Capillary Gravity Waves on the Free Surface of an Inviscid Fluid of Infinite Depth - Existence of Solitary Waves

Existence of solitary internal waves in a two layer uid of innnite depth. Preprint 1995. 17] J.M. Vanden{Broeck & F. Dias. Gravity{capillary solitary waves in water of innnite depth and related free surface ows. 17 The linear operator L im associated with the second equation has also a simple zero eigen-value with eigenfunction A which is even. Therefore, L im is also invertible on odd functions in C 1;2. All works in C 1;2 so A p 2 C 2 1;2 , hence e A and e B are in C 1;2. Tracing back to the form of the free surface = Z(), we have after a careful examination Z = tan (x) = O p 1 + x 2 at innnity and since @ @x 1 one has nally Z() = O p jj for jj ! 1: Summarizing we have proved Theorem 2 There exists a pair of reversible solitary wave solutions for the system (1.1) (1.2) (1.3) such that U ? 1 = O 3=2 j ln j=(1 + 1=2 jj) ; V = O ? p =(1 + 2) and Z = O ? p =jj as jj ! 1. Remark. It has been pointed out to the authors by J.C. Saut that these solitary waves do not decay exponentially. This results from the nonsmoothness of the Fourier symbol at the origin, while the Fourier transform of an exponentially decaying function is analytic in a strip containing the real axis. Because of (1.8) and cannot both decay exponentially at innnity, hence none of them decays exponentially due to (1.5). Let us mention that in the papers of Amick 1] and Sun 16] (the problem is diierent, but a similar method should work) it is found that the decay is like 1=x 2. The problem of the true rate here is diierent, since the principal part of the solitary wave coming from the normal form has an exponential decay. Here the non exponential decay comes from high order terms.