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An integrable shallow water equation with peaked solitons. (English) Zbl 0972.35521
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.

MSC:
35Q51Soliton-like equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25Solitary waves (inviscid fluids)