Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation.
In this paper, 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 1(-), 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 +/-ilambda 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) --> 1(+). When lambda(1 - rho) <or= 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.
Iooss
Gérard
G
Institute Universitaire de France, 103 bld Saint-Michel, 75005 Paris, France.
Lombardi
Eric
E
Sun
Shu Ming
SM
D006112:Gravitation; D009038:Motion; D017711:Nonlinear Dynamics; D010507:Periodicity; D012212:Rheology / Q000379:methods*; D012996:Solutions; D014867:Water; D014872:Water Movements
D012996:Solutions; D014867:Water