The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains

Rogue waves are rare giant, freak, monster or steep wave events in nonlinear deep water gravity waves which occasionally rise up to surprising heights above the background wave field. Holes are deep troughs which occur before and/or after the largest rogue crests. The dynamical behavior of these giant waves is here addressed as solutions of the nonlinear Schrodinger equation in both 1 + 1 and 3 + 1 dimensions. We discuss analytical results for 1 + 1 dimensions and demonstrate numerically, for certain sets of initial conditions, the ubiquitous occurrence of rogue waves and holes in 2 + 1 spatial dimensions. A typical wave field evidently consists of a background of stable wave modes punctuated by the intermittent upthrusting of unstable rogue waves. (C) 2000 Published by Elsevier Science B.V.