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

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 Korteweg–de 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.

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