Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods

In literature, nonlinear waves in elastic rods have been studied by many authors. Usually, the Navier-Bernoulli hypothesis (the assumption that plane cross-sections remain planar and normal to the rod axis) is used. Intuitively, one would expect that this would be a good approximation when one is mainly interested in longitudinal waves. However, there are no rigorous theoretical justifications available. Also, a defect of this assumption is that comparing with the exact three-dimensional theory the boundary conditions on the lateral surface can never be satisfied. Recently, three papers have been published to overcome this defect, but they contain some algebraic errors (which implies that the approach adopted there cannot be used to overcome this defect). So, this problem remains open. In this paper, we present our recent research results for this problem, and we have managed to establish asymptotically valid one-dimensional rod equations which are consistent with the lateral boundary conditions. Further, their dispersion relation can match with that of the exact three-dimensional field equations to any asymptotic order in the long-wave limit. For solitary waves in the far field, we derive to the leading order the KdV equation. Comparing its solitary-wave solution with that of the KdV equation obtained through the Navier-Bernoulli hypothesis, we find that the difference is very small. This provides some evidence to the validity of the assumption that plane cross sections remain planar and normal to the rod axis.