Rogue waves and downshifting in the presence of damping

Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011www.nat-hazards-earth-syst-sci.net/11/383/2011/doi:10.5194/nhess-11-383-2011© Author(s) 2011. CC Attribution 3.0 License.

Natural Hazardsand Earth

System Sciences

Rogue waves and downshifting in the presence of damping

A. Islas and C. M. Schober

Department of Mathematics, University of Central Florida, Orlando, FL, USA

Received: 2 October 2010 – Revised: 6 December 2010 – Accepted: 13 December 2010 – Published: 9 February 2011

Abstract. Recently Gramstad and Trulsen derived anew higher order nonlinear Schrodinger (HONLS) equationwhich is Hamiltonian (Gramstad and Trulsen, 2011). Weinvestigate the effects of dissipation on the developmentof rogue waves and downshifting by adding an additonalnonlinear damping term and a uniform linear dampingterm to this new HONLS equation. We find irreversibledownshifting occurs when the nonlinear damping is thedominant damping effect. In particular, when only nonlineardamping is present, permanent downshifting occurs for allvalues of the nonlinear damping parameterβ. Significantly,rogue waves do not develop after the downshifting becomespermanent. Thus in our experiments permanent downshiftingserves as an indicator that damping is sufficient to preventthe further development of rogue waves. We examine thegeneration of rogue waves in the presence of damping forsea states characterized by JONSWAP spectrum. Using theinverse spectral theory of the NLS equation, simulations ofthe NLS and damped HONLS equations using JONSWAPinitial data consistently show that rogue wave events are wellpredicted by proximity to homoclinic data, as measured bythe spectral splitting distanceδ. We defineδcutoff by requiringthat 95% of the rogue waves occur forδ < δcutoff. We findthatδcutoff decreases as the strength of the damping increases,indicating that for stronger damping the JONSWAP initialdata must be closer to homoclinic data for rogue waves tooccur. As a result when damping is present the proximityto homoclinic data and instabilities is more crucial for thedevelopment of rogue waves.

Correspondence to:C. M. Schober([email protected])

1 Introduction

Some theoretical models attribute the development ofrogue waves in deep water to a nonlinear focusing ofuncorrelated waves in a very localized region of thesea. This focusing may be related to the Benjamin-Feir (BF) instability, described to leading order by thenonlinear Schrodinger (NLS) equation (Henderson et al.,1999; Kharif and Pelinovsky, 2001, 2004). Homoclinicorbits of modulationally unstable solutions of the NLSequation which undergo large amplitude excursions awayfrom their target solution exhibit many of the observedproperties of rogue waves and can be used to model roguewaves (Osborne et al., 2000; Calini and Schober, 2002).

A more accurate description of the water wave dynamicsis obtained by retaining higher order terms in the asymptoticexpansion for the surface wave displacement. A commonlyused higher order NLS equation is the Dysthe equation(Dysthe, 1979). Homoclinic orbits of the unstable Stokeswave have been shown to be robust to the higher ordercorrections in the Dysthe equation. Laboratory experimentsconducted in conjunction with numerical simulations ofthe Dysthe equation established that the generic long-timeevolution of initial data near an unstable Stokes wavewith two or more unstable modes is chaotic (Ablowitzet al., 2000, 2001). Subsequent studies of the Dystheequation showed that for a rather general class of suchinitial data, the modulational instability leads to highamplitude waves, structurally similar to the optimal phasemodulated homoclinic solutions of the NLS equation, risingintermittently above a chaotic background (Calini andSchober, 2002; Schober, 2006). These earlier studies relatinghomoclinic solutions and rogue waves ignored the factthat the Dysthe equation is not Hamiltonian and neglecteddamping which, even when weak, can have a significanteffect on the wave dynamics.

Published by Copernicus Publications on behalf of the European Geosciences Union.

http://creativecommons.org/licenses/by/3.0/

384 A. Islas and C. M. Schober: Rogue waves and downshifting

Recently Gramstad and Trulsen (2011) brought the Dystheequation into Hamiltonian form obtaining a new higher ordernonlinear Schrodinger (HONLS) equation (Gramstad andTrulsen, 2011). In this paper we investigate the effectsof dissipation on the development of rogue waves anddownshifting in one space dimension by adding a nonlineardamping term (theβ-term which damps the mean flowKatoand Oikawa, 1995) and a uniform linear damping term to thisnew HONLS equation. The governing equations and theirproperties are set up in Sect. 2.

Downshifting in the evolution of nearly uniform planewaves was first observed byLake et al. (1977). Theirexperiment showed that after the wavetrain becomes stronglymodulated, it recurs as a nearly uniform wavetrain with thedominant frequency permanently downshifted. Permanentdownshifting has been confirmed in other laboratoryexperiments on the evolution of the Stokes wave (Su,1982; Huang et al., 1996). Conservative models have beeninadequate in describing permanent downshifting and severaldamped models have been suggested, e.g. where the dampingis modeling wave breaking or eddy viscosity to capturedownshifting (Trulsen and Dysthe, 1990; Hara and Mei,1991, 1994).

Our previous studies of rogue waves show that anε-neighborhood of the unstable plane wave (the same regimein which Lake examined downshifting) is effectively a roguewave regime for the Dysthe equation since the likelihoodof obtaining a rogue wave is extremely high. We re-examine this issue in Sect. 3 using the new HONLS equation.In Sect. 4, using initial data for an unstable modulatedwavetrain we examine whether irreversible downshifting(although here downshifting of the wave number isconsidered) occurs in our damped HONLS equation andwhat characterizes the damped wave train evolution on ashort and long time scale.

We find that rogue waves may emerge in both the linearand nonlinear damped regimes that were not present withoutdamping. Although damping decreases the growth ratesof the individual modes, the modes may coalesce due tochanges in their focusing times, thus resulting in larger wavesin the damped regime. Even so, on average, the strengthis smaller and fewer rogue waves occur when damping ispresent. While downshifting is not expected in the linearlydamped case, we examine the effect of linear damping onthe development of rogue waves due to its widespread use inmodeling dissipative processes.

The nonlinearβ-term models damping of the meanflow and since it is large only near the crest oftheenvelope, the damping is localized when the wavetrain isstrongly modulated. Due to the BF instability, irreversibledownshifting occurs when the nonlinear damping is thedominant damping effect. In particular, when only nonlineardamping is present, permanent downshifting occurs for allvalues of the nonlinear damping parameterβ, appearingabruptly for larger values ofβ. Significantly, we find that

after permanent downshifting occurs, rogue waves do notappear in the nonlinearly damped evolution. Thus in ourexperiments permanent downshifting serves as an indicatorthat there has been sufficient cumulative damping to inhibitthe further development of rogue waves.

Developing sea states, where nonlinear wave-waveinteractions continue to occur, are described by the JointNorth Sea Wave Project (JONSWAP) power spectrum. Aspectral quantity, the splitting distanceδ between simpleperiodic points of the Floquet spectrum of the associatedAKNS spectral problem of the initial condition may beused to measure the proximity in spectral space to unstablewaves and homoclinic data of the NLS equation. In (Islasand Schober, 2005; Schober, 2006), simulations of boththe NLS and Dysthe equations using JONSWAP initial dataconsistently show that rogue wave events are well predictedby proximity to homoclinic data, as measured byδ.

In Sect. 5 we examine the generation of rogue wavesin the presence of damping for sea states characterized byJONSWAP spectrum. Using the damped HONLS equationwe find that both the strength and likelihood of rogue wavesoccuring in a given simulation are typically smaller whendamping is present. We defineδcutoff by requiring that 95%of the rogue waves occur forδ < δcutoff. Significantly, wefind that δcutoff is generally decreasing as the strength ofthe damping increases. Thus when damping is present theJONSWAP initial data must be closer to instabilities andhomoclinic data for rogue waves to occur. The proximity tohomoclinic data and instabilities becomes more essential forthe development of rogue waves when damping is present.

2 Analytical background

2.1 A new damped higher order nonlinear Schrodingerequation

The equations for inviscid deep water waves, as well asthe nonlinear Schrodinger (NLS) equation, can be derivedfrom a Lagrangian. Ideally, a higher order NLS equationwould retain this feature. The commonly used higherorder NLS equation (Dysthe, 1979), often referred toas Dysthe’s equation, is not Hamiltonian and does notconserve momentum. Recently, Gramstad and TRulsenused the Zakharov equation enhanced with the Krasitskiikernel (Krasitskii, 1994) to bring the Dysthe equation intoHamiltonian form, obtaining a new higher order nonlinearSchrodinger (HONLS) equation (Gramstad and Trulsen,2011).

In order to examine rogue waves and downshifting in deepwater we consider the HONLS equation of Gramstad andTrulsen with periodic boundary conditions,u(x,t) = u(x +

L,t), and add damping as follows:

Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011 www.nat-hazards-earth-syst-sci.net/11/383/2011/

A. Islas and C. M. Schober: Rogue waves and downshifting 385

iut + uxx +2|u|2u+ i0u

+ iε

(1

2uxxx −8|u|

2ux −2ui(1+ iβ)[H

(|u|

2)]

x

)= 0, (1)

where H(f ) represents the Hilbert transform off .Throughout this paper “HONLS equation” refers to (1) with0 = 0 andβ = 0. The Hamiltonian for the HONLS equationis given by

H =

∫ L

0

{−i|ux |

2+ i|u|

4−

ε

4

(uxu

∗xx −u∗

xuxx

)+ 2ε|u|

2(u∗ux −uu∗

x

)+ iε|u|

2[H

(|u|

2)]

x

}dx. (2)

Uniform linear damping occurs for0 > 0 andβ = 0 and hasbeen used extensively to study damping of wave trains withcomparisons to physcial wave-tank experiments (Segur et al.,2005). Localized nonlinear damping of the mean flow occursfor ε,β > 0 and0 = 0 and it was introduced to investigatedownshifting in deep water waves (Kato and Oikawa, 1995).

2.1.1 Energy, energy flux, and the spectral center

The mass or wave energy,E, and the momentum or totalenergy flux,P , are defined by

E =

∫ L

0|u|

2dx, and P = i

∫ L

0

(u∗ux −uu∗

x

)dx .

To show damping occurs for positiveβ or 0, one finds that

dE

dt= −2

∫ L

0|u|

2(0+2εβ

[H

(|u|

2)]

x

)dx .

Expandingu and|u|2 in Fourier series,u =

∑∞

k=−∞uke

ikx ,|u|

2=

∑∞

k=0

(Bke

ikx+B∗

k e−ikx),

dE

dt= −2L

[0

∞∑k=−∞

|uk|2+2εβ

∞∑k=1

k|Bk|2

]. (3)

Thus the total energy is conserved for the HONLS equation(0 = β = 0) and is dissipated if either0 or β is positive.

The total energy flux is related to the symmetry of theFourier modes sinceP can be written as

P = −2L

∞∑k=1

k(|u−k|

2−|uk|

2)

.

Since the evolution of the total energy flux is

dP

dt= −2i

∫ L

0

(u∗ux −uu∗

x

)(0+2εβ

[H

(|u|

2)]

x

)dx ,

(4)

the momentum is conserved by the HONLS equation. Incontrast the momentum for the Dysthe equation exhibitedsmall oscillations in time (Islas and Schober, 2010).

Using the Fourier spectrum ofu(x,t), uk, two differentchoices of diagnostic frequencies can be used to measuredownshifting. On the one hand, the dominant mode orspectral peak intuitively corresponds to thek for which |uk|

achieves its maximum and is denoted askpeak. On the otherhand, Uchiyama and Kawahara defined the wave number forthe spectral center or mean frequency of the spectrum as(Uchiyama and Kawahara, 1994):

km = −1

2

P

E. (5)

The wave train is understood to experience a permanentfrequency downshift when the spectral centerkm decreasesmonotonically in “time” or there is a permanent downshift ofthe spectral peakkpeak(Trulsen and Dysthe, 1997a).

Given that the energy and momentum are conserved bythe HONLS equation, the spectral center, which is given bytheir ratio, is also conserved. In the linearly damped model,0 > 0,β = 0, from Eqs. (3) and (4) it follows that the energyand momentum have a simple time dependence:

E(t) = E(0)exp(−20t) (6)

P(t) = P(0)exp(−20t) . (7)

Thus the spectral centerkm is constant in time for the linearlydamped HONLS equation and downshifting will not occur.

2.1.2 Linear stability of the Stokes wave

The HONLS equation admits a uniform wave train solution,the Stokes wave,

ua(x,t) = ae2ia2t . (8)

Fourier analysis shows that this plane wave solution isunstable to sideband perturbations. When its amplitudea

is sufficiently large, for 0< πn/L < a, the solution of thelinearized NLS equation aboutua hasM linearly unstablemodes (UMs)ei(σnt+2πnx/L) with growth ratesσn given by

σ 2n = µ2

n

(µ2

n −4a2), µn = 2πn/L , (9)

whereM is the largest integer satisfying 0< M < aL/π. Werefer to this as the M-unstable mode regime.

