Normal forms for wave motion in fluid interfaces

The subject of this paper is the dynamics of wave motion in the two-dimensi onal Kelvin-Helmholtz problem for an interface between two immiscible fluid s of different densities. The difference of the mean flow between the two f luid bodies is taken to be zero, and the effects of surface tension are neg lected. We transform the problem to Birkhoff normal form, in which a precis e analysis can be made of classes of resonant solutions. This paper studies standing-wave solutions of the fourth-order normal form in particular deta il. We find that there are families of invariant resonant subsystems, which are nevertheless integrable. Within these families we describe the periodi c and the time quasi-periodic standing waves, and determine their stability or instability. In particular we show that for a certain range of densitie s, a basic time-periodic standing wave with principal wave number k is unst able to modes with principal wave numbers k/4 and 9k/4, and we calculate th e Lyapunov exponent of the instability. We furthermore show that the stable and unstable manifolds to these periodic solutions of the Birkhoff normal form are connected by a homoclinic orbit. This instability mechanism, as we ll as others that we describe, appears to be new, and its description is po ssible because of the precision afforded by the normal form. These results contrast with the case of the water wave problem described by Dyachenko and Zakharov [A.I. Dyachenko, V.E. Zakharov, Phys. Lett. A 190 (1994) 144-148] and Craig and Worfolk [W. Craig, P. Worfolk, Physica D 84 (1995) 513-531] where the fourth-order Birkhoff normal form is an integrable system, with a ll orbits undergoing stable almost-periodic motion, and instabilities arise only in normal forms to higher order. (C) 2000 Elsevier Science B.V. All r ights reserved.