## Abstract

Rogue waves observed in the ocean and elsewhere are often modelled by certain solutions of the nonlinear Schrodinger equation, describing the modulational instability of a plane wave and the subsequent development of multi-phase nonlinear wavetrains. In this paper, we describe how integrability and application of the inverse scattering transform can be used to construct a class of explicit asymptotic solutions that describe this process. We discuss the universal mechanism of the onset of multi-phase nonlinear waves (rogue waves) through the sequence of successive multi-breather wavetrains. Some applications to ocean waves and laboratory experiments are presented.

## 1. Introduction

The term rogue (or freak) waves is commonly identified with unexpectedly large water waves in the ocean [1,2]. Several mechanisms have been invoked, including essentially linear processes such as directional focusing or topographic and current focusing, and nonlinear processes such as modulational instability with the consequent nonlinear focusing of energy. Importantly, the nonlinear processes are often based on appropriate solutions of the focusing nonlinear Schrodinger (NLS) equation. These, and other proposed mechanisms, can be realized on other physical systems, leading to the speculation that rogue waves can occur there as well, see Akhmediev & Pelinovsky [3] and the articles that follow. In particular, the various breather solutions of the NLS equation have been invoked as models for rogue waves, with a special interest in the Peregrine breather, since this is spatially and temporally localized [4,5]. While there have been many numerical solutions of the NLS equation demonstrating that modulational instability of a plane wave can lead to rogue wave formation, there are relatively few works exploiting the analytical integrability of the NLS equation to analyse this process (e.g. [5] and references therein). In this paper, we show how the inverse scattering transform can be used to construct a class of explicit asymptotic solutions that demonstrate the development of a family of Peregrine breathers from a slowly modulated plane wave. Although we are not aware of any oceanic observations that demonstrate the development of a rogue wave from the modulation of a background wave, there are several laboratory experiments that examine this process, notably those by Osborne [2] and Onorato *et al.* [6–8]. Our theoretical results are then compared with observations from these experiments. We also note that the laboratory experiments of Chabchoub *et al.* [9,10] have demonstrated the presence of the Peregrine breather, and even higher order breathers in a wave tank.

The NLS equation for deep water waves is, in dimensional coordinates, see [2] for instance,
1.1Here, *A*(*x*,*t*) is the complex wave amplitude, defined so that at leading order in a weakly nonlinear asymptotic expansion, the surface elevation is given by
1.2Here, c.c. denotes the complex conjugate, and *c*_{g}=*g*/2*ω* is the group velocity. Note that *H*=4|*A*| is the crest–trough height.

The dominant terms in (1.1) are the first two terms. It is then convenient for the present purposes to cast this into the asymptotically equivalent form
1.3This is sometimes known as the ‘time-NLS’ equation [2]. Then, we impose the boundary condition, appropriate for a laboratory experiment, for instance,
1.4where *r*_{0}(*τ*) is a real-valued amplitude and *ϕ*_{0}(*τ*) is a real-valued phase. Note that at *x*=0, *τ*=−*c*_{g}*t*. Our aim in this paper is to estimate quantitatively how modulations in these boundary conditions can lead to the formation of rogue waves, as well as to describe the formation of the rogue waves from an analytical point of view.

In §2, we formulate the problem ((1.3), (1.4)) in scaled coordinates, revealing two small parameters, *α* and *ϵ*, measuring wave amplitude and modulation scale, respectively. We then discuss a universal mechanism for the onset of rogue waves within the NLS model. In §3, we present a particular class of boundary conditions (1.4) that demonstrate explicitly the development of a modulated plane wave into a so-called gradient collapse, where the spatial gradients are strongly localized, followed by the formation of a family of Peregrine breathers, see figures 1 and 2. The explicit formulae are applied to oceanic and laboratory situations. We conclude in §4. The appendix contains a summary of how the explicit expressions are obtained.

