On collinear steady-state gravity waves with an infinite number of exact resonances

In this paper, we investigate the nonlinear interaction of two primary progressive waves traveling in the same/opposite direction. Without loss of generality, two cases are considered: waves traveling in the same direction and waves traveling in the opposite direction. There exist an infinite number of resonant wave components in each case, corresponding to an infinite number of singularities in mathematical terms. Resonant wave systems with an infinite number of singularities are rather difficult to solve by means of traditional analytic approaches such as perturbation methods. However, this mathematical obstacle is easily cleared by means of the homotopy analysis method (HAM): the infinite number of singularities can be completely avoided by choosing an appropriate auxiliary linear operator in the frame of the HAM. In this way, we successfully gain steady-state systems with an infinite number of resonant components, consisting of the nonlinear interaction of the two primary waves traveling in the same/opposite direction. In physics, this indicates the general existence of so-called steady-state resonant waves, even in the case of an infinite number of resonant components. In mathematics, it illustrates the validity and potential of the HAM to be applied to rather complicated nonlinear problems that may have an infinite number of singularities.