2.2 Integrable theory of the nonlinear Schrodingerequation

In previous studies of sea states described by the JONSWAPpower spectrum we showed a nonlinear spectral decom-position of the data provides relevant information on thelikelihood of rogue waves. Here we briefly review thespectral theory of the nonlinear Schrodinger (NLS) equation,

iut +uxx +2|u|2u = 0 . (10)

www.nat-hazards-earth-syst-sci.net/11/383/2011/ Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011


The NLS equation is equivalent to the solvability conditionof the AKNS system, the pair of first-order linear systems(Zakharov and Shabat, 1972):

L(x)φ = 0, L(t)φ = 0 , (11)

for a vector-valued functionφ. The operatorL(x) is given by

L(x)=

(∂x + iλ −u

u∗ ∂x − iλ

)and depends onx and t through the potentialu and on thespectral parameterλ.

The nonlinear spectral decomposition of an NLS initialcondition (or in general of an ensemble of JONSWAPinitial data) is based on the inverse spectral theory of theNLS equation. For periodic boundary conditionsu(x +

L,t)= u(x,t), the Floquet spectrum associated with an NLSpotentialu (i.e. the spectrum of the linear operatorL(x) atu)can be described in terms of the Floquet discriminant ofu,defined as the trace of the transfer matrix of a fundamentalmatrix solution8 of (11) over the interval [0,L] (Ablowitzand Segur, 1981):

1(u;λ) = Trace(8(x,t;λ)−18(x +L,t;λ)

).

Then, the Floquet spectrum is defined as the region

σ(u) = {λ ∈ IC|1(u;λ) ∈ IR,−2≤ 1 ≤ 2}.

Points of the continuous spectrum ofu are those for whichthe eigenvalues of the transfer matrix have unit modulus, andtherefore1(u;λ) is real and between 2 and−2; in particular,the real line is part of the continuous spectrum. Points ofthe L-periodic/antiperiodic discrete spectrum ofu are thosefor which the eigenvalues of the transfer matrix are±1,equivalently1(u;λ) = ±2. Points of the discrete spectrumwhich are embedded in a continuous band of spectrum arecritical points for the Floquet discriminant (i.e., d1/dλ mustvanish at such points).

Since the transfer matrix only changes by conjugationwhen we shift inx or t , 1 is independent of those variables.An important consequence of this observation is that theFloquet discriminant is invariant under the NLS flow, andthus encodes an infinite family of constants of motion(parametrized byλ).

The continuous part of Floquet spectrum of a generic NLSpotential consists of the real axis and of complex bandsterminating in simple pointsλs

j (at which 1 = ±2,1′6=

0). The N-phase potentials are those characterized by afinite number of bands of continuous spectrum (or a finitenumber of simple points). Figure1a shows the spectrum ofa typical N-phase potential: complex critical points (usuallydouble points of the discrete spectrum for which1′

= 0 and1′′

6= 0), such as the one appearing in the figure, are ingeneral associated with linear instabilities ofu and label itshomoclinic orbits (Ercolani et al., 1990).

(a)


L(x) =

(

∂x + iλ −uu∗ ∂x− iλ

)

and depends onx andt through the potentialu and on thespectral parameterλ.

The nonlinear spectral decomposition of an NLS initialcondition (or in general of an ensemble of JONSWAP initialdata) is based on the inverse spectral theory of the NLS equa-tion. For periodic boundary conditionsu(x+L,t) = u(x,t),the Floquet spectrum associated with an NLS potentialu (i.e.the spectrum of the linear operatorL(x) atu) can be describedin terms of the Floquet discriminant ofu, defined as the traceof the transfer matrix of a fundamental matrix solutionΦ of(11) over the interval[0,L] (Ablowitz & Segur, 1981):

∆(u;λ)= Trace(

Φ(x,t;λ)−1Φ(x+L,t;λ))

.


σ(u)= {λ∈ IC|∆(u;λ)∈ IR,−2≤∆≤ 2}.

Points of the continuous spectrum ofu are those for whichthe eigenvalues of the transfer matrix have unit modulus, andtherefore∆(u;λ) is real and between2 and−2; in particular,the real line is part of the continuous spectrum. Points of theL-periodic/antiperiodic discrete spectrum ofu are those forwhich the eigenvalues of the transfer matrix are±1, equiva-lently ∆(u;λ) =±2. Points of the discrete spectrum whichare embedded in a continuous band of spectrum are criticalpoints for the Floquet discriminant (i.e., d∆/dλ must vanishat such points).

Since the transfer matrix only changes by conjugationwhen we shift inx or t, ∆ is independent of those vari-ables. An important consequence of this observation is thatthe Floquet discriminant is invariant under the NLS flow,and thus encodes an infinite family of constants of motion(parametrized byλ).

The continuous part of Floquet spectrum of a generic NLSpotential consists of the real axis and of complex bands ter-minating in simple pointsλs

j (at which∆=±2,∆′ 6= 0). TheN -phase potentials are those characterized by a finite numberof bands of continuous spectrum (or a finite number of sim-ple points). Fig. 1a shows the spectrum of a typicalN -phasepotential: complex critical points (usually double pointsofthe discrete spectrum for which∆′ = 0 and∆” 6= 0), suchas the one appearing in the figure, are in general associatedwith linear instabilities ofu and label its homoclinic orbits(Ercolani et al., 1990).

As a concrete example consider the plane wave solution,ua(x,t) = ae2ia2t, whose Floquet discriminant is given by∆(a,λ)= 2cos(

√a2 +λ2L). The resulting Floquet spectrum

(Fig. 1b) consists of the continuous bandsIR⋃

[−ia,ia], anda discrete part containing the simple periodic/antiperiodic

eigenvalues±ia, and the infinite sequence of double points

λ2j = (jπ/L)

2−a2, j ∈Z. (12)

The number of imaginary double points, obtained for0≤ j <aL/π, coincides with the number of unstable modes (eq. 9)computed directly from the linearization. Each imaginarydouble point “labels” the associated unstable mode (Calini& Schober, 2002).

planeλ−

λ − plane

Fig. 1: Spectrum of a) a generic unstable N-phase solutionand b) the plane wave solution. The simple periodic eigen-values are labeled by circles and the double points by crosses.

3 Rogue waves in the NLS and HONLS equations

3.1 Homoclinic solutions of the NLS equation as modelsfor rogue waves

The NLS equation admits modulationally unstable periodicsolutions with homoclinic orbits that can undergo large am-plitude excursions away from their target solution. Such ho-moclinic orbits can be used to model rogue waves. An exam-ple is provided by the unstable plane wave solutionua(x,t)=

ae2ia2t where the number of unstable modes (UMs) is pro-vided by eq. (9). For each UM there is a correspond-ing homoclinic orbit. A global representation of the homo-clinic orbits can be obtained by exponentiating the linear in-stabilities via Backlund transformations (Matveev & Salle,1991; McLaughlin & Schober, 1992). Moreover, for NLSpotentials with several UMs, iterated Backlund transforma-tions will generate their entire stable and unstable manifolds,comprised of homoclinic orbits of increasing dimension upto the dimension of the invariant manifolds. Such higher-dimensional homoclinic orbits are also known as combina-tion homoclinic orbits.

(b)


L(x) =

(

∂x + iλ −uu∗ ∂x− iλ

)

and depends onx andt through the potentialu and on thespectral parameterλ.

The nonlinear spectral decomposition of an NLS initialcondition (or in general of an ensemble of JONSWAP initialdata) is based on the inverse spectral theory of the NLS equa-tion. For periodic boundary conditionsu(x+L,t) = u(x,t),the Floquet spectrum associated with an NLS potentialu (i.e.the spectrum of the linear operatorL(x) atu) can be describedin terms of the Floquet discriminant ofu, defined as the traceof the transfer matrix of a fundamental matrix solutionΦ of(11) over the interval[0,L] (Ablowitz & Segur, 1981):

∆(u;λ)= Trace(

Φ(x,t;λ)−1Φ(x+L,t;λ))

.


σ(u)= {λ∈ IC|∆(u;λ)∈ IR,−2≤∆≤ 2}.

Points of the continuous spectrum ofu are those for whichthe eigenvalues of the transfer matrix have unit modulus, andtherefore∆(u;λ) is real and between2 and−2; in particular,the real line is part of the continuous spectrum. Points of theL-periodic/antiperiodic discrete spectrum ofu are those forwhich the eigenvalues of the transfer matrix are±1, equiva-lently ∆(u;λ) =±2. Points of the discrete spectrum whichare embedded in a continuous band of spectrum are criticalpoints for the Floquet discriminant (i.e., d∆/dλ must vanishat such points).

Since the transfer matrix only changes by conjugationwhen we shift inx or t, ∆ is independent of those vari-ables. An important consequence of this observation is thatthe Floquet discriminant is invariant under the NLS flow,and thus encodes an infinite family of constants of motion(parametrized byλ).

The continuous part of Floquet spectrum of a generic NLSpotential consists of the real axis and of complex bands ter-minating in simple pointsλs

j (at which∆=±2,∆′ 6= 0). TheN -phase potentials are those characterized by a finite numberof bands of continuous spectrum (or a finite number of sim-ple points). Fig. 1a shows the spectrum of a typicalN -phasepotential: complex critical points (usually double pointsofthe discrete spectrum for which∆′ = 0 and∆” 6= 0), suchas the one appearing in the figure, are in general associatedwith linear instabilities ofu and label its homoclinic orbits(Ercolani et al., 1990).

As a concrete example consider the plane wave solution,ua(x,t) = ae2ia2t, whose Floquet discriminant is given by∆(a,λ)= 2cos(

√a2 +λ2L). The resulting Floquet spectrum

(Fig. 1b) consists of the continuous bandsIR⋃

[−ia,ia], anda discrete part containing the simple periodic/antiperiodic

eigenvalues±ia, and the infinite sequence of double points

λ2j = (jπ/L)

2−a2, j ∈Z. (12)

The number of imaginary double points, obtained for0≤ j <aL/π, coincides with the number of unstable modes (eq. 9)computed directly from the linearization. Each imaginarydouble point “labels” the associated unstable mode (Calini& Schober, 2002).

planeλ−

λ − plane

Fig. 1: Spectrum of a) a generic unstable N-phase solutionand b) the plane wave solution. The simple periodic eigen-values are labeled by circles and the double points by crosses.



The NLS equation admits modulationally unstable periodicsolutions with homoclinic orbits that can undergo large am-plitude excursions away from their target solution. Such ho-moclinic orbits can be used to model rogue waves. An exam-ple is provided by the unstable plane wave solutionua(x,t)=

ae2ia2t where the number of unstable modes (UMs) is pro-vided by eq. (9). For each UM there is a correspond-ing homoclinic orbit. A global representation of the homo-clinic orbits can be obtained by exponentiating the linear in-stabilities via Backlund transformations (Matveev & Salle,1991; McLaughlin & Schober, 1992). Moreover, for NLSpotentials with several UMs, iterated Backlund transforma-tions will generate their entire stable and unstable manifolds,comprised of homoclinic orbits of increasing dimension upto the dimension of the invariant manifolds. Such higher-dimensional homoclinic orbits are also known as combina-tion homoclinic orbits.

Fig. 1. Spectrum of(a) a generic unstable N-phase solution and(b) the plane wave solution. The simple periodic eigenvalues arelabeled by circles and the double points by crosses.

As a concrete example consider the plane wave solution,ua(x,t) = ae2ia2t , whose Floquet discriminant is given by1(a,λ) = 2cos(

√a2+λ2L). The resulting Floquet spectrum

(Fig. 1b) consists of the continuous bands IR⋃

[−ia,ia], anda discrete part containing the simple periodic/antiperiodiceigenvalues±ia, and the infinite sequence of double points

λ2j = (jπ/L)2

−a2, j ∈ Z . (12)

The number of imaginary double points, obtained for 0≤ j <

aL/π , coincides with the number of unstable modes (Eq.9)computed directly from the linearization. Each imaginarydouble point “labels” the associated unstable mode (Caliniand Schober, 2002).



The NLS equation admits modulationally unstable periodicsolutions with homoclinic orbits that can undergo largeamplitude excursions away from their target solution. Suchhomoclinic orbits can be used to model rogue waves. Anexample is provided by the unstable plane wave solutionua(x,t) = ae2ia2t where the number of unstable modes(UMs) is provided by Eq. (9). For each UM there is acorresponding homoclinic orbit. A global representation ofthe homoclinic orbits can be obtained by exponentiating thelinear instabilities via Backlund transformations (Matveevand Salle, 1991; McLaughlin and Schober, 1992). Moreover,for NLS potentials with several UMs, iterated Backlundtransformations will generate their entire stable and unstablemanifolds, comprised of homoclinic orbits of increasingdimension up to the dimension of the invariant manifolds.Such higher-dimensional homoclinic orbits are also knownas combination homoclinic orbits.



Fig. 2. Amplitude plot of a homoclinic orbit of the Stokes wavewith a = 0.5, L = 2

√2π .

