Peregrine Breather Solution Research Articles

Peregrine breathers as design waves for wave-structure interaction

This paper introduces the Peregrine breather solution of the nonlinear Schrödinger-type equation as an innovative design wave. The major benefits are the potential to generate abnormal waves of certain frequency up to physically possible wave heights, the symmetrical abnormal wave shape and the availability of an analytical solution. The characteristics of Peregrine breathers as well as their impact on marine structures were investigated. To evaluate the applicability of the Peregrine breather solution as design wave, wave-structure investigations with a LNG carrier were performed in a set of Peregrine breathers at certain frequencies. The investigations comprised model tests as well as numerical simulations. For the purpose of comparison, a real-world abnormal wave reproduction was additionally investigated allowing the evaluation of breather-type abnormal waves regarding the applicability as design wave. Therefore, the famous ‘New Year Wave’, an abnormal wave recorded in the North Sea at the Draupner jacket platform on January 1st, 1995, was selected.

Fourth-order split-step pseudo-spectral method for the modified nonlinear Schrödinger equation

In the field of water waves, the modified nonlinear Schrödinger (mNLS) equation which models the wave propagation in water is numerically solved by using the split-step pseudo-spectral method. In the present paper, the fourth-order split-step pseudo-spectral method is introduced with better numerical results. The proposed method is based on a split-step method which decomposes the original equation into two parts, a linear problem and a nonlinear problem. In order to demonstrate the high accuracy and capability of the newly proposed method, a simple problem of periodic waves is presented to compare the traditional first-order method with the fourth-order method in terms of the conservation error and computational cost. Meanwhile, another numerical experiment concerning the Peregrine breather solution of the nonlinear Schrödinger (NLS) equation is presented by using the fourth-order split-step pseudo-spectral method. It is found that the fourth-order scheme can provide higher conservative accuracy and is computationally more efficient compared with the first-order scheme.

Twenty Two Parameter Deformations of the Twelfth Peregrine Breather Solutions to the NLS Equation

The twelfth Peregrine breather (P12 breather) solution to the focusing one dimensional nonlinear Schrodinger equation (NLS) with its twenty two real parameters deformations solutions to the NLS equation are explicitly constructed here. New families of quasi-rational solutions of the NLS equation in terms of explicit quotients of polynomials of degree 156 in x and t by a product of an exponential depending on t are obtained. The patterns of the modulus of these solutions in the $(x; t)$ plane, in function of the different parameters are studied in details.

Numerical study of nonlinear Peregrine breather under finite water depth

Though the focusing method can effectively generate waves which satisfy the definition of rogue waves at a specific position and moment, however, being inherently not nonlinear, the focusing model is still a controversial rogue wave generation method. Recently, nonlinear models are becoming more popular for rogue wave generation in physical and numerical tanks. In this paper, a weakly nonlinear model known as the Peregrine breather solution of the cubic Schrödinger equation is studied under finite water depth. In contrast with the focusing model, nonlinearity is considered throughout the simulation process, i.e., from the wave model to the generated waves. The numerical results are validated against theoretical solutions as well as experimental measurements. To further investigate their temporal-frequency characteristics, a wavelet analysis is performed on the generated Peregrine breather, and the concepts of life time and traveling distance are studied. The influence of higher order nonlinearity, i.e., the 2nd-order Stokes component of the perturbed expansion under finite water depth, is taken into account. We also discuss the influence of 2nd-order term on the life time, travel distance, and energy distribution.

Experimental investigation on rogue waves and their impacts on a vertical cylinder using the Peregrine breather model

This paper presents an experimental investigation on the rogue waves and the corresponding wave forces acting on a vertical cylinder. The waves are modelled using the Peregrine breather solution of the third-order nonlinear Schrödinger equation. The experimental wave elevations are compared with the theoretical solutions, and the behaviours of wave energy distribution and evolution are calculated using the fast Fourier transformation and wavelet transform (WT) methods. The resulting wave forces acting on the vertical cylinder are compared with the numerical results based on potential flow calculations. Moreover, coherence analyses are conducted for the rogue waves and their impact forces acting on the cylinder based on the WT.

A combined wave-dam-breaking model for rogue wave overtopping

Although great efforts have been devoted to revealing the formation mechanisms of rogue waves, the study of wave–structure interaction such as the green water phenomenon is still not sufficient. Since the traditional Ritter׳s solution of dam-breaking model gives improper estimations on the rogue wave overtopping, a combined wave-dam-breaking (CWDB) model is proposed. The CWDB model is deduced from the Ritter׳s solution, by including the influence of the rogue wave propagation as well as the delayed effect of the dam-breaking problem. Various rogue waves, which are based on the Peregrine breather solution of nonlinear Schrödinger (NLS) equation, are generated in a numerical wave flume. The rogue waves impinge onto a horizontal deck placed at the center part of the tank, creating green water on the deck surface. The CWDB model is validated by making comparisons with both numerical results and existing experimental measurements. Great improvement can be revealed by contrasting the results of the CWDB model with those of the Ritter׳s solution.

