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In this Letter we consider a new rod equation derived recently by Dai [H.-H. Dai, Acta Mech. 127 (1998) 193] for a compressible hyperelastic material. We explore various sufficient conditions of the initial data which guarantee the blow-up in finite time for periodic case. The focus of this Letter is on two particular classes of initial data, ∫Su0=0 or u0 is odd.
In this Letter an integrable shallow water equation derived by Dullin, Gottwald and Holm is investigated. Analogous to the Camassa–Holm equation, this equation possesses the blow-up phenomenon. The orbital stability of the peaked solitary waves is also proved.
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