Whitham modulation equations, coalescing characteristics, and dispersive Boussinesq dynamics

Daniel J. Ratliff, Thomas J. Bridges

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1+1 and 2+1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and do not depend on particular equations. One curious by-product of the analysis is that the theory can be used to confirm that the two-way Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein–Gordon equation is used to illustrate the theory.
Original languageEnglish
Pages (from-to)107-116
Number of pages10
JournalPhysica D: Nonlinear Phenomena
Early online date21 Jan 2016
Publication statusPublished - 15 Oct 2016
Externally publishedYes


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