On the standing wave problem in deep water
We present a new formulation of the classical two-dimensional standing wave problem which makes transparent the (seemingly mysterious) elimination of the quadratic terms made in the author's paper [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1989, 3513--3526 (1999; Zbl 0949.76014)]. Despite the presence of infinitely many resonances, corresponding to an infinite dimensional kernel of the linearized operator, we solve the infinite dimensional bifurcation equation by uncoupling the critical modes up to cubic order, via a Lyapunov-Schmidt like process. This is done without using a normalization of the cubic order terms as in the paper cited above, where the computation contains a mistake, although the conclusion was in the end correct. Then we give all possible bifurcating formal solutions, as power series of the amplitude, with an arbitrary number, possibly infinite, of dominant modes .
infinite dimensional bifurcation equation
scientific article; zbMATH DE number 1798860
scientific article