In this chapter, we shall assume the existence of appropriate asymptotic expansions, and derive asymptotic equations for hyperbolic PDEs. These equations display the qualitative effects of dissipation or dispersion balancing nonlinearity, and are easier to study analytically. In this way, the study of complicated models reduces to that of models of asymptotic approximations expressed by a hierarchy of equations, which facilitate numerical calculations; this is often the only way that progress can be made to analyze complicated systems. Essential ideas underlying these methods may be found in earlier publications; see for example Boillat [19], Taniuti, Asano and their coworkers ([7], [194]), Seymour and Varley [166], Germain [61], Roseau [155], Jeffrey and Kawahara [82], Fusco, Engelbrecht and their coworkers ([58], [59]), Cramer and Sen [43], Kluwick and Cox [94], and Cox and Kluwick [42]. An account of some of the rigorous results which deal with convergence of such expansions may be found in [47].