Study on the behavior of weakly nonlinear water waves in the presence of random wind forcing

Abstract

Specific solutions of the nonlinear Schrödinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is, whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrödinger equation is obtained for the case of wind forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrödinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather type solutions such as the Peregrine breather occur even in strong gusty wind conditions.

X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves

“…[41]. However, taking N = 4, we can derive a expression describing an interaction between two 2-resonance Y-type solitons for System (2), which are different from those in Ref. [41], i.e.,…”

“…+ A 34 e η 3 +η 4 + A 13 A 14 A 34 e η 1 +η 3 +η 4 + A 23 A 24 A 34 e η 2 +η 3 +η 4 , where ( For the shallow water waves, hybrid solutions composed of the 2-resonance Y-type solitons and first-order breathers, i.e., (14), rely on the coefficient α in System (2). Figure 4 exhibits the interaction between the 2-resonance Y-type soliton and first-order breather based on Solutions (14).…”

“…Figure 4 exhibits the interaction between the 2-resonance Y-type soliton and first-order breather based on Solutions (14). 4.3 Hybrid solutions consisting of the M -resonance Y-type solitons and higher-order lumps for System (2) Setting the following conditions on N -Soliton Solutions (5):…”

“…Waves, one of the most common natural phenomena, have attracted people's attention [1][2][3][4][5][6]. It has been known that mechanical waves consist of the water waves, seismic waves, sound waves in air, electromagnetic waves, quantum waves, etc [7][8][9][10][11][12].…”

The linearly damped nonlinear Schrödinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events

“…With such examples for an external driver, the damped and driven NLS (1) may become potentially relevant in the context of weakly nonlinear waves in the presence of wind forcing [23], [24], [25], [26], [27], [28]: Note that in this context, the damped and driven NLS incorporates linear dissipation with the damping strength being associated to the fluid's (e.g., water's) viscosity, but also linear forcing (competitive to damping) associated to the effect of wind forcing. Herein, the considered cases of drivers may serve as simplified phenomenological examples representing the effect of a spatiotemporal wave amplification due to the external forcing f (inducing "wind" effects); we also refer to the recent work [29] assuming the presence of a stochastic linear driving.…”