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The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations. (English) Zbl 1258.76047

Summary: The Serre equations are a pair of strongly nonlinear, weakly dispersive, Boussinesq-type partial differential equations. They model the evolution of the surface elevation and the depth-averaged horizontal velocity of an inviscid, irrotational, incompressible, shallow fluid. They admit a three-parameter family of cnoidal wave solutions with improved kinematics when compared to KdV theory. We examine their linear stability and establish that waves with sufficiently small amplitude/steepness are stable while waves with sufficiently large amplitude/steepness are unstable.

MSC:

76B25 Solitary waves for incompressible inviscid fluids
76E17 Interfacial stability and instability in hydrodynamic stability
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