper understanding, as well as a consistent development to higher approximations. This method for treating partial differential equations is analogous to the ‘multiple time scale' methods for ordinary differential equations in nonlinear vibration theory. It may also be regarded as a generalization of the geometrical optics approximation to nonlinear problems.
Jon C. Luke (1966) studied this question.