3.1.1 One unstable mode rogue wave solutions

A single (i.e. lowest dimensional) homoclinic orbit of theplane wave potential is given by

u(x,t)= ae−2ia2t 1+2cos(px)eσnt+2iφ+ρ+Ae2σnt+4iφ+2ρ

1+2cos(px)eσ1t+ρ +Ae2σnt+2ρ

(13)

whereA = 1/cos2φ, σn = ±p√

4a2−p2, φ = sin−1(p/2a),andp = µn = 2πn/L < a for some integern. Each UM hasan associated homoclinic orbit characterized by modep =

µn.

Figure 2 shows the space-time plot of the amplitude|u(x,t)| of a homoclinic orbit with one UM, fora = 0.5,L = 2

√2π andp = 2π/L. As t → ±∞, solution (13) limits

to the plane wave potential; in fact, the plane wave behaviordominates the dynamics of the homoclinic solution for mostof its lifetime. As t approachest0 = 0, nonlinear focusingoccurs due to the BF instability and the solution rises toa maximum height of 2.4a. Thus, the homoclinic solutionwith one UM can be regarded as the simplest model of roguewave.

An almost equally dramatic wave trough occurs close tothe crest of the rogue wave as a result of wave compressiondue to wave dislocation. The amplitude amplification factoris given by

Af =maxx∈[0,L],t∈IR|u(x,t)|

limt→±∞ |u(x,t)|≈ 2.4 . (14)

3.1.2 Phase modulated rogue waves

As the number of UMs increases, the space-time structure ofthe homoclinic solutions becomes more complex. When twoor more UMs are present the initial wave train can be phasemodulated to produce additional focusing.

The family of homoclinic orbits of the plane wavepotential with two UMs is given by an expression of the form

u(x,t) = ae2ia2t g(x,t)

f (x,t), (15)

where the expression forf (x,t) andg(x,t) depend on thetwo spatial modes cos(2nπx/L), cos(2mπx/L), and ontemporal exponential factors exp(σnt +ρn), exp(σmt +ρm),

with growth ratesσl = µl

√µ2

l −4a2, µl = 2πl/L. (Thecomplete formulas can be found inCalini et al., 1996; Caliniand Schober, 2002.)

As in the one-UM case, this combination homoclinicorbit decays to the plane wave potential ast → ±∞,and the associated rogue wave remains hidden beneaththe background plane wave for most of its lifetime. Thetemporal separation of the two spatial modes depends upon aparameterρ related to the differenceρn−ρm in the temporalphases (Calini et al., 1996; Calini and Schober, 2002).

In turn, ρ affects the amplitude amplification factor.Figure 3a–b shows the combination homoclinic orbit (15)obtained with all parameters set equal except forρ. InFig. 3a, ρ = .1, the modes are well separated, and theamplitude amplification factor is roughly three. In Fig.3b,the value ofρ is approximately−0.65, corresponding to thetwo UMs being simultaneously excited or coalesced. For thisvalue ofρ the amplitude amplification factor is maximal andthe rogue wave rises to a height of 4.1 times the height of thecarrier wave. The surface plot of|u(x,t)| has been rotated inFig. 3b to facilitate comparison with Fig.4a.

Figure 3a shows focusing due to only weak amplitudemodulation of the initial wave train; the growth in amplitudebeginning att ≈ 10 and att ≈ 25 is due to the BF instability.However, in Fig.3b focusing due to both amplitude andphase modulation occurs. The amplitude growth att ≈ 10is due to the BF instability, while the additional very rapidfocusing att ≈ 18.4 is due to the phase modulation. Ingeneral it is possible to select the phases in a combinationhomoclinic orbit withN spatial modes so that any numbern

(2≤ n ≤ N ) of modes coalesce at some fixed time.

3.2 Rogue waves in the higher order NLS equation

In this section we examine the robustness of homoclinicsolutions of the NLS equation, as well as frequencydownshifting, when higher order terms are included inthe wave dynamics. The HONLS equation (1) is solvednumerically using a smoothed exponential time differencingintegrator. The details of the method are provided in Sect. 4



Fig. 3. Amplitude plots of the combination homoclinic orbit:(a)the phases are selected to produce well separated spatial modes;(b)shows a coalesced homoclinic orbit.

and in references (Cox and Matthews, 2002; Khaliq et al.,2009). As expected, we find permanent downshifting doesnot occur in the HONLS equation. We find that the chaoticbackground increases the likelihood and amplitude of roguewaves as compared to predictions obtained with the NLSequation.

We choose initial data that are small perturbations of theunstable plane wave potential,

u(x,0) = a(1+0.1cosµx), (16)

where the amplitudea is varied to be in the two or threeunstable mode regime withµ = 2π/L and L = 4

√2π .

Figure 4a illustrates a prominent rogue wave solution ofEq. (1), ε = 0.05,β = 0 = 0 for initial data (16) in the two-UM regime with a = 0.5, 456< t < 461. The solutionrapidly becomes chaotic (aroundt = 27) with rogue wavesemerging intermittently afterwards. For example, att ≈

−50

5

243

244

245

2460

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3

4

Fig. 4. Rogue waves solutions for the HONLS equation (ε = .05,0 = β = 0) when(a) two and(b) three unstable modes are present.

459 a rogue wave rises with a maximum wave amplitudeUmax ≈ 2.15. Comparing Fig.4a with Fig. 3b one findsthat the structure of this rogue wave is similar to that ofthe combination homoclinic solution (15) with coalescedspatial modes obtained whenρ = −0.65, although the waveamplification factor is slightly smaller. The symmetrybreaking effects of the higher order terms in the HONLSequation prevent a complete spatial coalescence of thenonlinear modes.

Numerical simulations of the HONLS equation in thethree-UM regime, e.g. initial condition (16) with a = 0.7,show a similar phenomenon: after the onset of chaoticdynamics, rogue waves rise intermittently above the chaoticbackground (see Fig.4b). At t ≈ 245 a rogue wave develops,Umax ≈ 3.59, which is close to the coalesced homoclinicsolution of the NLS equation in the three-UM regime.

Extensive numerical experiments were performed for theHONLS equation in the two- and three-UM regime, varying



the perturbation strengthε, the amplitudea and adding insome random phasesφi ’s in the initial data. In all cases, thecoalesced homoclinic NLS solution emerges generically as astructurally stable feature of the perturbed dynamics.

The occurence of rogue wave in the HONLS equationis distinct from their occurence in the NLS or the dampedHONLS equations. In the HONLS equation, rogue waveswill occur intermittently throughout very long time series. Inthe NLS equation they occur only once or twice, depndingon whether the modes are coalesced or not. In the dampedHONLS, they occur irregularly over a short time period buteventually are damped out.

The chaotic regime produces additional focusing byeffectively selecting optimal phase modulations and thechaotic dynamics singles out the maximally coalescedhomoclinic solutions of the unperturbed NLS equation asphysically observable rogue waves. Moreover, the likelihoodof larger amplitude waves (see e.g. Fig.4) increases forthe HONLS equation, as substantiated by the diagnosticsdeveloped in Sect.5, correlating wave strengths in the NLSand HONLS models to proximity to homoclinic data. Thus,for initial data in the neighborhood of an unstable plane wavethe underlying chaotic dynamics of the HONLS equationfavors the occurrence of large amplitude rogue waves, ascompared to predictions obtained from the NLS equation.

To investigate downshifting we varied the perturbationstrengthε in the HONLS equation and the parameters in theinitial data in the two- and three-UM regime. Typical resultsobtained with the HONLS equation (1) are shown in Fig.5for initial data

u(x,0) = 0.7(1+0.1(cosµx +0.2(cos2µ(x −L/4)

+ 0.38cos3µ(x −L/3)))), (17)

with L = 4√

2π andε = 0.05.Figure 5a shows the time evolution of the strength of

u(x,t),

S(t) =Umax(t)

Hs(t), (18)

whereUmax(t) = maxx∈[0,L]|u(x,t)| andHs(t) is the signifi-cant wave height, defined as 4 times the standard deviationof the surface elevation. Rogue waves occur throughoutthe time series whenever the strength exceeds the thresholdcriteria of 2.2. Figure5b shows the time evolution of themain Fourier modes|Ak(t)| for k = 0, ±1,...,±4. Duringeach of the modulation stages the zeroth mode loses energyas the upper and lower higher harmonics become excited.For 0< t < 200 the total energy and the total energy flux areconstant. This allows the energy to flow back to the zerothmode keeping the spectral centerkm constant (Fig.5c). Theplot of kpeak (Fig. 5d) confirms that permanent frequencydownshifting does not occur. As expected, even when steeperwaves are obtained, permanent frequency downshifting doesnot occur in the HONLS numerical experiments, indicatingthe necessity of a dissipative or nonconservative process.

the NLS equation they occur only once or twice, depndingon whether the modes are coalesced or not. In the dampedHONLS, they occur irregularly over a short time period buteventually are damped out.

The chaotic regime produces additional focusing by effec-tively selecting optimal phase modulations and the chaoticdynamics singles out the maximally coalesced homoclinicsolutions of the unperturbed NLS equation as physically ob-servable rogue waves. Moreover, the likelihood of larger am-plitude waves (see e. g. Fig. 4) increases for the HONLSequation, as substantiated by the diagnostics developed insection 5, correlating wave strengths in the NLS and HONLSmodels to proximity to homoclinic data. Thus, for initial datain the neighborhood of an unstable plane wave the under-lying chaotic dynamics of the HONLS equation favors theoccurrence of large amplitude rogue waves, as compared topredictions obtained from the NLS equation.

To investigate downshifting we varied the perturbationstrengthǫ in the HONLS equation and the parameters in theinitial data in the two- and three-UM regime. Typical resultsobtained with the HONLS equation (1) are shown in Fig. 5for initial data

u(x,0) = 0.7(1+0.1(cosµx+0.2(cos2µ(x−L/4)

+0.38cos3µ(x−L/3)))), (17)

with L = 4√

2π andǫ = 0.05.Figure 5a shows the time evolution of the strength of

u(x,t),

S(t)=Umax(t)

Hs(t), (18)

whereUmax(t)= maxx∈[0,L]|u(x,t)| andHs(t) is the signif-icant wave height, defined as 4 times the standard deviation ofthe surface elevation. Rogue waves occur throughout the timeseries whenever the strength exceeds the threshold criteria of2.2. Figure 5b shows the time evolution of the main Fouriermodes|Ak(t)| for k = 0,±1,...,±4. During each of the mod-ulation stages the zeroth mode loses energy as the upper andlower higher harmonics become excited. For0< t < 200 thetotal energy and the total energy flux are constant. This al-lows the energy to flow back to the zeroth mode keepingthe spectral centerkm constant (Fig. 5c). The plot ofkpeak

(Fig. 5d) confirms that permanent frequency downshiftingdoes not occur. As expected, even when steeper waves areobtained, permanent frequency downshifting does not occurin the HONLS numerical experiments, indicating the neces-sity of a dissipative or nonconservative process.

4 Rogue waves and downshifting in the presence ofdamping

In the last section we saw that anǫ-neighborhood of the un-stable plane wave with two or more UMS is effectively a

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Fig. 5: The HONLS eq.,Γ = 0, ǫ = 0.05, β = 0: time evolu-tion of (a) the strengthS(t), (b) the main Fourier mode, (c)the spectral centerkm, and (d) the spectral peakkpeak.

Fig. 5. The HONLS equation,0 = 0, ε = 0.05, β = 0: timeevolution of (a) the strengthS(t), (b) the main Fourier mode,(c)the spectral centerkm, and(d) the spectral peakkpeak.



4 Rogue waves and downshifting in the presenceof damping

In the last section we saw that anε-neighborhood of theunstable plane wave with two or more UMS is effectivelya rogue wave regime for the HONLS equation as theprobability of obtaining a rogue wave is high. In thenumerical experiments we choose initial data that are smallperturbations of an unstable plane wave to study the effectsof linear and nonlinear damping on the development of roguewaves and downshifting. We consider the damped HONLSequation in non-dimensional form with the damping termson the same order or smaller than the higher order NLSterms. Consequently we vary the damping coefficients to be0 < εβ, 0 < O(ε), whereε is the coefficient of the higherorder NLS terms.

In the numerical experiments Eq. (1) is solved usinga smoothing fourth-order exponential time differencingintegrator (Cox and Matthews, 2002; Khaliq et al., 2009) thatavoids inaccuracies when inverting matrix polynomials byusing Pade approximations. We useN = 256 Fourier modesin space and a fourth-order Runge-Kutta discretizationin time (1t = 10−3). The high frequency modes wereeliminated at every time step to avoid aliasing by settinguk =

0 for |k| > 120. Eliminating the high frequencies does nothave a significant effect on the exchange of energy betweenthe dominant low wave number modes and thus does notimpact our results related to frequency downshifting androgue waves.

This temporal and spatial resolution allows for the threeintegral invariants of the conservative HONLS equation,E,P , andH, to be conserved with an accuracy ofO(10−8),O(10−2), O(10−2), respectively. The nonlinear modecontent of the data is numerically computed using the directspectral transform described above, i.e. the system of ODEs(11) is numerically solved to obtain the discriminant1. Thezeros of1±2 are then determined with a root solver basedon Muller’s method (Ercolani et al., 1990). The spectrumis computed with an accuracy ofO(10−6), whereas thespectral quantities we are interested in range fromO(10−3)

toO(10−1).

