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Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods. (English) Zbl 1074.74541
Summary: In this paper, we study an integrable nonlinear evolution equation which arises in the context of nonlinear dispersive waves in hyperelastic rods. To consider bounded travelling-wave solutions, we conduct a phase plane analysis. A new feature is that there is a vertical singular line in the phase plane. By considering equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. The trajectories which represent bounded travelling-wave solutions are studied one by one. In total, we find there are 12 types of bounded travelling waves, both supersonic and subsonic. While in literature solutions for only two types of travelling waves are known, here we provide explicit solution expressions for all 12 types of travelling waves. Also, it is noted for the first time that peakons can have applications in a real physical problem.
MSC:
74H45Vibrations (dynamical problems in solid mechanics)
74B20Nonlinear elasticity
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
35Q72Other PDE from mechanics (MSC2000)
74H05Explicit solutions (dynamical problems in solid mechanics)