Addition of dispersive terms to the method of averaged Lagrangian

Abstract

Whitham's method of averaged Lagrangian is applied to the problem of Stokes waves on the surface of a layer of ideal fluid. We derive a Lagrangian which contains additional terms with \documentclass[12pt]{minimal}\begin{document}$a_ {x} ^ {2}$\end{document}ax2 and aaxx besides nonlinear terms of Whitham with a2 and a4, a being the wave amplitude. These terms with derivatives appear after taking into account (1) additional terms in the Stokes expansions with the same approximation as the one was taken into consideration by Whitham; (2) supplementary slow (in comparison to the rapid phase oscillations) dependence of the amplitudes of harmonics on coordinates and time. We demonstrate the need for the account of such terms in the Lagrangian for obtaining the correct coefficients of dispersive terms of evolution equations from the corresponding variational equations.