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Solitary-wave interactions in elastic rods. (English) Zbl 0606.73028

The paper studies the propagation of longitudinal deformation waves modelled by the equation \[ u_{tt}-u_{ttxx}-u_{xx}-1/p(u^ p)_{xx}=0 \] with \(p=3\) or 5. The equation is first derived subject to certain constraints relating to the material of the rod and the class of solutions to be considered. It is then shown that this equation fails the Painlevé test, and so is probably not completely integrable. In conclusion the head-on collision of two equal solitary waves is studied both numerically and asymptotically for the case of small and large amplitude waves.
Reviewer: A.Jeffrey

MSC:

74J99 Waves in solid mechanics
35Q99 Partial differential equations of mathematical physics and other areas of application
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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