It is well known that the water-wave problem with weak surface tension hassmall-amplitude line solitary-wave solutions which to leading order aredescribed by the nonlinear Schr\"odinger equation. The present paper containsan existence theory for three-dimensional periodically modulated solitary-wavesolutions which have a solitary-wave profile in the direction of propagationand are periodic in the transverse direction; they emanate from the linesolitary waves in a dimension-breaking bifurcation. In addition, it is shownthat the line solitary waves are linearly unstable to long-wavelengthtransverse perturbations. The key to these results is a formulation of thewater wave problem as an evolutionary system in which the transverse horizontalvariable plays the role of time, a careful study of the purely imaginaryspectrum of the operator obtained by linearising the evolutionary system at aline solitary wave, and an application of an infinite-dimensional version ofthe classical Lyapunov centre theorem.
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