Tenth Peregrine breather solution to the NLS equation

In this paper we construct a particularly important solution to the focusing NLS equation, namely a Peregrine breather of the rank 10 which we call, P10 breather. The related explicit formula is given by the ratio of two polynomials of degree 110 with integer coefficients times trivial exponential factor. This formula drastically simplifies for the “initial values” namely for t=0 or x=0. This formula confirms a general conjecture saying that between all quasi-rational solutions of the rank N fixed by the condition that its absolute value tends to 1 at infinity and its highest maximum is located at the point (x=0,t=0), the PN breather is distinguished by the fact that PN(0,0)=2N+1 and, in the aforementioned class of quasi-rational solutions, it is an absolute maximum. At the end we also make a few remarks concerning the rational deformations of P10 breather involving 2N−2 free real parameters chosen in a way that PN breather itself corresponds to the zero values of these parameters although we have no intention to discuss the properties of these deformations here.

Ten-parameter deformations of the sixth-order Peregrine breather solutions of the NLS equation

In this paper, we construct new deformations of the Peregrine breather of order 6 with ten real parameters. We obtain new families of quasi-rational solutions of the one-dimensional focusing nonlinear Schrödinger (NLS) equation. With this method, we construct new patterns of different types of rogue waves. We get, as already found for the lower order, the triangular configurations and rings isolated. Moreover, one sees for certain values of the parameters the appearance of new configurations of concentric rings.

Erratum: Deformations of third-order Peregrine breather solutions of the nonlinear Schrödinger equation with four parameters [Phys. Rev. E88, 042903 (2013)

Received 5 November 2013DOI:https://doi.org/10.1103/PhysRevE.88.069901©2013 American Physical Society

Deformations of third-order Peregrine breather solutions of the nonlinear Schrödinger equation with four parameters

We present a new representation of solutions of the one-dimensional nonlinear focusing Schrödinger equation (NLS) as a quotient of two determinants. This formulation gives in the case of the order 3, new solutions with four parameters. This gives a very efficient procedure to construct families of quasirational solutions of the NLS equation and to describe the apparition of multirogue waves. With this method, we construct analytical expressions of four-parameters solutions; when all these parameters are equal to 0, we recover the Peregrine breather of order 3. It makes possible with this four-parameters representation, to generate all the types of patterns for the solutions, like the triangular configurations or the ring structures.

Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model

Being considered as a prototype for description of oceanic rogue waves, the Peregrine breather solution of the nonlinear Schrödinger equation has been recently observed and intensely investigated experimentally in particular within the context of water waves. Here, we report the experimental results showing the evolution of the Peregrine solution in the presence of wind forcing in the direction of wave propagation. The results show the persistence of the breather evolution dynamics even in the presence of strong wind and chaotic wave field generated by it. Furthermore, we have shown that characteristic spectrum of the Peregrine breather persists even at the highest values of the generated wind velocities thus making it a viable characteristic for prediction of rogue waves.

Six-parameters deformations of fourth order Peregrine breather solutions of the nonlinear Schrödinger equation

We construct solutions of the focusing nonlinear Schrödinger (NLS) equation as a quotient of two determinants. This formulation gives in the case of the order 4, new deformations of the Peregrine breather with 6 real parameters. We construct families of quasi-rational solutions of the NLS equation and describe the apparition of multi-rogue waves. With this method, we construct the analytical expressions of deformations of the Peregrine breather of order 4 with 6 real parameters and plot different types of rogue waves.

Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test

Under suitable assumptions, the nonlinear dynamics of surface gravity waves can be modeled by the one-dimensional nonlinear Schrödinger equation. Besides traveling wave solutions like solitons, this model admits also breather solutions that are now considered as prototypes of rogue waves in ocean. We propose a novel technique to study the interaction between waves and ships/structures during extreme ocean conditions using such breather solutions. In particular, we discuss a state of the art sea-keeping test in a 90-meter long wave tank by creating a Peregrine breather solution hitting a scaled chemical tanker and we discuss its potential devastating effects on the ship.

Two Parameters Deformations of Ninth Peregrine Breather Solution of the NLS Equation and Multi-Rogue Waves

This paper is a continuation of a recent paper on the solutions of the focusing NLS equation. The representation in terms of a quotient of two determinants gives a very efficient method of determination of famous Peregrine breathers and their deformations. Here we construct Peregrine breathers of order and multi-rogue waves associated by deformation of parameters. The analytical expression corresponding to Peregrine breather is completely given.

Experimental study of spatiotemporally localized surface gravity water waves

We present experimental results on the study of spatiotemporally localized surface wave events on deep water that can be modeled using the Peregrine breather solution of the nonlinear Schrödinger equation. These are often considered as prototypes of oceanic rogue waves that can focus wave energy into a single wave packet. For small steepness values of the carrier gravity waves the Peregrine breathers are relatively wide, thus providing an excellent agreement between the theory and experimental results. For larger steepnesses the focusing leads to temporally and spatially shorter events. Nevertheless, agreement between measurements and the Peregrine breather theory remains reasonably good, with discrepancies of modulation gradients and spatiotemporal symmetries being tolerable. Lifetimes and travel distances of the spatiotemporally localized wave events determined from the experiment are in good agreement with the theory.