The existence of internal solitary waves in a two-fluid system near the KdV limit

Abstract

Abstract Two fluid layers of constant density lying one over the other on top of a rigid horizontal lower boundary with either a free upper surface or a rigid upper boundary can support solitary waves. The existence of a unique branch of such waves emanating from the horizontal flow at a critical speed U ∗ is demonstrated in both cases by use of the Nash—Moser implicit function theorem. These results complement the global results of Amick and Turner (1986) and are analogous to the work of Friedrichs and Hyers (1954) and Beale (1977) for surface waves. It is also noted that the most obvious variational principle which characterizes these waves as constrained extremals (Benjamin, 1984) is of indefinite type, having a Hessian with infinitely many positive and infinitely many negative eigenvalues.