A note on the Lagrangian method for nonlinear dispersive waves

This paper makes a few remarks on the method of the averaged Lagrangian developed by Whitham to describe slow variations of nonlinear wave trains. The concept of multiple scales is incorporated into the variational formalism, and a consistent development to higher approximation is suggested as a formal perturbation based on the variational principle. The propagation of weakly dispersive long waves is also reconsidered in relation to the Lagrangian method. It is demonstrated that the nonlinear Schrodinger equation and the Korteweg-de Vries equation can be derived from the Euler-Lagrange equations of the perturbed Lagrangian.