4.1 The HONLS equation with linear damping

We begin by examining the evolution of damped uniformwavetrains with initially one to three pairs of sidebandsexcited. It is important to note in comparisons of thedamped and undamped dynamics that atypical cases mayarise. Figure6a–b shows the strength of the wavetrain(solid line) andUmax (dashed line) for the undamped andlinearly damped (0 = 0.005) HONLS equation (ε = 0.05),respectively, for initial data (16) in the two-UM regime:a = 0.5, µ = 2π/L andL = 4

√2π . Contrary to what one

might expect, Fig.6b shows a rogue wave occuring att ≈ 23in the damped case with an enhanced strength≈ 2.8 that

rogue wave regime for the HONLS equation as the probabil-ity of obtaining a rogue wave is high. In the numerical experi-ments we choose initial data that are small perturbations ofanunstable plane wave to study the effects of linear and nonlin-ear damping on the development of rogue waves and down-shifting. We consider the damped HONLS equation in non-dimensional form with the damping terms on the same orderor smaller than the higher order NLS terms. Consequently wevary the damping coefficients to be0 < ǫβ, Γ < O(ǫ), whereǫ is the coefficient of the higher order NLS terms.

In the numerical experiments equation (1) is solved usinga smoothing fourth-order exponential time differencing in-tegrator (Cox & Matthews, 2002; Khaliq et al., 2009) thatavoids inaccuracies when inverting matrix polynomials byusing Pade approximations. We useN = 256 Fourier modesin space and a fourth-order Runge-Kutta discretization intime (∆t = 10−3). The high frequency modes were elimi-nated at every time step to avoid aliasing by settinguk = 0for |k|> 120. Eliminating the high frequencies does not havea significant effect on the exchange of energy between thedominant low wave number modes and thus does not im-pact our results related to frequency downshifting and roguewaves.

This temporal and spatial resolution allows for the threeintegral invariants of the conservative HONLS equation,E,P , andH , to be conserved with an accuracy ofO(10−8),O(10−2), O(10−2), respectively. The nonlinear mode con-tent of the data is numerically computed using the direct spec-tral transform described above, i.e. the system of ODEs (11)is numerically solved to obtain the discriminant∆. The ze-ros of ∆± 2 are then determined with a root solver basedon Muller’s method (Ercolani et al., 1990). The spectrum iscomputed with an accuracy ofO(10−6), whereas the spec-tral quantities we are interested in range fromO(10−3) toO(10−1).

4.1 The HONLS Equation with Linear Damping

We begin by examining the evolution of damped uniformwavetrains with initially one to three pairs of sidebands ex-cited. It is important to note in comparisons of the dampedand undamped dynamics that atypical cases may arise. Fig-ures 6a-b show the strength of the wavetrain (solid line) andUmax (dashed line) for the undamped and linearly damped(Γ = 0.005) HONLS equation (ǫ = 0.05), respectively, forinitial data (16) in the two-UM regime:a = 0.5, µ = 2π/LandL = 4

√2π. Contrary to what one might expect, Fig. 6b

shows a rogue wave occuring att≈ 23 in the damped casewith an enhanced strength≈ 2.8 that was not present in theundamped evolution (Fig. 6a). This typically occurs for smallvalues ofΓ or β where the smaller growth rates of the indi-vidual modes due to damping can be offset by a coalescenceof the modes due to changes in their focusing times.

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Fig. 6: StrengthS(t) (solid line) andUmax (dashed line) for(a) HONLS Equationǫ = 0.05; (b) HONLS Equation withLinear Damping,ǫ = 0.05, Γ = 0.006, for IC u0 = 0.5(1+0.1cosµx).

We examined the HONLS equation with linear dampingfor 0.005≤Γ≤ 0.05. Although individual simulations mayyield larger amplification factors in the damped regime, wefind that on average, in the presence of linear damping (seesec. 5.2 for confirmation with the nonlinear spectral diag-nostics), the strength is smaller and fewer rogue waves oc-cur. Figure 7 shows the results obtained with equation (1)for ǫ = 0.05, Γ = 0.01, β = 0 for initial condition (17) for0 < t < 200. In the linearly damped evolution, Figs. 7a, thelast rogue wave occurs att≈ 10; in contrast, in Fig. 5a roguewaves occur throughout the unperturbed HONLS time se-ries. The total energy exponentially decays (eq. 6) and theFourier modes are uniformly damped. The total energy fluxis constant, keeping the spectral centerkm zero. Permanentdownshifting does not occur for the linearly damped HONLSequation.

Fig. 6. StrengthS(t) (solid line) andUmax (dashed line) for(a) HONLS equationε = 0.05; (b) HONLS equation with lineardamping,ε = 0.05,0 = 0.006, for ICu0 = 0.5(1+0.1cosµx).

was not present in the undamped evolution (Fig.6a). Thistypically occurs for small values of0 or β where the smallergrowth rates of the individual modes due to damping can beoffset by a coalescence of the modes due to changes in theirfocusing times.

We examined the HONLS equation with linear dampingfor 0.005≤ 0 ≤ 0.05. Although individual simulations mayyield larger amplification factors in the damped regime, wefind that on average, in the presence of linear damping(see Sect. 5.2 for confirmation with the nonlinear spectraldiagnostics), the strength is smaller and fewer rogue wavesoccur. Figure7 shows the results obtained with Eq. (1)for ε = 0.05, 0 = 0.01, β = 0 for initial condition (17) for0< t < 200. In the linearly damped evolution, Fig.7a, thelast rogue wave occurs att ≈ 10; in contrast, in Fig.5arogue waves occur throughout the unperturbed HONLS timeseries. The total energy exponentially decays (Eq.6) and theFourier modes are uniformly damped. The total energy fluxis constant, keeping the spectral centerkm zero. Permanentdownshifting does not occur for the linearly damped HONLSequation.



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Fig. 7: The HONLS eq. with linear damping:Γ = 0.01, ǫ =0.05, β = 0. The time evolution of (a) the strengthS(t) (b)the main Fourier modes,0 < t < 200.

4.2 The HONLS Equation with Nonlinear Damping

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Fig. 8: Spatial distribution of|u(x,t∗)| (solid line) and of theβ-term (dashed line) att∗ = 22.7 for initial condition (16),for ǫ = 0.05, β = 0.1, Γ = 0.

The perturbation of the Hilbert tranform term in equation(1),βu[H

(

|u|2)

]x. is referred to as the “β-term” and modelsnonlinear damping of the mean flow. Figure (8) shows thespatial distribution of|u(x,t∗)| (solid line) and of theβ-term

(dashed line ) of the nonlinearly damped HONLS equationfor ǫ = 0.05, β = 0.1, Γ = 0, for initial condition (16). Att = 22.7 the wavetrain is strongly modulated and we find thattheβ-term is significant only near the maximum crest of theenvelope. In general, steeper waves result in stronger damp-ing and the effects of the nonlinearβ-term are concentratedin space-time when the wavetrain is strongly modulated.

To illustrate the effects of nonlinear damping on down-shifting we consider the solution of equation (1) withǫ =0.05, β = 0.5 andΓ = 0, for initial data (17),0 < t < 50. Figs.9(a-b) show the surface amplitude|u(x,t)| and the time evo-lution of the strength, respectively. Two rogue waves occurearly in the time series,t < 10, before damping has a chanceto significantly alter the growth rate of the modes and pre-vent further rogue waves from forming. Comparing Fig. 9band Fig. 9c, the onset of downshifting occurs when the firstrogue wave appears at approximatelyt≈ 6 when theβ-termis large. This produces an abrupt rapid decay in the energy(Fig. 9d) and growth in the flux (Fig. 9e), resulting in arapid shift in the spectral centerkm (Fig. 9f). The downshift-ing becomes irreversible aftert≈ 17 when the wavetrain isdemodulated and the lower modesk =−1,−4 become dom-inant. In the demodulation stage theβ-term is small so thatthe energy and flux change at a much slower rate. Thuskm

remains negative and does not upshift back.Permanent downshifting is obtained for all values ofβ.

As an example consider the solution of equation (1) withǫ = 0.05, β = 0.04, Γ = 0 for initial data (17),0 < t < 200.For smallβ there is a slow gradual decay in the total en-ergy and flux rather than the abrupt rapid decay obtained forlargerβ. As a result the last rogue wave appears att≈ 51 (Fig. 10a), much later than forβ = 0.5. The plot of the evolu-tion of the main Fourier modes (Fig. 10b) shows permanentdownshifting occurs att≈ 65. Figure 11, which shows theevolution ofkpeak for the nonlinearly damped HONLS equa-tion (ǫ = 0.05, Γ = 0) with β = 0.04 or β = 0.5, provides analternate view of permanent downshifting forβ 6= 0 and itsoccurence at later times for smaller values ofβ.

When both linear and nonlinear damping are present andthe nonlinear damping is dominant, the mechanism for per-manent downshifting is the same. Fig. (12) shows the so-lution of equation (1) withǫ = 0.05, β = 0.5, Γ = 0.005 forinitial data (17),0 < t < 50. Comparing to Fig. (9), the roguewaves occur at approximately the same time with slightlysmaller amplitudes. An abrupt rapid decay in the energy andgrowth in the flux occurs as before leading to a rapid shift inthe spectral centerkm.

Characterization of the nonlinearly damped evolution:To characterize the effects of nonlinear damping on roguewaves we fixǫ = 0.05 andΓ =0 in equation (1) and vary thenonlinear damping coefficient,0 < β < 0.75, as well as theamplitude of the initial data in the three-UM regime:

u(x,0)= a(1+0.01cosµx), (19)

Fig. 7. The HONLS eq. with linear damping:0 = 0.01, ε = 0.05,β = 0. The time evolution of(a) the strengthS(t) (b) the mainFourier modes, 0< t < 200.

4.2 The HONLS Equation with nonlinear damping

The perturbation of the Hilbert tranform term in Eq. (1),βu

[H

(|u|

2)]

x. is referred to as the “β-term” and models

nonlinear damping of the mean flow. Figure8 shows thespatial distribution of|u(x,t∗)| (solid line) and of theβ-term(dashed line) of the nonlinearly damped HONLS equationfor ε = 0.05, β = 0.1, 0 = 0, for initial condition (16).At t = 22.7 the wavetrain is strongly modulated and wefind that theβ-term is significant only near the maximumcrest of the envelope. In general, steeper waves result instronger damping and the effects of the nonlinearβ-term areconcentrated in space-time when the wavetrain is stronglymodulated.

To illustrate the effects of nonlinear damping on down-shifting we consider the solution of Eq. (1) with ε =

0.05, β = 0.5 and0 = 0, for initial data (17), 0< t < 50.Figure 9a–b shows the surface amplitude|u(x,t)| and thetime evolution of the strength, respectively. Two rogue wavesoccur early in the time series,t < 10, before damping hasa chance to significantly alter the growth rate of the modesand prevent further rogue waves from forming. ComparingFig. 9b and9c, the onset of downshifting occurs when thefirst rogue wave appears at approximatelyt ≈ 6 when theβ-

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Fig. 7: The HONLS eq. with linear damping:Γ = 0.01, ǫ =0.05, β = 0. The time evolution of (a) the strengthS(t) (b)the main Fourier modes,0 < t < 200.

4.2 The HONLS Equation with Nonlinear Damping

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Fig. 8: Spatial distribution of|u(x,t∗)| (solid line) and of theβ-term (dashed line) att∗ = 22.7 for initial condition (16),for ǫ = 0.05, β = 0.1, Γ = 0.

The perturbation of the Hilbert tranform term in equation(1),βu[H

(

|u|2)

]x. is referred to as the “β-term” and modelsnonlinear damping of the mean flow. Figure (8) shows thespatial distribution of|u(x,t∗)| (solid line) and of theβ-term

(dashed line ) of the nonlinearly damped HONLS equationfor ǫ = 0.05, β = 0.1, Γ = 0, for initial condition (16). Att = 22.7 the wavetrain is strongly modulated and we find thattheβ-term is significant only near the maximum crest of theenvelope. In general, steeper waves result in stronger damp-ing and the effects of the nonlinearβ-term are concentratedin space-time when the wavetrain is strongly modulated.

To illustrate the effects of nonlinear damping on down-shifting we consider the solution of equation (1) withǫ =0.05, β = 0.5 andΓ = 0, for initial data (17),0 < t < 50. Figs.9(a-b) show the surface amplitude|u(x,t)| and the time evo-lution of the strength, respectively. Two rogue waves occurearly in the time series,t < 10, before damping has a chanceto significantly alter the growth rate of the modes and pre-vent further rogue waves from forming. Comparing Fig. 9band Fig. 9c, the onset of downshifting occurs when the firstrogue wave appears at approximatelyt≈ 6 when theβ-termis large. This produces an abrupt rapid decay in the energy(Fig. 9d) and growth in the flux (Fig. 9e), resulting in arapid shift in the spectral centerkm (Fig. 9f). The downshift-ing becomes irreversible aftert≈ 17 when the wavetrain isdemodulated and the lower modesk =−1,−4 become dom-inant. In the demodulation stage theβ-term is small so thatthe energy and flux change at a much slower rate. Thuskm

remains negative and does not upshift back.Permanent downshifting is obtained for all values ofβ.

