Progressing waves in an infinite nonlinear string

We wish to determine the types of waves which propagate in this medium at constant speed, and their propagation speeds, for various values of a and P. The problem reduces to that of interpreting the solutions of the nonlinear ordinary differential equation (5); the properties of these solutions, and the method of obtaining them, are discussed in §2. We proceed, in §§3, 4 and 5, to illustrate the noteworthy features of the wave solutions of (1) by considering particular combinations of a and p. Waves which travel at speeds greater (less) than one will be called supercritical (subcritical). In §3 it is shown that, for given positive a and P, all continuous wave solutions are of the form (13), where the amplitude a and (supercritical) velocity c may be chosen arbitrarily. In contrast, wave solutions of the simple wave equation (2) uu — uxx = 0 may have arbitrary shapes, but travel only with velocities +1. In §4 we note a duality principle between sub- and supercritical waves when the signs of a and P are changed. Finally, in §5, we see that when aP