## 2. Formulation and mathematical background

First, the NLS equation (1.3) is rescaled to a canonical form. Thus, set
2.1where asterisk denotes the complex conjugate. Then, (1.1) takes the form, in non-dimensional variables,
2.2Note that here, *α* and *ϵ* are two dimensionless free parameters, representing wave steepness and modulation scale, respectively. In the asymptotic theory that we exploit here, it is assumed that 0<*ϵ*≪1, and we recall that 0<*α*≪1 is required for the validity of the NLS equation (1.1). The initial condition (1.4) becomes
2.3
where
2.4Thus, the initial condition corresponds to an amplitude of *O*(*α*), that is slowly modulated, on a scale *O*(1/*kϵα*) and a phase that is also slowly modulated, but on a scale of *O*(1/*kα*), in the original dimensional variables. It will be further assumed that *R*_{0}(*X*)>0,*Φ*_{0}(*X*) are real-valued analytic functions and that *R*_{0}(*X*) has a sufficient decay as .

Up to the point of the gradient catastrophe, that is, in the modulated plane-wave region, seek a solution of the form
2.5which leads to the system
2.6and
2.7Then, omitting the term proportional to *ϵ*^{2} yields the nonlinear elliptic system
2.8and
2.9The solution generically develops a singularity at some finite point *T*=*T*_{0},*X*=*X*_{0}, the point of gradient catastrophe. This point can be obtained as a solution of a system of transcendental equations written in terms of *R*_{0} and *Φ*_{0}, subject to some further ‘technical restraints’. In particular, we require that has finite limits at infinity,
2.10where approaches the limits *ν*_{±} sufficiently fast as . In the asymptotic limit *ϵ*→0, the wave evolution before and beyond the point of gradient collapse can be determined using a nonlinear steepest descent method applied to a Riemann–Hilbert version of the inversion scattering transform. The details can be found, for example, in Tovbis *et al.* [11] and are summarized in appendix A.

Numerical simulations of the solution of the focusing NLS (2.2) with the initial data *Q*(*X*,0)=sech *X* and are presented in figures 1 and 2. As shown in the figures, the *X*- and *T*-planes are subdivided into the regions where the solution is asymptotically (as ) described by either modulated plane waves (smooth region) or by two-phase nonlinear waves, which can be expressed in terms of the Riemann theta function (oscillatory region).

The key features of the wave evolution, shown in figures 1 and 2, can be extended to generic decaying analytical one-hump initial data. Here are some highlights, see [12],

— Regions of different asymptotic regimes are separated by

*breaking curves*or*nonlinear caustics*in the*X*- and*T*-planes that do not depend on*ϵ*. Equations for these breaking curves are given by (A8) in the appendix. A detailed description of the transitional behaviour at the breaking curve can be found in Bertola & Tovbis [13].— The tip of the breaking curve (

*X*_{0},*T*_{0}) is the point of gradient catastrophe for the plane-wave approximation. Immediately behind this tip, the solution suddenly bursts into rapid-amplitude oscillations (spikes). Each spike within the vicinity of (*X*_{0},*T*_{0}) has the height |*Q*(*X*_{0},*T*_{0})|(3+*O*(*ϵ*^{1/5})) and the shape of a scaled Peregrine breather (an explicit rational solution to the NLS). Here, |*Q*(*X*_{0},*T*_{0})| is the height of the ‘background’ waves at the point of the gradient catastrophe, which can also be found in terms of*R*_{0}and*Φ*_{0}.— The centres of the spikes correspond to the poles of the special

*tritronquée*solution to the first Painlevé equation*y*′′(*v*)=6*y*^{2}(*v*)−*v*through the near-linear map between an*O*(*ϵ*^{4/5}) neighbourhood of (*X*_{0},*T*_{0}) and a bounded disc of the complex*v*-plane, given by 2.11where and the non-zero constant*C*is explicitly defined by (A10).— The family of initial data (3.1) with

*ν*≥0 may (*ν*∈[0,2)) or may not (*ν*≥2) contain an ensemble of*O*(1/*ϵ*) stationary solitons centred at*X*=0, whose interaction will lead to more complicated behaviour for larger*T*(beyond the values shown in figures 1 and 2. The slopes of the breaking curve for this family are near the tip and ±1/2*ν*as [14].