As an example consider the solution of equation (1) withǫ = 0.05, β = 0.04, Γ = 0 for initial data (17),0 < t < 200.For smallβ there is a slow gradual decay in the total en-ergy and flux rather than the abrupt rapid decay obtained forlargerβ. As a result the last rogue wave appears att≈ 51 (Fig. 10a), much later than forβ = 0.5. The plot of the evolu-tion of the main Fourier modes (Fig. 10b) shows permanentdownshifting occurs att≈ 65. Figure 11, which shows theevolution ofkpeak for the nonlinearly damped HONLS equa-tion (ǫ = 0.05, Γ = 0) with β = 0.04 or β = 0.5, provides analternate view of permanent downshifting forβ 6= 0 and itsoccurence at later times for smaller values ofβ.

When both linear and nonlinear damping are present andthe nonlinear damping is dominant, the mechanism for per-manent downshifting is the same. Fig. (12) shows the so-lution of equation (1) withǫ = 0.05, β = 0.5, Γ = 0.005 forinitial data (17),0 < t < 50. Comparing to Fig. (9), the roguewaves occur at approximately the same time with slightlysmaller amplitudes. An abrupt rapid decay in the energy andgrowth in the flux occurs as before leading to a rapid shift inthe spectral centerkm.

Characterization of the nonlinearly damped evolution:To characterize the effects of nonlinear damping on roguewaves we fixǫ = 0.05 andΓ =0 in equation (1) and vary thenonlinear damping coefficient,0 < β < 0.75, as well as theamplitude of the initial data in the three-UM regime:

u(x,0)= a(1+0.01cosµx), (19)

Fig. 8. Spatial distribution of|u(x,t∗)| (solid line) and of theβ-term(dashed line) att∗ = 22.7 for initial condition (16), for ε = 0.05,β = 0.1, 0 = 0.

term is large. This produces an abrupt rapid decay in theenergy (Fig.9d) and growth in the flux (Fig.9e), resultingin a rapid shift in the spectral centerkm (Fig. 9f). Thedownshifting becomes irreversible aftert ≈ 17 when thewavetrain is demodulated and the lower modesk = −1,−4become dominant. In the demodulation stage theβ-term issmall so that the energy and flux change at a much slowerrate. Thuskm remains negative and does not upshift back.

Permanent downshifting is obtained for all values ofβ. Asan example consider the solution of Eq. (1) with ε = 0.05,β = 0.04, 0 = 0 for initial data (17), 0< t < 200. For smallβ there is a slow gradual decay in the total energy and fluxrather than the abrupt rapid decay obtained for largerβ. As aresult the last rogue wave appears att ≈ 51 (Fig.10a), muchlater than forβ = 0.5. The plot of the evolution of the mainFourier modes (Fig.10b) shows permanent downshiftingoccurs att ≈ 65. Figure11, which shows the evolution ofkpeakfor the nonlinearly damped HONLS equation (ε = 0.05,0 = 0) with β = 0.04 orβ = 0.5, provides an alternate viewof permanent downshifting forβ 6= 0 and its occurence atlater times for smaller values ofβ.

When both linear and nonlinear damping are presentand the nonlinear damping is dominant, the mechanism forpermanent downshifting is the same. Figure12 shows thesolution of Eq.1 with ε = 0.05,β = 0.5,0 = 0.005 for initialdata (17), 0< t < 50. Comparing to Fig.9, the rogue wavesoccur at approximately the same time with slightly smalleramplitudes. An abrupt rapid decay in the energy and growthin the flux occurs as before leading to a rapid shift in thespectral centerkm.

Characterization of the nonlinearly damped evolution

To characterize the effects of nonlinear damping on roguewaves we fixε = 0.05 and0 = 0 in Eq. (1) and vary thenonlinear damping coefficient, 0< β < 0.75, as well as the



(a)

0 5 10 15 20 25 30 35 40 45 501

1.2

1.4

1.6

1.8

2

2.2

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STR

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TH

TIME

(b)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

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0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES

TIME

(c)

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

ENER

GY

TIME

(d)

Fig. 9: a-d Nonlinearly damped HONLS eq.ǫ = 0.05, β =0.5, Γ = 0: a) Surface amplitude|u(x,t)|, and the time evo-lution of (b) the strengthS(t), (c) the main Fourier modes,(d) the total energy, for initial data (17).

0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

FLU

X

TIME

(e)

0 5 10 15 20 25 30 35 40 45 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

SPEC

TRAL

CEN

TER

TIME

(f)

Fig. 9: e-f Nonlinearly damped HONLS eq.ǫ = 0.05, β =0.5, Γ = 0: the time evolution of (e) the total energy flux, and(f) the spectral centerkm for initial data (17).

with 0.57 < a < 0.67, µ = 2π/L, L = 4√

2π. Figure 13shows, as a function ofβ: a) the final value of the spectralcenterkm, b) the maximum strength maxt∈[0,200]S(t), c) thenumber of rogue waves obtained for0 < t < 200, and d) thetime of the last rogue wave. The solid curve represents theaverages over the initial data. In the numerical experimentswe observe the following:

1. The spectral centerkm(t) is a decreasing function oftfor all β > 0. Figure 13a is not showing the time evo-lution of km, (see e.g. Fig. 9f for a typical example),but rather the final value ofkm at t = 200, where smallervalues mean a downshift to a lower mode. Thus, per-manent downshifting is observed for allβ > 0 with theonset occuring rapidly for larger values ofβ.

2. Figures 13b-c show that for small values ofβ the aver-age maximum strength in time and average number ofrogue waves exhibit irregular behavior. We may obtainmore or larger rogue waves in the damped regime thanwithout damping for this time frame due to changes inthe focusing time and coalescence of the modes. How-ever for largeβ, the maximum strength in time and num-ber of rogue waves are, in general, decreasing.

(a)

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(b)

0 5 10 15 20 25 30 35 40 45 500

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1

FOU

RIE

R M

OD

ES

TIME

(c)

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1

ENER

GY

TIME

(d)


0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

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0.8

1

FLU

X

TIME

(e)

0 5 10 15 20 25 30 35 40 45 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

SPEC

TRAL

CEN

TER

TIME

(f)


with 0.57 < a < 0.67, µ = 2π/L, L = 4√




(a)

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(b)

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1

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TIME

(c)

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GY

TIME

(d)


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0

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1

FLU

X

TIME

(e)

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0

0.1

SPEC

TRAL

CEN

TER

TIME

(f)


with 0.57 < a < 0.67, µ = 2π/L, L = 4√




(a)

0 5 10 15 20 25 30 35 40 45 501

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TIME

(b)

0 5 10 15 20 25 30 35 40 45 500

0.1

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1

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RIE

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OD

ES

TIME

(c)

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

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0.8

1

ENER

GY

TIME

(d)


0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

FLU

X

TIME

(e)

0 5 10 15 20 25 30 35 40 45 50−0.5

−0.4

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−0.2

−0.1

0

0.1

SPEC

TRAL

CEN

TER

TIME

(f)


with 0.57 < a < 0.67, µ = 2π/L, L = 4√




Fig. 9. Nonlinearly damped HONLS equationε = 0.05, β = 0.5,0 = 0: (a) surface amplitude|u(x,t)|, and the time evolution of(b)the strengthS(t), (c) the main Fourier modes,(d) the total energy,for initial data (17), the time evolution of(e) the total energy flux,and(f) the spectral centerkm for initial data (17).

(a)

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1

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TIME

(c)

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1

ENER

GY

TIME

(d)


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1

FLU

X

TIME

(e)

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−0.4

−0.3

−0.2

−0.1

0

0.1

SPEC

TRAL

CEN

TER

TIME

(f)


with 0.57 < a < 0.67, µ = 2π/L, L = 4√




(a)

0 5 10 15 20 25 30 35 40 45 501

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TIME

(b)

0 5 10 15 20 25 30 35 40 45 500

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1

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OD

ES

TIME

(c)

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1

ENER

GY

TIME

(d)


0 5 10 15 20 25 30 35 40 45 50

0

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1

FLU

X

TIME

(e)

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0

0.1

SPEC

TRAL

CEN

TER

TIME

(f)


with 0.57 < a < 0.67, µ = 2π/L, L = 4√




Fig. 9. Continued.

amplitude of the initial data in the three-UM regime:

u(x,0) = a(1+0.01cosµx), (19)

with 0.57< a < 0.67, µ = 2π/L, L = 4√

2π . Figure 13shows, as a function ofβ: (a) the final value of the spectralcenterkm, (b) the maximum strength maxt∈[0,200]S(t), (c) thenumber of rogue waves obtained for 0< t < 200, and (d) thetime of the last rogue wave. The solid curve represents theaverages over the initial data. In the numerical experimentswe observe the following:

1. The spectral centerkm(t) is a decreasing function oft for all β > 0. Figure13a is not showing the timeevolution ofkm, (see e.g. Fig.9f for a typical example),but rather the final value ofkm at t = 200, where smallervalues mean a downshift to a lower mode. Thus,permanent downshifting is observed for allβ > 0 withthe onset occuring rapidly for larger values ofβ.

2. Figure 13b–c shows that for small values ofβ theaverage maximum strength in time and average numberof rogue waves exhibit irregular behavior. We mayobtain more or larger rogue waves in the damped regimethan without damping for this time frame due to changesin the focusing time and coalescence of the modes.However for largeβ, the maximum strength in time andnumber of rogue waves are, in general, decreasing.



0 10 20 30 40 50 60 70 80 90 1001

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3ST

REN

GTH

TIME

(a)

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES, k

= 0

, −4

TIME

(b)

Fig. 10: The HONLS eq. with nonlinear dampingǫ = 0.05,β = 0.04, Γ = 0: time evolution of (a) the strengthS(t) and(b) the main Fourier modes for initial data (17).

3. For all values of the initial amplitude, the first roguewave is only slightly delayed (t≈ 6) for all β considered(not shown). The time of the last rogue wave,Tl, mayslightly increase for smallβ. As β increases,Tl signif-icantly decreases, indicating fewer rogue waves occur.For β > β∗ only the initial pair of rogue waves develop(Fig. 13d) as largeβ values trigger sufficient nonlineardamping that inhibit further rogue wave formation.

Relation of Downshifting and Rogue Waves :It is sig-nificant in Figs. 9b-c and 10a-b that permanent downshiftingoccurs after the last rogue wave in the time series. In otherwords, rogue waves do not occur after the downshifting be-comes irreversible in these examples. To generalize this re-sult we analyze the previous series of numerical experimentsusing initial data (19) with0.57 < a < 0.67 to determine 1)the time of the last rogue wave and 2) the last timekpeak isequal tok0, indicating permanent downshifting, as the per-turbation strengthβ is varied.

Figure 14 compares the time at which downshifting be-comes irreversible (labeled by an x) with the time at whichthe last rogue wave occurs (labeled by a box) for the HONLSequation with only nonlinear damping,ǫ = 0.05,Γ = 0 and0 < β < 0.75. Fig. 14a shows the comparison for initialdata (19) witha = 0.63 : we observe that for all values of

0 10 20 30 40 50 60 70 80 90 100−5

−4

−3

−2

−1

0

1

2

3

4

5

SPEC

TRAL

PEA

K

TIME

(a)

0 10 20 30 40 50 60 70 80 90 100−5

−4

−3

−2

−1

0

1

2

3

4

5

SPEC

TRAL

PEA

K

TIME

(b)

Fig. 11: The evolution ofkpeak for the nonlinearly dampedHONLS equation withǫ = 0.05, Γ = 0 and (a)β = 0.04 or (b)β = .5 for initial data (17).

β rogue waves do not occur after the downshifting becomesirreversible. Further, rogue waves do not develop after per-manent downshifting occurs for any other pair of parame-ter values (a,β) considered in the experiments. This is sum-marized in Fig. 14b which compares, for0 < β < 0.75, theaverage time of the last rogue wave (box) with the averagetime at which downshifting is irreversible (x), where the av-erages are over the six simulations with initial data amplitude0.57 < a < 0.67. These results imply permanent downshift-ing serves as an indicator that the cumulative effects of damp-ing are sufficient to prevent the further development of roguewaves.

Finally we consider the case when both linear and nonlin-ear damping are present and the nonlinear damping is dom-inant to allow permanent downshifting to occur. Figure 15compares the times for permanent downshifting and the lastrogue wave for the damped HONLS eqn. withǫ = 0.05,Γ=0.005 and0 < β < 0.75. The same relation between down-shifting and the last rogue wave are observed to hold, i.e.rogue waves do not occur after the downshifting is perma-nent. However, due to the additional linear damping there isa longer time lag between the last rogue wave and permanentdownshifting.

Fig. 10. The HONLS equation with nonlinear dampingε = 0.05,β = 0.04,0 = 0: time evolution of(a) the strengthS(t) and(b) themain Fourier modes for initial data (17).

