Nonlinear superposition and absorption of delta waves in one space dimension

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In the study of the initial value problem for linear hyperbolic partial differential equations it is usually possible to extend the solution operator from classical to distribution initial data. This extension is useful for several reasons. First, the existence of fundamental solutions allows one to construct other solutions by the superposition principle. Second, distribution data are idealizations of certain kinds of classical initial data: The delta functon is an idealization of a high localized peak, the derivative of a delta function is an idealization of a single sharp localized oscillation, and so forth. If one can understand how the idealized data are propagated, then, with suitable continuity, one has understood some of the essential features of the classical solutions with data which are close to the distribution data. In the linear theory there are many theorems which give precise analytical content to this idea. In the nonlinear case the first reason for studying distribution data is not operative since there is no general superposition principal for nonlinear equations. The second reason is certainly appropriate, however, since one is very interested in the propagation of high peaks, sharp jumps, and