Multiple time scale formalism and its application to long water waves

In the present work, by employing the multiple time scaling method, we studied the nonlinear waves in shallow-water problem and obtained a set of Korteweg–deVries equations governing the various order terms in the perturbation expansion. By seeking a travelling wave type of solutions for the evolution equations, we have obtained a set of wave speeds associated with each time parameter. It is shown that the speed corresponding to the lowest order time parameter given the wave speed of the conventional reductive perturbation method, whereas the wave speeds corresponding to the higher order time parameters give the speed correction terms. The result obtained here is exactly the same with that of Demiray [H. Demiray, Modified reductive perturbation method as applied to long water waves: Korteweg–deVries hierarchy, Int. J. Nonlinear Sci. 6 (2008) 11–20] who employed the modified reductive perturbation method.