3. For all values of the initial amplitude, the first roguewave is only slightly delayed (t ≈ 6) for allβ considered(not shown). The time of the last rogue wave,Tl ,may slightly increase for smallβ. As β increases,Tlsignificantly decreases, indicating fewer rogue wavesoccur. For β > β∗ only the initial pair of roguewaves develop (Fig.13d) as largeβ values triggersufficient nonlinear damping that inhibit further roguewave formation.

Relation of downshifting and rogue waves

It is significant in Figs. 9b–c and 10a–b that permanentdownshifting occurs after the last rogue wave in the timeseries. In other words, rogue waves do not occur afterthe downshifting becomes irreversible in these examples.To generalize this result we analyze the previous series ofnumerical experiments using initial data (19) with 0.57<

a < 0.67 to determine (1) the time of the last rogue wave and(2) the last timekpeak is equal tok0, indicating permanentdownshifting, as the perturbation strengthβ is varied.

Figure 14 compares the time at which downshiftingbecomes irreversible (labeled by an x) with the time at whichthe last rogue wave occurs (labeled by a box) for the HONLSequation with only nonlinear damping,ε = 0.05,0 = 0 and

0 10 20 30 40 50 60 70 80 90 1001

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

STR

ENG

TH

TIME

(a)

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES, k

= 0

, −4

TIME

(b)

Fig. 10: The HONLS eq. with nonlinear dampingǫ = 0.05,β = 0.04, Γ = 0: time evolution of (a) the strengthS(t) and(b) the main Fourier modes for initial data (17).

3. For all values of the initial amplitude, the first roguewave is only slightly delayed (t≈ 6) for all β considered(not shown). The time of the last rogue wave,Tl, mayslightly increase for smallβ. As β increases,Tl signif-icantly decreases, indicating fewer rogue waves occur.For β > β∗ only the initial pair of rogue waves develop(Fig. 13d) as largeβ values trigger sufficient nonlineardamping that inhibit further rogue wave formation.

Relation of Downshifting and Rogue Waves :It is sig-nificant in Figs. 9b-c and 10a-b that permanent downshiftingoccurs after the last rogue wave in the time series. In otherwords, rogue waves do not occur after the downshifting be-comes irreversible in these examples. To generalize this re-sult we analyze the previous series of numerical experimentsusing initial data (19) with0.57 < a < 0.67 to determine 1)the time of the last rogue wave and 2) the last timekpeak isequal tok0, indicating permanent downshifting, as the per-turbation strengthβ is varied.

Figure 14 compares the time at which downshifting be-comes irreversible (labeled by an x) with the time at whichthe last rogue wave occurs (labeled by a box) for the HONLSequation with only nonlinear damping,ǫ = 0.05,Γ = 0 and0 < β < 0.75. Fig. 14a shows the comparison for initialdata (19) witha = 0.63 : we observe that for all values of

0 10 20 30 40 50 60 70 80 90 100−5

−4

−3

−2

−1

0

1

2

3

4

5

SPEC

TRAL

PEA

K

TIME

(a)

0 10 20 30 40 50 60 70 80 90 100−5

−4

−3

−2

−1

0

1

2

3

4

5

SPEC

TRAL

PEA

K

TIME

(b)

Fig. 11: The evolution ofkpeak for the nonlinearly dampedHONLS equation withǫ = 0.05, Γ = 0 and (a)β = 0.04 or (b)β = .5 for initial data (17).

β rogue waves do not occur after the downshifting becomesirreversible. Further, rogue waves do not develop after per-manent downshifting occurs for any other pair of parame-ter values (a,β) considered in the experiments. This is sum-marized in Fig. 14b which compares, for0 < β < 0.75, theaverage time of the last rogue wave (box) with the averagetime at which downshifting is irreversible (x), where the av-erages are over the six simulations with initial data amplitude0.57 < a < 0.67. These results imply permanent downshift-ing serves as an indicator that the cumulative effects of damp-ing are sufficient to prevent the further development of roguewaves.

Finally we consider the case when both linear and nonlin-ear damping are present and the nonlinear damping is dom-inant to allow permanent downshifting to occur. Figure 15compares the times for permanent downshifting and the lastrogue wave for the damped HONLS eqn. withǫ = 0.05,Γ=0.005 and0 < β < 0.75. The same relation between down-shifting and the last rogue wave are observed to hold, i.e.rogue waves do not occur after the downshifting is perma-nent. However, due to the additional linear damping there isa longer time lag between the last rogue wave and permanentdownshifting.

Fig. 11.The evolution ofkpeakfor the nonlinearly damped HONLSequation withε = 0.05, 0 = 0 and(a) β = 0.04 or (b) β = .5 forinitial data (17).

0< β < 0.75. Figure14a shows the comparison for initialdata (19) with a = 0.63: we observe that for all valuesof β rogue waves do not occur after the downshiftingbecomes irreversible. Further, rogue waves do not developafter permanent downshifting occurs for any other pair ofparameter values (a, β) considered in the experiments. Thisis summarized in Fig.14b which compares, for 0< β <

0.75, the average time of the last rogue wave (box) with theaverage time at which downshifting is irreversible (x), wherethe averages are over the six simulations with initial dataamplitude 0.57< a < 0.67. These results imply permanentdownshifting serves as an indicator that the cumulativeeffects of damping are sufficient to prevent the furtherdevelopment of rogue waves.

Finally we consider the case when both linear andnonlinear damping are present and the nonlinear dampingis dominant to allow permanent downshifting to occur.Figure 15 compares the times for permanent downshiftingand the last rogue wave for the damped HONLS equationwith ε = 0.05, 0 = 0.005 and 0< β < 0.75. The samerelation between downshifting and the last rogue wave areobserved to hold, i.e. rogue waves do not occur after thedownshifting is permanent. However, due to the additionallinear damping there is a longer time lag between the lastrogue wave and permanent downshifting.



0 5 10 15 20 25 30 35 40 45 501

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STR

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(a)

0 5 10 15 20 25 30 35 40 45 500

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(b)

Fig. 12: Nonlinearly damped HONLS eq.ǫ = 0.05, β = 0.5,Γ = 0.005: the time evolution of (a) the strengthS(t), (b) themain Fourier modes, for initial data (17).

5 Damped random oceanic sea states

In this section we examine the generation of rogue waves inthe presence of damping for sea states characterized by theJoint North Sea Wave Project (JONSWAP) spectrum:

S(f)=αg2

(2πf)5e−

5

4 (f0

f )4

γe−

1

2 (f−f0σf0

)2

, σ =

{

0.07 f ≤ f0

0.09 f > f0

Heref is spatial frequency,fn = kn/2π, f0 is the dominantfrequency determined by the wind speed at a specified heightabove the sea surface andg is gravity. Developing storm dy-namics are governed by this spectrum for a range of param-etersγ andα. Hereγ is the peakedness parameter andαis related to the amplitude and energy content. The putativeregion of validity for the NLS equation and its higher ordergeneralizations is2 < γ < 8 and0.008 < α< 0.02

The initial data for the surface elevation is of the form(Onorato et al., 2001)

η(x,0)=

N∑

n=1

Cncos(knx−φn), (20)

wherekn = 2πn/L, the random phasesφn are uniformlydistributed on(0,2π) and the spectral amplitudesCn =

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.25

−0.2

−0.15

−0.1

−0.05

0

β

Spec

tral C

ente

r

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

β

Max

imum

Stre

ngth

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

2

3

4

5

6

7

8

9

β

Num

ber o

f Rog

ue W

aves

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

Tim

e of

Las

t Rog

ue W

ave

β

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(d)

Fig. 13: The HONLS eqn. with nonlinear damping0 < β <0.75, u0 = a(1+0.01cosµx), 0.57 < a < 0.67, (a) spectralcenterkm at t = 200, (b) maxt∈[0,200]S(t), (c) number ofrogue waves, and (d) time of last rogue wave as a functionof β. The solid curves represent the averages.

Fig. 12. Nonlinearly damped HONLS equationε = 0.05, β = 0.5,0 = 0.005: the time evolution of(a) the strengthS(t), (b) the mainFourier modes, for initial data (17).

0 5 10 15 20 25 30 35 40 45 501

1.2

1.4

1.6

1.8

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2.4

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ENG

TH

TIME

(a)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES

TIME

(b)




S(f)=αg2

(2πf)5e−

5

4 (f0

f )4

γe−

1

2 (f−f0σf0

)2

, σ =

{

0.07 f ≤ f0

0.09 f > f0



η(x,0)=

N∑

n=1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.25

−0.2

−0.15

−0.1

−0.05

0

Spec

tral C

ente

r

β

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

β

Max

imum

Stre

ngth

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

2

3

4

5

6

7

8

9

β

Num

ber o

f Rog

ue W

aves

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

Tim

e of

Las

t Rog

ue W

ave

β

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(d)


Fig. 13. The HONLS equation with nonlinear damping 0< β <

0.75, u0 = a(1+0.01cosµx), 0.57< a < 0.67, (a) spectral centerkm at t = 200, (b) maxt∈[0,200]S(t), (c) number of rogue waves,and(d) time of last rogue wave as a function ofβ. The solid curvesrepresent the averages.

0 5 10 15 20 25 30 35 40 45 501

1.2

1.4

1.6

1.8

2

2.2

2.4

STR

ENG

TH

TIME

(a)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES

TIME

(b)




S(f)=αg2

(2πf)5e−

5

4 (f0

f )4

γe−

1

2 (f−f0σf0

)2

, σ =

{

0.07 f ≤ f0

0.09 f > f0



η(x,0)=

N∑

n=1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

(b)

β

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

Max

imum

Stre

ngth

β9

a = 0.57a = 0.59a = 0.61

−0.25

−0.2

−0.15

−0.1

−0.05

0

Spec

tral C

ente

r

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

2

3

4

5

6

7

8

β

Num

ber o

f Rog

ue W

aves

a = 0.63a = 0.65a = 0.67

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

Tim

e of

Las

t Rog

ue W

ave

β

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(d)


0 5 10 15 20 25 30 35 40 45 501

1.2

1.4

1.6

1.8

2

2.2

2.4

STR

ENG

TH

TIME

(a)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES

TIME

(b)




S(f)=αg2

(2πf)5e−

5

4 (f0

f )4

γe−

1

2 (f−f0σf0

)2

, σ =

{

0.07 f ≤ f0

0.09 f > f0



η(x,0)=

N∑

n=1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.25

−0.2

−0.15

−0.1

−0.05

0

β

Spec

tral C

ente

r

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

β

Max

imum

Stre

ngth

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

2

3

4

5

6

7

8

9

Num

ber o

f Rog

ue W

aves

β

(c)

150a = 0.57

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

Tim

e of

Las

t Rog

ue W

ave

β

a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(d)


0 5 10 15 20 25 30 35 40 45 501

1.2

1.4

1.6

1.8

2

2.2

2.4

STR

ENG

TH

TIME

(a)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES

TIME

(b)




S(f)=αg2

(2πf)5e−

5

4 (f0

f )4

γe−

1

2 (f−f0σf0

)2

, σ =

{

0.07 f ≤ f0

0.09 f > f0



η(x,0)=

N∑

n=1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.25

−0.2

−0.15

−0.1

−0.05

0

β

Spec

tral C

ente

r

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

β

Max

imum

Stre

ngth

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

2

3

4

5

6

7

8

9

β

Num

ber o

f Rog

ue W

aves

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

Tim

e of

Las

t Rog

ue W

ave

β

(d)


0 5 10 15 20 25 30 35 40 45 501

1.2

1.4

1.6

1.8

2

2.2

2.4

STR

ENG

TH

TIME

(a)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES

TIME

(b)




S(f)=αg2

(2πf)5e−

5

4 (f0

f )4

γe−

1

2 (f−f0σf0

)2

, σ =

{

0.07 f ≤ f0

0.09 f > f0



η(x,0)=

N∑

n=1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.25

−0.2

−0.15

−0.1

−0.05

0

β

Spec

tral C

ente

r

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

β

Max

imum

Stre

ngth

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

2

3

4

5

6

7

8

9

β

Num

ber o

f Rog

ue W

aves

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

Tim

e of

Las

t Rog

ue W

ave

β

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(d)


0 5 10 15 20 25 30 35 40 45 501

1.2

1.4

1.6

1.8

2

2.2

2.4

STR

ENG

TH

TIME

(a)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FOU

RIE

R M

OD

ES

TIME

(b)




S(f)=αg2

(2πf)5e−

5

4 (f0

f )4

γe−

1

2 (f−f0σf0

)2

, σ =

{

0.07 f ≤ f0

0.09 f > f0



η(x,0)=

N∑

n=1



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−0.25

−0.2

−0.15

−0.1

−0.05

0

β

Spec

tral C

ente

r

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

β

Max

imum

Stre

ngth

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

2

3

4

5

6

7

8

9

β

Num

ber o

f Rog

ue W

aves

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

Tim

e of

Las

t Rog

ue W

ave

β

a = 0.57a = 0.59a = 0.61a = 0.63a = 0.65a = 0.67

(d)


Fig. 13. Continued.



