Variational methods and applications to water waves

This paper reviews various uses of variational methods in the theory of nonlinear dispersive waves, with details presented for water waves. The appropriate variational principle for water waves is discussed first, and used to derive the long-wave approximations of Boussinesq and Korteweg & de Vries. The resonant near-linear interaction theory is presented briefly in terms of the Lagrangian function of the variational principle. Then the author’s theory of slowly varying wavetrains and its application to Stokes’s waves are reviewed. Luke’s perturbation theory for slowly varying wavetrains is also given. Finally, it is shown how more general dispersive relations can be formulated by means of integro-differential equations; an important application of this, developed with some success, is towards resolving longstanding difficulties in understanding the breaking of water waves.