Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity

Abstract

The existence of two-dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established. When formulated as a second-order equation for a real-valued function w on the 2-torus and a positive parameter μ, the problem is fully nonlinear (the highest order x-derivative appears in the nonlinear term but not in the linearization at 0) and completely resonant (there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter μ). Moreover, for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial. Using a Nash-Moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions, the existence question can be reduced to one of estimating the inverses of linearized operators at non-zero points. After changing coordinates these operators become first order and non-local in space and second order in time. After further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing, or quasi-one-dimensional in the sense of [22]. The operator can then be inverted for two reasons. First, the explicit formula for the approximate solution means that, restricted to the infinite-dimensional kernel of the linearization at zero, the inverse exists and can be estimated. Second, the small-divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their Diophantine properties. A parameter-dependent version of the Nash-Moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth, corresponding to a set of values of the parameter μ>1 which is dense at 1. Unimodal means that the term of smallest order in the amplitude is cos x cos t, which is one of many eigenfunctions of the completely resonant linearized problem.