S(f ) =αg2

(2πf )5e−

54

(f0f

)4

γ e−

12

(f −f0σf0

)2

, σ =

{0.07 f ≤ f0

0.09 f >f0

Heref is spatial frequency,fn = kn/2π , f0 is the dominant



0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(b)

Fig. 14: The nonlinear damped HONLS eqn. withǫ =0.05,Γ= 0 and0 < β < 0.75. The time downshifting is irre-versible (x) and the time the last rogue wave occurs (box) asa function ofβ, u0 = a(1+0.01cosµx), where (a)a = 0.63and (b) averaged over0.57 < a < 0.67.

√

S(fn)/2L. The relation ofη to the solution of thenonlinear Schrodinger equationu(x,t) is given by η =

1√2k

Re{

iueikx}

.

5.1 Nonlinear spectral diagnostic for determining roguewaves

In previous work we proposed a nonlinear spectral decompo-sition of the JONSWAP data and introduced the splitting dis-tance between two simple points,δ(λ+,λ−) = |λ+−λ−| tomeasure the proximity to homoclinic data of the NLS equa-tion (Islas & Schober, 2005; Schober, 2006). Briefly, ho-moclinic solutions arise as an appropriate degeneration ofafinite gap solution, i.e. asδ(λ+,λ−)→ 0 the resulting doublepoint is complex. Such a potential possesses no linear unsta-ble modes (simple points and real double points are in generalassociated with neutrally stable modes), although the solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(b)

Fig. 15: The damped HONLS eqn. withǫ = 0.05,Γ= 0.005and0 < β < 0.75. The time downshifting is irreversible (x)and the time the last rogue wave occurs (box) as a function ofβ andΓ = 0.005, u0 = a(1+0.01cosµx), where (a)a = 0.63and (b) averaged over0.57 < a < 0.67.

is “close” in the spectral sense to an unstable solution. Con-sequently,δ measures the proximity in the spectral plane tocomplex double points and their corresponding instabilitiesand can be used as a criterium for predicting the strength andoccurrence of rogue waves (Islas & Schober, 2005).

We begin by determining the nonlinear spectrum of JON-SWAP initial data given by (20) for combinations ofα =0.008,0.012,0.016,0.02, andγ = 1,2,4,6,8. For each suchpair (γ,α), fifty simulations were performed, each with adifferent set of randomly generated phases. As expected,the spectral configuration depends on the energyα and thepeakednessγ. However, the phase information is just as im-portant to the final determination of the spectrum. Typical ex-amples of the numerically computed nonlinear spectrum forJONSWAP initial data (γ = 4, α = 0.016) with two differentrealizations of the random phases are shown in Figs. 16a and17a. Since the NLS spectrum is symmetric with respect to the

Fig. 14. The nonlinear damped HONLS equation withε =

0.05,0 = 0 and 0< β < 0.75. The time downshifting is irreversible(x) and the time the last rogue wave occurs (box) as a function ofβ,u0 = a(1+0.01cosµx), where(a) a = 0.63 and(b) averaged over0.57< a < 0.67.

frequency determined by the wind speed at a specified heightabove the sea surface andg is gravity. Developing stormdynamics are governed by this spectrum for a range ofparametersγ andα. Hereγ is the peakedness parameter andα is related to the amplitude and energy content. The putativeregion of validity for the NLS equation and its higher ordergeneralizations is 2< γ < 8 and 0.008< α < 0.02


η(x,0) =

N∑n=1

Cncos(knx −φn), (20)

where kn = 2πn/L, the random phasesφn are uniformlydistributed on(0,2π) and the spectral amplitudesCn =√

S(fn)/2L. The relation ofη to the solution of the nonlinearSchrodinger equationu(x,t) is given byη = 1

√2k

Re{iueikx

}.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(b)

Fig. 14: The nonlinear damped HONLS eqn. withǫ =0.05,Γ= 0 and0 < β < 0.75. The time downshifting is irre-versible (x) and the time the last rogue wave occurs (box) asa function ofβ, u0 = a(1+0.01cosµx), where (a)a = 0.63and (b) averaged over0.57 < a < 0.67.

√

S(fn)/2L. The relation ofη to the solution of thenonlinear Schrodinger equationu(x,t) is given by η =

1√2k

Re{

iueikx}

.


In previous work we proposed a nonlinear spectral decompo-sition of the JONSWAP data and introduced the splitting dis-tance between two simple points,δ(λ+,λ−) = |λ+−λ−| tomeasure the proximity to homoclinic data of the NLS equa-tion (Islas & Schober, 2005; Schober, 2006). Briefly, ho-moclinic solutions arise as an appropriate degeneration ofafinite gap solution, i.e. asδ(λ+,λ−)→ 0 the resulting doublepoint is complex. Such a potential possesses no linear unsta-ble modes (simple points and real double points are in generalassociated with neutrally stable modes), although the solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

20

40

60

80

100

120

140

160

180

200

β

Tim

e

Last RogueDS onset

(b)

Fig. 15: The damped HONLS eqn. withǫ = 0.05,Γ= 0.005and0 < β < 0.75. The time downshifting is irreversible (x)and the time the last rogue wave occurs (box) as a function ofβ andΓ = 0.005, u0 = a(1+0.01cosµx), where (a)a = 0.63and (b) averaged over0.57 < a < 0.67.

is “close” in the spectral sense to an unstable solution. Con-sequently,δ measures the proximity in the spectral plane tocomplex double points and their corresponding instabilitiesand can be used as a criterium for predicting the strength andoccurrence of rogue waves (Islas & Schober, 2005).

We begin by determining the nonlinear spectrum of JON-SWAP initial data given by (20) for combinations ofα =0.008,0.012,0.016,0.02, andγ = 1,2,4,6,8. For each suchpair (γ,α), fifty simulations were performed, each with adifferent set of randomly generated phases. As expected,the spectral configuration depends on the energyα and thepeakednessγ. However, the phase information is just as im-portant to the final determination of the spectrum. Typical ex-amples of the numerically computed nonlinear spectrum forJONSWAP initial data (γ = 4, α = 0.016) with two differentrealizations of the random phases are shown in Figs. 16a and17a. Since the NLS spectrum is symmetric with respect to the

Fig. 15. The damped HONLS eqn. withε = 0.05, 0 = 0.005 and0< β < 0.75. The time downshifting is irreversible (x) and the timethe last rogue wave occurs (box) as a function ofβ and0 = 0.005,u0 = a(1+0.01cosµx), where(a) a = 0.63 and(b) averaged over0.57< a < 0.67.


In previous work we proposed a nonlinear spectral decom-position of the JONSWAP data and introduced the splittingdistance between two simple points,δ(λ+,λ−) = |λ+ −λ−|

to measure the proximity to homoclinic data of the NLSequation (Islas and Schober, 2005; Schober, 2006). Briefly,homoclinic solutions arise as an appropriate degenerationof a finite gap solution, i.e. asδ(λ+,λ−) → 0 the resultingdouble point is complex. Such a potential possesses nolinear unstable modes (simple points and real double pointsare in general associated with neutrally stable modes),although the solution is “close” in the spectral sense to anunstable solution. Consequently,δ measures the proximityin the spectral plane to complex double points and their



.0

0.2

0.1

0.3

0.4

0.5

0.6

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.40 0.5

..

..

.

.

λλ+

_

(a)

0 10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

UM

AX

& 2

.2*S

H

(b)

Fig. 16: (a) Spectrum and (b) evolution ofUmax “near” tohomoclinic data.

real axis only the spectrum in the upper half complexλ-planeis displayed.

We find that JONSWAP data may correspond to “semi-stable” solutions, i.e. JONSWAP data can be viewed as per-turbations ofN -phase solutions with one or more unstablemodes (compare Fig. 16a with Fig. 1a, the spectrum of anunstableN -phase solution). In Fig. 16a the splitting dis-tanceδ(λ+,λ−)≈ 0.07, while in Fig. 17aδ(λ+,λ−)≈ 0.2.Thus, for fixedα andγ as the phases are randomly varied,the spectral distanceδ of typical JONSWAP data from ho-moclinic data varies significantly and can be quite “near” asin Fig. 16a, or “far” from homoclinic data as in Fig. 17a.

The most striking feature is that irrespective of the valuesof α andγ, in simulations of the NLS equation (10) extremewaves develop for JONSWAP initial data that is “near” NLShomoclinic data, whereas the JONSWAP data that is “far”from NLS homoclinic data typically does not generate ex-treme waves. Figs. 16b and 17b show the corresponding evo-lution of Umax (Umax = maxx∈[0,L]|u(x,t)|), obtained withthe NLS equation.Umax is given by the solid curve andas a reference the threshold for a rogue wave,2.2 times thethe significant wave height,2.2HS, is given by the dashed

.

....

..

0

0.2

0.1

0.3

0.4

0.5

0.6

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.40 0.5

λ

λ+

_

(a)

0 10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

UM

AX

& 2

.2*S

H

(b)

Fig. 17: (a) Spectrum and (b) evolution ofUmax “far” fromhomoclinic data.

curve. Fig. 16b shows that when the nonlinear spectrum isnear homoclinic data,Umax exceeds2.2HS (a rogue wavedevelops att≈ 40). Fig. 17b shows that when the nonlinearspectrum is far from homoclinic data,Umax is significantlybelow2.2HS and a rogue wave does not develop. As a resultwe can correlate the occurrence of rogue waves characterizedby JONSWAP spectrum with the proximity to homoclinic so-lutions of the NLS equation.

5.2 JONSWAP experiments with the NLS and HONLSequations

The results of hundreds of simulations of the NLS andHONLS equations consistently show that proximity to ho-moclinic data is a crucial indicator of rogue wave events.Figure 18 provides a synthesis of 250 random simula-tions of (a) the NLS equation and of (b-c) the HONLSequation (β = Γ = 0) for ǫ = 0.025 and ǫ = 0.05, respec-tively, using JONSWAP initial data for different(γ,α) pairswith γ = 1,2,4,6,8 andα = 0.008,0.012,0.016,0.02 and ran-domly generated phases. Our considerations are restrictedto semi-stableN -phase solutions near to unstable solutions

Fig. 16. (a)Spectrum and(b) evolution ofUmax “near” tohomoclinic data.

corresponding instabilities and can be used as a criterium forpredicting the strength and occurrence of rogue waves (Islasand Schober, 2005).

We begin by determining the nonlinear spectrum ofJONSWAP initial data given by (20) for combinations ofα = 0.008,0.012,0.016,0.02, andγ = 1,2,4,6,8. For eachsuch pair (γ , α), fifty simulations were performed, each witha different set of randomly generated phases. As expected,the spectral configuration depends on the energyα and thepeakednessγ . However, the phase information is just asimportant to the final determination of the spectrum. Typicalexamples of the numerically computed nonlinear spectrumfor JONSWAP initial data (γ = 4, α = 0.016) with twodifferent realizations of the random phases are shown inFigs. 16a and17a. Since the NLS spectrum is symmetricwith respect to the real axis only the spectrum in the upperhalf complexλ-plane is displayed.

.0

0.2

0.1

0.3

0.4

0.5

0.6

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.40 0.5

..

..

.

.

λλ+

_

(a)

0 10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

UM

AX

& 2

.2*S

H

(b)

Fig. 16: (a) Spectrum and (b) evolution ofUmax “near” tohomoclinic data.

real axis only the spectrum in the upper half complexλ-planeis displayed.

We find that JONSWAP data may correspond to “semi-stable” solutions, i.e. JONSWAP data can be viewed as per-turbations ofN -phase solutions with one or more unstablemodes (compare Fig. 16a with Fig. 1a, the spectrum of anunstableN -phase solution). In Fig. 16a the splitting dis-tanceδ(λ+,λ−)≈ 0.07, while in Fig. 17aδ(λ+,λ−)≈ 0.2.Thus, for fixedα andγ as the phases are randomly varied,the spectral distanceδ of typical JONSWAP data from ho-moclinic data varies significantly and can be quite “near” asin Fig. 16a, or “far” from homoclinic data as in Fig. 17a.

The most striking feature is that irrespective of the valuesof α andγ, in simulations of the NLS equation (10) extremewaves develop for JONSWAP initial data that is “near” NLShomoclinic data, whereas the JONSWAP data that is “far”from NLS homoclinic data typically does not generate ex-treme waves. Figs. 16b and 17b show the corresponding evo-lution of Umax (Umax = maxx∈[0,L]|u(x,t)|), obtained withthe NLS equation.Umax is given by the solid curve andas a reference the threshold for a rogue wave,2.2 times thethe significant wave height,2.2HS, is given by the dashed

.

....

..

