A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves

Abstract

The existence of a line solitary-wave solution to the water-wave problem with strong surface-tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg and G. de Vries and rigorously confirmed a century later by C. J. Amick and K. Kirchgässner in 1989. A model equation derived by B. B. Kadomtsev and V. I. Petviashvili in 1970 suggests that the Kortewegde Vries line solitary wave belongs to a family of periodically modulated solitary waves which have a solitary-wave profile in the direction of motion and are periodic in the transverse direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. It is shown that the Korteweg-de Vries solitary wave undergoes a dimension-breaking bifurcation that generates a family of periodically modulated solitary waves. The term dimension-breaking phenomenon describes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.