Interacting nonlinear wave envelopes and rogue wave formation in deep water

Abstract

A rogue wave formation mechanism is proposed within the framework of a coupled nonlinear Schrödinger (CNLS) system corresponding to the interaction of two waves propagating in oblique directions in deep water. A rogue condition is introduced that links the angle of interaction with the group velocities of these waves: different angles of interaction can result in a major enhancement of rogue events in both numbers and amplitude. For a range of interacting directions it is found that the CNLS system exhibits significantly more extreme wave amplitude events than its scalar counterpart. Furthermore, the rogue events of the coupled system are found to be well approximated by hyperbolic secant functions; they are vectorial soliton-type solutions of the CNLS system, typically not considered to be integrable. Overall, our results indicate that crossing states provide an important mechanism for the generation of rogue water wave events.PACS numbers: 02.60.Cb Even in the age of modern ship building and innovative navigation equipment, ship accidents are an important area of maritime concern. In recent years researchers have been investigating a topic which heretofore had been reserved for marine folklore: giant waves that seem to appear from nowhere in high seas that can lead to disastrous outcomes. These waves have, nowadays, been documented to exist and are usually referred to as rogue or freak waves.The conditions that cause such waves to grow enormously in size are a topic of great interest with many different hypotheses being proposed 1,2 . An important one is the nonlinear mechanism of the self-wave interactions, such as modulation instability; in water wave physics this is called the Benjamin-Feir instability 3 and has been found under certain conditions to produce significant wave amplification.The envelope of nonlinear water waves, under suitable conditions, is modeled by a nonlinear Schrödinger equation (NLS) 4 . Interestingly enough, the NLS not only gives a suitable description of these waves 2,5-9 , it is also an important equation used to investigate propagation of pulses in many other physical systems such as nonlinear optical fibers 10-13 , Bose-Einstein condensates 14 , magnetic spin waves 15 and many others. The state of the sea in which rogue waves form is often complex 16-18 with certain key wave interactions dominating the wave structure. Here we focus on a coupled NLS (CNLS) system, derivable directly from the Euler equations. Underlying the wave phenomena are separate wave trains, propagating in different directions which interact at different angles; this provides a useful model of crossing sea states 16,[19][20][21][22] . The scalar NLS equation would be insufficient to describe these phenomena (rogue events associated with 2D scalar NLS equations are discussed a) Electronic mail: Corresponding author: horikis@uoi.gr in Ref.23 ). As with the scalar problem the CNLS system exhibits modulation instability (MI), which as mentioned above, is an important aspect of rogue wave dynamics. Indeed, rec...

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