Camassa, Roberto; Holm, Darryl D. An integrable shallow water equation with peaked solitons. (English) Zbl 0972.35521 Phys. Rev. Lett. 71, No. 11, 1661-1664 (1993). Summary: We derive a new completely integrable dispersive shallow water equation that is bi-Hamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler’s equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak. Cited in 6 ReviewsCited in 1874 Documents MSC: 35Q51 Soliton equations 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 76B25 Solitary waves for incompressible inviscid fluids Keywords:bi-Hamiltonian; conservation laws; asymptotic expansion; soliton solution × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] R. Camassa, Physica (Amsterdam) 60D pp 1– (1992) [2] A. E. Green, J. Fluid Mech. 78 pp 237– (1976) · Zbl 0351.76014 · doi:10.1017/S0022112076002425 [3] D. D. Holm, Phys. Fluids 31 pp 2371– (1988) · Zbl 0643.76007 · doi:10.1063/1.866587 [4] G. B. Whitham, in: Linear and Nonlinear Waves (1974) · Zbl 0373.76001 [5] T. B. Benjamin, Philos. Trans. R. Soc. London A 227 pp 47– (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032 [6] P. J. Olver, Phys. Lett. A 126 pp 501– (1988) · doi:10.1016/0375-9601(88)90047-3 [7] P. J. Olver, Contemp. Math. 28 pp 231– (1984) [8] B. Fornberg, Philos. Trans. R. Soc. London A 289 pp 373– (1978) · Zbl 0384.65049 · doi:10.1098/rsta.1978.0064 [9] P. J. Olver, in: Applications of Lie Groups to Differential Equations (1986) · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2 [10] M. J. Ablowitz, in: Solitons and the Inverse Scattering Transform (1981) · Zbl 0472.35002 · doi:10.1137/1.9781611970883 [11] R. Camassa, Phys. Trans. R. Soc. London A 337 pp 429– (1991) · Zbl 0748.76058 · doi:10.1098/rsta.1991.0133 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.