The nth-order self-resonance law of single wave train for surface capillary-gravity waves in deep water

Wave-wave resonance mechanism plays a fundamental and prominent role in the process of energy transmission and distribution in whether microscopic or macroscopic materials. For the most extensive and intuitive ocean surface wave motion on earth, it is bound to be even more so. Can we extract the general wave-wave resonance law from it? Especially the most special and brief resonance one for single wave train. To this end, according to a set of classical methods proposed by Phillips for initiating modern water wave dynamics with the specific 4-wave resonance conditions, and starting from the basic governing equations of ocean deep-water surface capillary-gravity waves, the first-order differential equation of the Fourier component of free surface displacement and the second-, third- and fourth-order integral differential ones which are becoming more and more complex but tend to be complete are given in turn by the Fourier-Stieltjes transformation and perturbation method. Under a set of symbol system which are self-created, self-evident and concise, these equations are solved in turn to obtain the first-order free surface displacement of single wave train, the Fourier coefficients of the second-, third- and fourth-order non-resonant and resonant free surface ones and the second-, third- and fourth-order resonant conditions, thus leading to the general nth-order self-resonance law of single wave train. This completely reveals the rich connotation of single wave resonance dynamics of ocean surface capillary-gravity waves, effectively expands the application range of the classical single wave resonance solutions given by Phillips for ocean surface gravity waves, lays the foundation for depicting single and multiple resonance interaction mechanisms of double and multi-wave trains of ocean surface waves, and so provides a typical example for the exploration of single-wave resonance law in all wave fields.