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On the integrability of a class of nonlinear dispersive wave equations. (English) Zbl 1089.35522

Summary: 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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References:

[1] Degasperis A Procesi M Asymptotic Integrability, in Symmetry and Perturbation Theory, Editors: Degasperis A and Gaeta G, World Scientific, Singapore, 1999, 23–37
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