Waves in a liquid containing bubbles, distributed in it with densityn(r),and waves in an elastic structure coupled with a number of oscillators are analyzed, with the principal question being the influence of nonlinearity on the form of the wave. This form (a soliton, for example) is very sensitive to a change of nonlinear parameter of the structure. The change occurs when the structure alters its local elastic properties during its operation (when fractures originate, for example). Stationary nonlinear waves in media with resonant dispersion (MRD) are nonstandard compared with nonlinear waves in media with weak dispersion – KdV solitons and cnoidal waves. The governing equation, the Resonant Dispersion Equation (RDE), is formulated. Its solutions have important differences compared with those of KdV and Burgers Equations. They include solitons with vertical fronts and shock waves of a complicated form. These forms are very sensitive to the nonlinear and dissipative parameters of the structure. ©2000 American Institute of Physics.