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We investigate the integrability of a class of 1+1 dimensional models describing nonlinear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.
R. Ivanov, On the Integrability of a Class of Nonlinear Dispersive Wave Equations, Journal of Nonlinear Mathematical Physics 12, 2005, 462-468