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For arbitrary values of a parameter λ ∈ R, finite-time blowup of solutions to the generalized, inviscid Proudman-Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem. Mathematics Subject Classification (2010). 35B44, 35B10, 35B65, 35Q35.

The propagation of longitudinal deformation waves in an elastic rod is modelled by the nonlinear partial differential equationwith p = 3 or 5. This equation is first derived under a range of possible constraints. We then show that this equation and even certain generaliza.ions do not pass the Painleve test, and hence are probably not completely integrable. Finally, we study the head-on collision of two equal solitary waves numerically and also asymptotically for small and large amplitude.

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