Wave and Basin Structure in Spatially Coupled magneto-Elastic Beam System - Transitions between Coexisting wave solutions

The standing and traveling waves are well known phenomena of the coupled ordinary differential equations in many fields. The wave solutions of the coupled system are considered to be similar to the partial differential equation of the system. In this paper, the waves which appear in a coupled magneto-elastic beam system are discussed theoretically and numerically. The physical system is continuous elastically and discrete magnetically. There are several classes of models describing the system behavior. The Galerkin method is one of the powerful methods used to analyze the dynamics of the spatially distributed structure. The numerical solutions appeared in the coupled ordinary differential equation must show the spatially discrete characteristics even in the distributed system. However, most of the results obtained in the coupled systems are not more than the numerical approximation of the related partial differential equations. The large number of oscillations are given for the approximation. In this paper, the relationship between the coupled magneto-elastic beam system and the modified KdV equation is established by using the long wave approximation. However, in the short wave length range, the approximation to the partial differential equation has no physical rationality. Therefore, the analysis of the differencedifferential equation provides an important place of knowledge filling up the gap between the characteristics of the physical model and the numerical approximation. Keyword : coupled Duffing’s equation, elastic wave, modified KdV equation, attractor basin, heteroclinic structure, wave propagation