On three-dimensional generalizations of the Boussinesq and Korteweg-de Vries equations

Infeld, E.

doi:10.1090/qam/592196

Author: E. Infeld
Journal: Quart. Appl. Math. 38 (1980), 277-287
MSC: Primary 35Q20; Secondary 76B25
DOI: https://doi.org/10.1090/qam/592196
MathSciNet review: 592196
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Abstract: Three-dimensional generalizations of two different forms of the Boussinesq equation are derived. They are investigated for stability of slowly varying nonlinear wavetrains. The results obtained are then compared with the stability properties following from the full water wave equations. Agreement is found to be good for ${h_0}{k_0}$ (depth times wavenumber) of order one. This is very satisfactory, as the Boussinesq equations are only supposed to be valid for small ${h_0}{k_0}$. In particular, one version of the Boussinesq equation is found to yield instability with respect to one-dimensional perturbations for ${h_0}{k_0} > 1.5$ (as against 1.36 for the full equations). Finally, a similar comparison is performed for the three-dimensional Korteweg—de Vries equation.

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