Abstract
Whitham's averaged Lagrangian method is generalized for the three-dimensional (3D) case of the Stokes waves on the surface of an ideal fluid layer. We derive the extended Lagrangian with the explicit terms , , and which were only partially and implicitly present in the Whitham Lagrangian by the term with the nonlinear frequency, where is the complex-valued amplitude of the wave envelope. The and terms generate the additional terms with and in the dispersive part of the extended Lagrangian compared to Whitham's terms with a2 and a4, where a is the real wave amplitude. The variation of our extended Lagrangian produces the evolutional Davey–Stewartson equations for the wave envelope and velocity potential which were originally obtained by the method of multiple scales.
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