Normal forms for wave motion in fluid interfaces

Abstract

The subject of this paper is the dynamics of wave motion in the two-dimensional Kelvin–Helmholtz problem for an interface between two immiscible fluids of different densities. The difference of the mean flow between the two fluid bodies is taken to be zero, and the effects of surface tension are neglected. We transform the problem to Birkhoff normal form, in which a precise analysis can be made of classes of resonant solutions. This paper studies standing-wave solutions of the fourth-order normal form in particular detail. We find that there are families of invariant resonant subsystems, which are nevertheless integrable. Within these families we describe the periodic and the time quasi-periodic standing waves, and determine their stability or instability. In particular we show that for a certain range of densities, a basic time-periodic standing wave with principal wave number k is unstable to modes with principal wave numbers k/4 and 9 k/4, and we calculate the 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 well as others that we describe, appears to be new, and its description is possible 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 all orbits undergoing stable almost-periodic motion, and instabilities arise only in normal forms to higher order.