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Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation. Recent developments in the mathematical theory of water waves (Oberwolfach, 2001). (English) Zbl 1152.76335

Summary: We study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness \(h\). We assume that the flow is potential and the dimensionless parameters are the ratio between densities \(\rho = \rho_2/\rho_1\) and \(\lambda = gh/c^2\). We study special values of the parameters such that \(\lambda(1 -\rho)\) is near \(^-\), where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where \(U = 0\) corresponds to a uniform state (velocity \(c\) in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with, in addition, a double eigenvalue in 0, a pair of simple imaginary eigenvalues \(\pm i\lambda\) at a distance \(O(1)\) from 0, and for \(\lambda(1 -\rho)\) above \(1\), another pair of simple imaginary eigenvalues tending towards 0 as \(\lambda(1-\rho)\to 1^+\). When \(\lambda(1-\rho)\leq 1\), this pair disappears into the essential spectrum. The rest of the spectrum lies at a distance at least \(O(1)\) from the imaginary axis. We show in this paper that for \(\lambda(1-\rho)\) close to \(1^-\) there is a family of periodic solutions similar to that in the Lyapunov-Devaney theorem (despite the resonance due to the point 0 in the spectrum). Moreover, showing that the full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillation, we also prove the existence of a family of homoclinic connections to these periodic orbits, provided that these ones are not too small.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
47J99 Equations and inequalities involving nonlinear operators
76E99 Hydrodynamic stability
35Q35 PDEs in connection with fluid mechanics
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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