The Instabilities of Periodic Traveling Water Waves with Respect to Transverse Perturbations

Using an exact reformulation of the classical surface water wave problem due to Ablowitz, Fokas and Musslimani, we investigate the instabilities of one-dimensional stationary periodic waves, with respect to transverse perturbations. Such perturbations have trigonometric dependence on the transverse variable, and are bounded (typically quasi periodic) in the longitudinal direction. Using the new formulation, we examine waves in both deep and shallow water, confirming previous results about their instabilities due to McLean and Francius & Kharif, among others. Not only do we confirm these known results, but we demonstrate that the new formulation allows for greatly improved accuracy, at a reduced cost. As a consequence, we show that the regions of instability in the eigenfunction parameter space are smaller than previously ascertained.