S. A. Rybak
Yu. I. Skrynnikov
N. N. Andreev Acoust. Inst. of Russian Acad. of Sci., Shvernik str. 4, 117036, Moscow, Russia
To describe the propagation of nonlinear longitudinal strain waves in a rod that curves into an arc of radius R use is made of the nonlinear Klein--Gordon equation in the form u[sub tt]-c[sup 2]u[sub xx]+(c[sup 2]/R[sup 2])u=((beta)/2(rho))(u[sup 2])[sub xx]. Here, (beta) and (rho) are the nonlinearity parameter and the density of the material, and u is the x derivative (the x axis is directed along the rod) of the longitudinal component of the displacement vector, i.e., the longitudinal strain; c is the velocity of linear longitudinal waves. The steady-state exact solution in the form of solitary wave was found. The wave has two symmetrically located vertical edges. The tails of the wave obtained, i.e., the parts that decay infinitely in either direction, have the same structure as the tails of a Korteweg--de Vries soliton. As in the case of the Korteweg--de Vries soliton, the amplitude A and the width (Delta) of the wave depend on its velocity. However, a different relation exists between the amplitude and the width: (Delta)~A[sup 1/2].