Nonlinear stability of resonant capillary-gravity waves. (English) Zbl 0745.76022
Summary: The method of multiple scales, in both space and time, is used to examine the stability, under very long perturbations, of the capillar-gravity which arise as a result of second harmonic resonance. A pair of coupled nonlinear partial differential equations for the amplitude of the wave and its second harmonic are derived. These describe the evolution of the wavetrain up to cubic order and may be regarded as the counterparts of the single nonlinear Schrödinger equation which occurs in the nonresonant case. Regions of instability are identified.
MSC:
| 76E30 | Nonlinear effects in hydrodynamic stability |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B45 | Capillarity (surface tension) for incompressible inviscid fluids |
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