The most striking feature of the microlocal analysis around the point of gradient catastrophe [12] is the *universal mechanism of transition from modulated plane wave to a two-phase nonlinear wave solution at the point of the gradient catastrophe.* This universal mechanism includes the appearance of successive wavetrains of *n* Peregrine breathers, starting with *n*=1. These wavetrains seam into the two-phase nonlinear wave, given by the Riemann theta functions, as . The location of the centres of the breathers are governed by the poles of the *tritronquée* solution to the first Painlevé equation. We note that recently Erkintalo *et al.* [15] described a similar-looking structure of successive breathers using a Darboux transformation to generate a higher order breather solution, see their figs 2 and 3. In §3, we present details of some specific cases based on the analysis presented here.

## 3. Applications

We consider a particular family of boundary data,
3.1The NLS evolution of this data was analytically studied in [14], while the particular case *ν*=0 (that is, no phase modulation) and *ϵ*=1 (that is, not in the semi-classical limit considered in this paper) was studied by Satsuma & Yajima [16], using the inverse scattering transform.

Using (2.1), this corresponds to
3.2Since *ϵ*,*α*, albeit 0<*ϵ*,*α*≪1, remain free parameters, this example illustrates the general rule that a wave with steepness *O*(*α*), modulated on a length scale of , with a phase of order 1/*ϵ* modulated on the same length scale, produces a rogue wave after a distance of order 1/*kϵα*^{2}(*ν* + 2). Here, *ν* is a second free parameter, measuring the magnitude of the phase modulation. Note that the Benjamin–Feir index (BFI) is defined as , where *K* is the modulation wavenumber [2]. Here, we interpret , so that BFI=1/*ϵ*. Thus, the BFI depends only on *ϵ*, and as *ϵ*→0, confirming that many rogue waves will be generated. Note that because the present theory is for an infinite spatial domain, the definition of a modulation wavenumber is not unique; with this present choice, *r*_{0}(*τ*=2*π*/*K*) is reduced by a factor of 0.09.

First, we examine the implication of these formulae in an oceanic setting. For instance, consider a time series at *x*=0 of a periodic wave with amplitude 4 m with a period *P*=2*π*/*ω*=12 s and hence a wavelength of *Λ*=2*π*/*k*=225 m; this corresponds to *α*=0.11. Suppose that this is then modulated on a length scale of 6*Λ*, that is, ; this corresponds to *ϵ*=0.54 and BFI=1.87. The modulation is in the variable *τ*, which corresponds to a modulation time scale of 2*π*/*Kc*_{g}=12*P*=144 s. This then produces rogue waves after a distance *x*_{c}=12.3(*ν*+2)^{−1} km. Then, the height at the point of the gradient catastrophe is *A*_{c}=8(*ν*+2)^{1/2}*m*, and hence the generated Peregrine breathers have a height 3*A*_{c}=24(*ν*+2)^{1/2}*m*. Thus, as *ν* increases, that is, the magnitude of the phase modulation *ϕ*_{0}(*x*) is increased, the distance to the gradient collapse is decreased, while the amplitude of the generated Peregrine breathers increases. On the other hand, as *ν*→−2, the time of the gradient collapse increases to infinity, and the amplitude of the generated Peregrine breathers decreases to zero. When *ν*=0, so that only the initial amplitude is modulated, *x*_{c}=6.15 km and the height of the generated Peregrine breathers is 3*A*_{c}=34 m. Since a commonly accepted criterion for rogue waves is that they exceed the background waves by a factor of around 3, this would imply that (*ν*+2)^{1/2}>1, that is, *ν*>−1 is sufficient.

However, as noted in §1, since we are not aware of any oceanic observations that directly connect a rogue wave with a prior modulation, we turn next to the laboratory experiments reported by Osborne [2] and Onorato *et al.* [6–8]. These were conducted in the Marintek wave flume, which is 270 m long, with a programmable wavemaker. The four experiments reported had the parameters as set out in table 1.

