Enhanced rise of rogue waves in slant wave groups

Numerical simulations of fully nonlinear equations of motion for long-crested waves at deep water demonstrate that in elongate wave groups the formation of extreme waves occurs most intensively if in an initial state the wave fronts are oriented obliquely to the direction of the group. An “optimal” angle, resulting in the highest rogue waves, depends on initial wave amplitude and group width, and it is about 18–28 degrees in a practically important range of parameters.     

Rogue waves in a multistable system

Clear evidence of rogue waves in a multistable system is revealed by experiments with an erbium-doped fiber laser driven by harmonic pump modulation. The mechanism for the rogue wave formation lies in the interplay of stochastic processes with multistable deterministic dynamics. Low-frequency noise applied to a diode pump current induces rare jumps to coexisting subharmonic states with high-amplitude pulses perceived as rogue waves. The probability of these events depends on the noise filtered frequency and grows up when the noise amplitude increases. The probability distribution of spike amplitudes confirms the rogue wave character of the observed phenomenon. The results of numerical simulations are in good agreement with experiments.     

Edge-wave-driven durable variations in the thickness of the surfactant film and concentration of surface floats

By employing a simple model for small-scale linear edge waves propagating along a homogeneous sloping beach, we demonstrate that certain combinations of linear wave components may lead to durable changes in the thickness of the surfactant film, equivalently, in the concentration of various substances (debris, litter) floating on the water surface. Such changes are caused by high-amplitude transient elevations that resemble rogue waves and occur during dispersive focusing of wave fields with a continuous spectrum. This process can be treated as an intrinsic mechanism of production of patches in the surface layer of an otherwise homogeneous coastal environment impacted by linear edge waves.     

Rogue waves in the basin of intermediate depth and the possibility of their formation due to the modulational instability

The properties of rogue waves in the basin of intermediate depth are discussed in comparison with known properties of rogue waves in deep waters. Based on observations of rogue waves in the ocean of intermediate depth we demonstrate that the modulational instability can still play a significant role in their formation for basins of 20 m and larger depth. For basins of smaller depth, the influence of modulational instability is less probable. By using the rational solutions of the nonlinear Schrodinger equation (breathers), it is shown that the rogue wave packet becomes wider and contains more individual waves in intermediate rather than in deep waters, which is also confirmed by observations.     

The run-up of nonlinearly deformed sea waves on the coast of a bay with a parabolic cross-section

The run-up of long waves on the coast of a bay with a parabolic cross-section, where the region of constant depth along the principal axis of the bay is connected with the linearly inclined segment, is considered. The study is carried out analytically in the framework of the nonlinear shallow-water theory under the approximation that the height of the initial wave is small compared to the basin depth, and the reflection from the inflection point of the bottom is negligibly small. Three types of incident waves, viz., a sinusoidal wave and solitary waves of positive and negative polarities, are considered in detail. It is shown that a sinusoidal wave undergoes nonlinear deformation at a segment of constant depth faster than solitary waves of positive and negative polarities. Solitary waves of negative polarity steepen somewhat faster than solitary waves of positive polarity. Waves of positive polarity steepen at wave front, while waves of negative polarity steepen at wave rear. These differences in steepness may become crucial at the wave run-up stage, since the wave run-up height on the coast of a bay with a parabolic cross-section is directly proportional to the steepness of a wave that arrives at the slope and can lead to the anomalous run-up of waves on the coast.     

Observationally constrained modeling of sound in curved ocean internal waves: examination of deep ducting and surface ducting at short range

A study of 400 Hz sound focusing and ducting effects in a packet of curved nonlinear internal waves in shallow water is presented. Sound propagation roughly along the crests of the waves is simulated with a three-dimensional parabolic equation computational code, and the results are compared to measured propagation along fixed 3 and 6 km source/receiver paths. The measurements were made on the shelf of the South China Sea northeast of Tung-Sha Island. Construction of the time-varying three-dimensional sound-speed fields used in the modeling simulations was guided by environmental data collected concurrently with the acoustic data. Computed three-dimensional propagation results compare well with field observations. The simulations allow identification of time-dependent sound forward scattering and ducting processes within the curved internal gravity waves. Strong acoustic intensity enhancement was observed during passage of high-amplitude nonlinear waves over the source/receiver paths, and is replicated in the model. The waves were typical of the region (35 m vertical displacement). Two types of ducting are found in the model, which occur asynchronously. One type is three-dimensional modal trapping in deep ducts within the wave crests (shallow thermocline zones). The second type is surface ducting within the wave troughs (deep thermocline zones).     

Conformal variables in the numerical simulations of long-crested rogue waves

The recently suggested theoretical model for highly nonlinear potential long-crested water waves is discussed, where weak three-dimensional effects are included as small corrections to exact two-dimensional equations written in terms of the conformal variables [V.P. Ruban, Phys. Rev. E 71, 055303(R) (2005)]. Some numerical results based on this theory are presented, which describe spontaneous formation of rogue waves on the deep water for different initial conditions. In particular, the given numerical examples describe: (i) nonlinear stage of the modulational instability, (ii) breathing rogue wave in a random wave field, and (iii) freak wave in a weakly crossing sea state.     

Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber

In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.     

New Patterns of the Two-Dimensional Rogue Waves:(2+1)-Dimensional Maccari System

The ocean rogue wave is one kind of puzzled destructive phenomenon that has not been understood thoroughly so far. The two-dimensional nature of this wave has inspired the vast endeavors on the recognizing new patterns of the rogue waves based on the dynamical equations with two-spatial variables and one-temporal variable,which is a very crucial step to prevent this disaster event at the earliest stage. Along this issue, we present twelve new patterns of the two-dimensional rogue waves, which are reduced from a rational and explicit formula of the solutions for a(2+1)-dimensional Maccari system. The extreme points(lines) of the first-order lumps(rogue waves) are discussed according to their analytical formulas. For the lower-order rogue waves, we show clearly in formula that parameter b_2 plays a significant role to control these patterns.     

Nonlinear stage of the Benjamin-Feir instability: three-dimensional coherent structures and rogue waves

A specific, genuinely three-dimensional mechanism of rogue wave formation, in a late stage of the modulational instability of a perturbed Stokes deep-water wave, is recognized through numerical experiments. The simulations are based on fully nonlinear equations describing weakly three-dimensional potential flows of an ideal fluid with a free surface in terms of conformal variables. Spontaneous formation of zigzag patterns for wave amplitude is observed in a nonlinear stage of the instability. If initial wave steepness is sufficiently high (ka>0.06), these coherent structures produce rogue waves. The most tall waves appear in turns of the zigzags. For ka<0.06, the structures decay typically without formation of steep waves.     

Predicting rogue waves

Observational data regarding anomalously high waves on the sea’s surface (freak or rogue waves) are reviewed. The objectives of the research are identified, and the difficulties encountered are noted. The main physical mechanisms employed in explaining rogue waves are listed, and possible approaches to predicting marine hazards are discussed. Principles for ongoing short-term forecasting of extreme waves (within tens of wave periods or wavelengths) are proposed. Some preliminary results are presented.     

Giant waves in weakly crossing sea states

The formation of rogue waves in sea states with two close spectral maxima near the wave vectors k ₀ ± Δk/2 in the Fourier plane is studied through numerical simulations using a completely nonlinear model for long-crested surface waves [24]. Depending on the angle θ between the vectors k ₀ and Δk, which specifies a typical orientation of the interference stripes in the physical plane, the emerging extreme waves have a different spatial structure. If θ ≲ arctan(1/√2), then typical giant waves are relatively long fragments of essentially two-dimensional ridges separated by wide valleys and composed of alternating oblique crests and troughs. For nearly perpendicular vectors k ₀ and Δk, the interference minima develop into coherent structures similar to the dark solitons of the defocusing nonlinear Schroedinger equation and a two-dimensional killer wave looks much like a one-dimensional giant wave bounded in the transverse direction by two such dark solitons.     

Rogue Wave with a Controllable Center of Nonlinear Schrödinger Equation

The rogue waves with a controllable center are reported for the nonlinear Schrödinger equation in terms of rational-like functions by using a direct method. The position of these solutions can be controlled by choosing different center parameters and this may describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, Bose-Einstein condensates respectively.     

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.     

Shallow water rogue wavetrains in nonlinear optical fibers

In addition to deep-water rogue waves which develop from the modulation instability of an optical CW, wave propagation in optical fibers may also produce shallow water rogue waves. These extreme wave events are generated in the modulationally stable normal dispersion regime. A suitable phase or frequency modulation of a CW laser leads to chirp-free and flat-top pulses or flaticons which exhibit a stable self-similar evolution. Upon collision, flaticons at different carrier frequencies, which may also occur in wavelength division multiplexed transmission systems, merge into a single, high-intensity, temporally and spatially localized rogue pulse.     

The effects of background fields on vector financial rogue wave pattern

We study the structure of financial rogue waves with variable stock volatility. We find that there are mainly three types of rogue wave patterns in the coupled system, namely eye-shaped, anti-eye-shaped, and four-petaled. The transition between them can be induced by relative wave vector and amplitude difference between the two background fields. The results will help us achieve a deep understanding of financial rogue waves and suggest a number of implications.     

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schrodinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.     

Rogue waves at low Benjamin-Feir indices: Numerical study of the role of nonlinearity

The nonlinear interaction between waves in incoherent sea states is weaker than their dispersion. In this situation, random space-time focusing is the main mechanism of the formation of rogue waves. The numerical simulation has indicated that nonlinearity becomes important at the final stage of focusing and can significantly change predictions of the so-called second-order theory concerning the parameters of rogue waves. The elongation of the crest of a rogue wave as compared to that predicted by the second-order theory is an important effect promoting the “weighting of the tails” of the distribution function of the vertical deviation of the free surfaces.