Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations

Abstract

In a recent paper Zabusky has given an accurate estimate of the time interval in which solutions of the nonlinear string equation y_tt = c²(1 + εy_x)y_xx exist. A previous numerical study of solutions of this equation disclosed an anomaly in the partition of energy among the various modes; Zabusky's estimate shows that at the time when the anomaly was observed the solution does not exist. The proof of Zabusky uses the hodograph method; in this note we give a much simpler derivation of the same result based on an estimate given some years ago by the author.