Here, the first line is the experiment reported by Osborne [2], in which a carrier wave with a period of 1.3 s was modulated with the stated parameters *A*_{0},*α*,*K*/*k* and BFI. The remaining parameters *ϵ*,*x*_{0},*A*_{c} in the table are deduced using our formulae as above, with *ν*=0, as there was no phase modulation in these experiments. The remaining three lines are from experiments reported by Onorato *et al.* [6–8], in which the wave maker was driven with the Jonswap spectrum, with a peak period of 1.5 s. Here, 2*A*_{0} is the significant wave height in the experiment, and *K*/*k* measures the spectral bandwidth. Significantly, our parameter *ϵ* is not small for any of these experiments. Nevertheless, the computed values of *x*_{c} and 3*A*_{c} are in reasonably good qualitative agreement with the laboratory experiments, see fig. 29.2 in Osborne [2] and fig. 1 in Onorato *et al.* [7].

## 4. Discussion

Since oceanic rogue waves are often modelled by the NLS equation, in this paper, we have described how its integrability, and application of the inverse scattering transform, can be used to construct a class of explicit asymptotic solutions that describe how a modulated plane wave will collapse at a point, the gradient catastrophe, and then develop into a nonlinear wavetrain, characterized by the successive formation of families of Peregrine breathers. In particular, we have shown that for a special class of initial conditions describing a modulated plane wave, we can provide explicit formulae for the point of gradient collapse, the height of the Peregrine breathers, the location of the nonlinear wavetrain, and many other features of the solution shown in figures 1 and 2. These expressions contain two parameters, one, *α*, measuring wave amplitude, and the other, *ν*, is a measure of the strength of the phase modulation. Thus, we can describe a large family of modulated plane waves. In the text, we have presented an example of an oceanic application, and a comparison with the laboratory experiments of Osborne [2] and Onorato *et al.* [6–8], to indicate the potential to make precise quantitative predictions in practical situations. However, it needs to be noted that the present results are limited to non-breaking small-amplitude waves, and to one horizontal spatial dimension.

## Acknowledgements

We would like to thank Marco Bertola for providing the figures shown in this paper.

## Appendix A. Detailed analyses near the point of gradient catastrophe

It is well known that the inverse scattering transform method allows one to construct the NLS evolution for given initial data. Let denote the direct scattering transform for equation (2.2) and , where is the scattering data, with *z* being a spectral parameter. Since the time evolution of is simple and known explicitly, the solution *Q*=*Q*(*X*,*T*) to (2.2) is given by .

Consider analytic, rapidly decaying (as ) modulated plane-wave initial data *Q*_{0}, given by (2.3). To simplify our considerations, assume that *Q*_{0} is purely radiational (no solitons) initial data. The case when solitons are present can also be incorporated within the presented framework. Then, the scattering data for *Q*_{0} consists of the reflection coefficient *ρ*=*ρ*(*z*,*ϵ*) only, which can be represented as
A1We will call the approximate reflection coefficient. Under our assumptions, and *f*_{0} can be analytically continued into the upper complex half-plane . In the lower half-plane , *f*_{0} is defined by the Schwarz symmetry .

It is well known that the inverse scattering problem can be cast as a (multiplicative) matrix Riemann–Hilbert problem (RHP) with the jump on , whose jump matrix is given in terms of *ρ*(*z*,*ϵ*). Let us consider the approximate reflection coefficient *ρ*_{0}(*z*,*ϵ*). Using the nonlinear steepest descent method for asymptotic (small *ϵ* limit) solution of this RHP, one can obtain the system of equations
A2for the unknown branchpoint *β*=*β*(*X*,*T*)=*a*(*X*,*T*)+i*b*(*X*,*T*), where is normalized by
A3and denotes a negatively oriented loop around a simple Schwarz-symmetrical arc , which intersect only at *z*=*ν*_{+}. The branchcut of *P* coincides with *γ*_{m}. Here, we assumed that
A4

The system (A2) is known as a system of modulation equations (or implicit solutions to the Whitham equations) that defines the modulated plane-wave representation (2.5),
for the solution of the Cauchy problem (2.2) and (2.3). Indeed, the system (A2) can be viewed as an implicit solution of the Cauchy problem for (2.6), with
A5The corresponding modulated plane wave approximates the evolution of the initial data with accuracy *O*(*ϵ*) in the modulated plane-wave (genus zero) region, i.e. until the leading order approximation undergoes a phase transition. According to the nonlinear steepest descent method, the *O*(*ϵ*) accuracy is uniform on compact subsets of the genus zero region.

