Application Of Homotopy Analysis Method In Steady-state Resonant Waves

Since the phenomenon of wave resonance exits in many ?elds, such as ocean engineering, optics, ?nance and so on, the study of wave resonance hence is important and has theoretic and practical meanings. In the ?eld of ocean engineering, there are a lot of studies about the linear and nonlinear wave resonance. Most of them focused on the evolution of the wave system’s amplitude and the corresponding energy transportation among different wave components. Using the Homotopy Analysis Method,this dissertation is devoted to a resonant wave system where there is no energy transportation among various wave modes at all. Three typical problems are investigated in detail and described brie?y as follows:(1) The resonant wave system propagating over a ?at bottom in the water of ?nite depth is investigated. The nonlinear governing equations of the nonlinear wave system are solved using the Homotopy Analysis Method. It is found that there are multiple steady-state solutions with time-independent wave amplitude, namely that they have the time-independent energy spectrum. In addition, it is found as well that the resonant wave component has less energy than the primary ones in some steady-state solutions, while it has the most energy of the wave system in another steady-state solutions. The HAM results are veri?ed by solving the Zakharov equation. The relationship between the steady and unsteady solutions is found as well.(2) The wave-bottom resonance between the surface wave system with single primary wave and bottom with in?nite sinusoidal ripples is investigated. This isthe simplest wave-bottom resonance, called the class-I Bragg resonance. The steady-state solutions with no energy transportation is focused as well. It is found that there exist two totally different wave systems, both of which have time-independent energy spectrum. The primary and resonant modes together have most of the wave energy. They share the same energy in the ?rst steadystate solution while different energy in the other steady-state solution. It is found for the ?rst time that there are bifurcations for these two steady-state solutions with respect to the propagation angle, mean water depth, bottom slope and nonlinearity.(3) The steady-state wave system with in?nite resonances in shallow water is studied. In the context of the Homotopy Analysis Method, the Korteweg-de Vries(Kd V) equation is solved. It is found that the proper choice of the auxiliary linear operator for the problem is important. Using HAM, we overcome the in?nite singularities which are caused by the resonances, and ?nally the steady-state solution is obtained.