Two-Dimensional Instabilities of Weakly Nonlinear Capillary Gravity Waves of Permanent Form near the Fourth Harmonic Resonance

Three systems of weakly nonlinear envelope equations for a fundamental mode and its fourth harmonic of two-dimensional capillary gravity waves on deep water are derived. These systems are used to investigate stability of progressive periodic waves of permanent form. It is found that the permanent waves in which the fundamental mode and its fourth harmonic are of the same order are unstable. Modulational instability in which the dominant mode in the eigenfunction is near the fourth harmonic is found in the third order approximation. Furthermore, instability related to the fourth harmonic resonance is found in the fourth order approximation.