LOCAL AND GLOBAL ASPECTS OF THE (1, N) MODE INTERACTION FOR CAPILLARY GRAVITY-WAVES

The structure of the solutions of the capillary-gravity wave problem in a neighbourhood of a (1, n) mode interaction with n greater-than-or-equal-to 4 are considered, extending the results of Jones and Toland in this important case. These results are based on the known general form of the bifurcation equations which arises due to the symmetry of the problem and also the universal unfolding of a pitchfork bifurcation. It is shown how the higher-order terms of the bifurcation equation unfold a secondary pitchfork bifurcation which occurs in the truncated bifurcation equations, obtained by ignoring the higher-order terms. The global effect of these local mode interactions is then considered and it is shown how repeated mode interactions result in secondary loops of solutions being formed which intersect three primary branches of solutions. An example of this behaviour in the capillary-gravity wave problem is given.