Although the (highly nonlinear) scattering transform is (with a few exceptions) not constructive, the small *ϵ* limit of is, in fact, given by a ‘linear’ integral transform. Assume, for simplicity, that *Φ*′_{0}(*X*) is a monotonic function. Then, the graph of is a simple analytic curve with the endpoints *ν*_{−} and *ν*_{+}. Let *x*(*β*) denote the inverse function to *β*(*X*) on *Σ*_{0}. Then, the small *ϵ* limit of is given by Tovbis & Venakides [17],
A6Because of the analyticity of *Q*_{0}(*X*) and, thus, of *x*(*u*), the function *f*_{0}(*z*) can be analytically continued from *Σ*_{0} into .

The time evolution *Σ*_{T} of the curve *Σ*_{0} is defined by the second modulation equation (A2), whereas its *X*-parametrization, defined by the first equation (A2), yields the functions *R*(*X*,*T*) and *Φ*′(*X*,*T*). (Note that the integral transform (A6) of *x*(*β*) along *Σ*_{0}, *β*∈*Σ*_{0}, is inverse to the transform for *X*=*X*(*β*) given by the first equation in (A2).) The point of gradient catastrophe is a special branchpoint *β*_{0} in the spectral plane, as well as the corresponding values *X*_{0},*T*_{0} in the space–time plane, where the curve *Σ*_{T} develops a singularity (casp). The point *β*_{0} is defined by a system of two (real) equations [11],
A7whereas the corresponding *X*_{0} and *T*_{0} are defined by (A2) with *β*=*β*_{0}. The amplitude *R*(*X*_{0},*T*_{0})=*b*_{0}.

Once the point of gradient catastrophe *X*_{0},*T*_{0} is defined, the (two) branches of the breaking curve, emanating from *X*_{0},*T*_{0}, are given by
A8where *z* is a point on the spectral plane (emanating from *β*_{0} as we move along the breaking curve),
A9and the point *z* is inside the loop *γ*_{m}. Note that (A8) is the system of three real equations for four (*X*,*T*,*z*,*z*) real unknowns. The constant *C*, mentioned in (2.11), is given by
A10

Finally, consider the following illustrative example. Assume that *w*(*z*)=sign (*ν*_{+}−*z*)*f*_{0}(*z*), , is a continuous piece-wise linear function with *w*(*z*)>0 on (*ν*_{−},*ν*_{+}) and *w*(*z*)<0 outside [*ν*_{−},*ν*_{+}]. For our purposes, it is convenient to consider the derivative
A11where *v*_{0}>0, and *H*(*z*) denotes the Heaviside function. We also assume that all *μ*_{j}∈(*ν*_{−},*ν*_{+}) (the reflection coefficient *ρ*_{0}(*z*,*ϵ*) is exponentially small outside (*ν*_{−},*ν*_{+}), so that the jumps outside (*ν*_{−},*ν*_{+}) have a very small impact on the problem). Then, , where *δ* is the Dirac delta function, and equations (A7) for the point of gradient *β*_{c} catastrophe become
A12where denote the branch of the radical that is positive on . The corresponding values of *X*_{c},*T*_{c} can be obtained from
A13where we substitute *β*=*β*_{c}. The corresponding initial data can be obtained by solving the system (A13) with *T*=0 for *β*.

The particular case , *N*=1, *μ*_{1}=0 and *ν*_{±}=±1 was studied in Tovbis *et al.* [14]. They obtained *R*_{0}=sech *X*, and (see (3.1) with *ν*=2) *β*_{c}=2*i*, . The height at the point of the gradient catastrophe is 2, and so the height of the spikes is 6.

- Received February 11, 2013.
- Accepted June 4, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.