0

0.2

0.1

0.3

0.4

0.5

0.6

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.40 0.5

λ

λ+

_

(a)

0 10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

UM

AX

& 2

.2*S

H

(b)

Fig. 17: (a) Spectrum and (b) evolution ofUmax “far” fromhomoclinic data.

curve. Fig. 16b shows that when the nonlinear spectrum isnear homoclinic data,Umax exceeds2.2HS (a rogue wavedevelops att≈ 40). Fig. 17b shows that when the nonlinearspectrum is far from homoclinic data,Umax is significantlybelow2.2HS and a rogue wave does not develop. As a resultwe can correlate the occurrence of rogue waves characterizedby JONSWAP spectrum with the proximity to homoclinic so-lutions of the NLS equation.


The results of hundreds of simulations of the NLS andHONLS equations consistently show that proximity to ho-moclinic data is a crucial indicator of rogue wave events.Figure 18 provides a synthesis of 250 random simula-tions of (a) the NLS equation and of (b-c) the HONLSequation (β = Γ = 0) for ǫ = 0.025 and ǫ = 0.05, respec-tively, using JONSWAP initial data for different(γ,α) pairswith γ = 1,2,4,6,8 andα = 0.008,0.012,0.016,0.02 and ran-domly generated phases. Our considerations are restrictedto semi-stableN -phase solutions near to unstable solutions

Fig. 17. (a) Spectrum and(b) evolution of Umax “far” fromhomoclinic data.

We find that JONSWAP data may correspond to “semi-stable” solutions, i.e. JONSWAP data can be viewed asperturbations of N-phase solutions with one or more unstablemodes (compare Fig.16a with Fig. 1a, the spectrum ofan unstable N-phase solution). In Fig.16a the splittingdistanceδ(λ+,λ−) ≈ 0.07, while in Fig. 17a δ(λ+,λ−) ≈

0.2. Thus, for fixedα and γ as the phases are randomlyvaried, the spectral distanceδ of typical JONSWAP data fromhomoclinic data varies significantly and can be quite “near”as in Fig.16a, or “far” from homoclinic data as in Fig.17a.

The most striking feature is that irrespective of the valuesof α andγ , in simulations of the NLS equation (10) extremewaves develop for JONSWAP initial data that is “near”NLS homoclinic data, whereas the JONSWAP data that is“far” from NLS homoclinic data typically does not generateextreme waves. Figures16b and17b show the correspondingevolution of Umax (Umax = maxx∈[0,L]|u(x,t)|), obtainedwith the NLS equation.Umax is given by the solid curveand as a reference the threshold for a rogue wave, 2.2 times



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Fig. 18: Maximum strength versus the splitting distanceδ(λ+,λ−) for a) the NLS equation and the HONLS equationwith b) ǫ = 0.025, and c)ǫ = 0.05 .

of the NLS with one UM. Each circle represents the maxi-mum strength, maxt∈[0,20]S(t), attained in time during onesimulation, as a function of the splitting distanceδ(λ+,λ−).A horizontal line atUmax/HS = 2.2 indicates the referencestrength for rogue wave formation. We identify two criti-cal valuesδ1(ǫ) andδ2(ǫ) that clearly show that (i) ifδ < δ1

(near homoclinic data) the likelihood of a rogue wave is ex-tremely high; (ii) if δ1 < δ < δ2, the likelihood of obtainingrogue waves decreases asδ increases and, (iii) ifδ > δ2 thelikelihood of a rogue wave occurring is extremely small.

As α andγ are varied the behavior is robust. The maxi-mum wave strength and the occurrence of rogue waves arewell predicted by the proximity to homoclinic solutions. Theindividual plots of the strength vs. the splitting distanceδfor particular pairs(γ,α) are qualitatively the same regard-less of the pair chosen. Comparing Figs. 18a-b we find thatenhanced focusing and increased wave strength occur in thechaotic HONLS evolution as compared with the NLS evo-lution. Figure 18 shows that as the perturbation strengthǫincreases, the maximum wave strength and the likelihood ofrogue waves occuring in a given simulation increases.

5.3 JONSWAP experiments with the damped HONLSequations

In this section we show that, in the presence of damping, thenonlinear spectrum and the proximity to homoclinic data islikewise an important indicator of the maximum strength intime and likelihood of rogue waves.

Figure 19 shows the maximum wave strength vs. thesplitting distanceδ(λ+,λ−) for 250 random simulations ofthe damped HONLS equation (ǫ = 0.05), with either (a)linear damping,β = 0,Γ = 0.025, or with (b) nonlineardampingβ = 0.5,Γ = 0. As before, JONSWAP initial datawas used for different(γ,α) pairs with γ = 1,2,4,6,8 andα = 0.008,0.012,0.016,0.02and randomly generated phases.Each circle represents the maximum strength attained duringone simulation,0 < t < 20.

In Fig. 19 it is evident that the strength and likelihoodof rogue waves occuring in a given simulation is typicallysmaller when damping is present (compare to results of theHONLS equation (Fig. 18c). A cutoff vaue ofδ exists suchthat rogue waves typically do not occur ifδ > δcutoff . Wedetermineδcutoff by requiring that95% of the rogue wavesoccur forδ < δcutoff . For example, the cutoff values for thelinear (Γ =0.02) and nonlinear (ǫβ = 0.02) damped HONLSequation areδdamped

cutoff ≈ 0.16,0.17, respectively, which areless than the cutoff value for the undamped HONLS equationwhere,δundamped

cutoff ≈ 0.2. Significantly, this implies that whendamping is present the JONSWAP initial data must be closerto homoclinic data and instabilities to obtain rogue waves.

Figure 20 showsδdampedcutoff as a function of the damping

parametersΓ (circles) andǫβ (boxes). For each of the tenvalues ofΓ and of ǫβ, 0 < Γ,ǫβ < 0.04, 250 simulations

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Fig. 18. Maximum strength versus the splitting distanceδ(λ+,λ−)

for (a) the NLS equation and the HONLS equation with(b)ε=0.025, and(c) ε = 0.05.

the the significant wave height, 2.2Hs, is given by thedashed curve. Figure16b shows that when the nonlinearspectrum is near homoclinic data,Umax exceeds 2.2Hs (arogue wave develops att ≈ 40). Figure17b shows thatwhen the nonlinear spectrum is far from homoclinic data,Umax is significantly below 2.2Hs and a rogue wave doesnot develop. As a result we can correlate the occurrence ofrogue waves characterized by JONSWAP spectrum with theproximity to homoclinic solutions of the NLS equation.


The results of hundreds of simulations of the NLS andHONLS equations consistently show that proximity tohomoclinic data is a crucial indicator of rogue wave events.Figure 18 provides a synthesis of 250 random simulationsof (a) the NLS equation and of (b–c) the HONLS equation(β = 0 = 0) for ε = 0.025 andε = 0.05, respectively, usingJONSWAP initial data for different (γ , α) pairs with γ =

1,2,4,6,8 and α = 0.008,0.012,0.016,0.02 and randomlygenerated phases. Our considerations are restricted tosemi-stable N-phase solutions near to unstable solutionsof the NLS with one UM. Each circle represents themaximum strength, maxt∈[0,20]S(t), attained in time duringone simulation, as a function of the splitting distanceδ(λ+,λ−). A horizontal line atUmax/Hs = 2.2 indicates thereference strength for rogue wave formation. We identifytwo critical valuesδ1(ε) andδ2(ε) that clearly show that (i)if δ < δ1 (near homoclinic data) the likelihood of a roguewave is extremely high; (ii) ifδ1 < δ < δ2, the likelihood ofobtaining rogue waves decreases asδ increases and, (iii) ifδ > δ2 the likelihood of a rogue wave occurring is extremelysmall.

As α and γ are varied the behavior is robust. Themaximum wave strength and the occurrence of rogue wavesare well predicted by the proximity to homoclinic solutions.The individual plots of the strength vs. the splitting distanceδ for particular pairs (γ , α) are qualitatively the sameregardless of the pair chosen. Comparing Fig.18a–b wefind that enhanced focusing and increased wave strengthoccur in the chaotic HONLS evolution as compared with theNLS evolution. Figure18 shows that as the perturbationstrengthε increases, the maximum wave strength and thelikelihood of rogue waves occuring in a given simulationincreases.



Figure 19 shows the maximum wave strength vs. thesplitting distanceδ(λ+,λ−) for 250 random simulationsof the damped HONLS equation (ε = 0.05), with either(a) linear damping,β = 0, 0 = 0.025, or with (b) nonlineardampingβ = 0.5,0 = 0. As before, JONSWAP initial datawas used for different (γ , α) pairs with γ = 1,2,4,6,8and α = 0.008,0.012,0.016,0.02 and randomly generatedphases. Each circle represents the maximum strengthattained during one simulation, 0< t < 20.



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Fig. 19: Maximum strength vs. splitting distanceδ(λ+,λ−)for the HONLS equation (ǫ = 0.05) with either (a) lineardamping,β = 0, Γ = 0.025, or (b) nonlinear dampingβ =0.5, Γ = 0.

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Fig. 20: The damped HONLS equation (ǫ = 0.05): δdampedcutoff

as a function ofΓ (circles) andǫβ (boxes)

of the damped HONLS equation were carried out. We findthat δdamped

cutoff is generally decreasing as the strength of thedamping increases. The cutoff value inδ for the nonlin-early damped case is not monotonically decreasing as atyp-ical cases exist where a small increase in damping may beoffset by a coalescence of the modes due to changes in thefocusing times. The overall decay inδdamped

cutoff indicates thatfor rogue waves to occur the JONSWAP initial data must liein a shrinking neighborhood of the homoclinic data. Thusthe proximity to instabilities and homoclinic data is more es-sential for the development of rogue waves when damping ispresent.

The observations from the nonlinear spectral diagnosticcomplement the earlier results characterizing the nonlineardamped evolution. In section 4, the numerical simulationsstart very close to homolinic data (for initial data (19) thesplitting distance is on the order of10−6, an effective roguewave regime). The first pair of rogue waves were not sig-nificantly altered by the damping. However after permanentdownshifting occured any further rogue waves were inhib-ited. Here we are generalizing to JONSWAP initial datawhich are not necessarily close to homoclinic data. We aremonitoring the maximum strength in time as a function ofδ which is a separate issue from the total number of roguewaves in a given simulation.

Finally we remark that we recently examined rogue wavesand downshifting in the damped Dysthe equation (Islas &Schober, 2010). While the Dysthe equation does not con-serve momentum, we have obtained qualitatively similar re-sults. For the nonlinearly damped Dysthe equation, irre-versible downshifting also occurs for all values ofβ, althoughon a slightly longer timescale. Rogue waves do not occur inthe time series after the downshifting is permanent. Similarly,δdampedcutoff generally decreases as the strength of the damping

increases suggesting the heightened importance of proxim-ity to homoclinic data when obtaining rogue waves in thedamped Dysthe regime.

References

Ablowitz, M.J., Hammack, J., Henderson, D., Schober, C.M.(2000). Modulated Periodic Stokes Waves in Deep Water. Phys.Rev. Lett. 84:887–890.

Ablowitz, M.J., Hammack, J., Henderson, D., Schober, C.M.(2001). Long time dynamics of the modulational instability ofdeep water waves. Physica D 152-153:416–433.

Ablowitz, M.J., Segur, H. (1981). Solitons and the Inverse Scatter-ing Transform. SIAM.

Calini, A., Ercolani, N.M., McLaughlin, D.W., Schober, C.M.(1996). Mel’nikov analysis of numerically induced chaos in thenonlinear Schrodinger equation. Physica D 89:227–260.

Calini, A., Schober, C.M. (2002). Homoclinic chaos increases thelikelihood of rogue waves. Phys. Lett. A 298:335–349.

Cox, S.M., Matthews, P.C. (2002). Exponential time differencingfor stiff systems. Journal of Computational Physics 176:430–455.

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0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040.1

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0.25

Γ or εβ

δ cuto

ff

Γβ









References


Ablowitz, M.J., Hammack, J., Henderson, D., Schober, C.M.(2001). Long time dynamics of the modulational instability ofdeep water waves. Physica D 152-153:416–433.


Calini, A., Ercolani, N.M., McLaughlin, D.W., Schober, C.M.(1996). Mel’nikov analysis of numerically induced chaos in thenonlinear Schrodinger equation. Physica D 89:227–260.



Fig. 19. Maximum strength vs. splitting distanceδ(λ+,λ−) for theHONLS equation (ε = 0.05) with either(a) linear damping,β = 0,0 = 0.025, or(b) nonlinear dampingβ = 0.5, 0 = 0.

In Fig. 19 it is evident that the strength and likelihoodof rogue waves occuring in a given simulation is typicallysmaller when damping is present (compare to results of theHONLS equation, Fig.18c). A cutoff vaue ofδ exists suchthat rogue waves typically do not occur ifδ > δcutoff. Wedetermineδcutoff by requiring that 95% of the rogue wavesoccur forδ < δcutoff. For example, the cutoff values for thelinear (0 = 0.02) and nonlinear (εβ = 0.02) damped HONLSequation areδdamped


cutoff ≈ 0.2. Significantly, this implies that whendamping is present the JONSWAP initial data must becloser to homoclinic data and instabilities to obtain roguewaves.


parameters0 (circles) andεβ (boxes). For each of the tenvalues of0 and of εβ, 0< 0, εβ < 0.04, 250 simulationsof the damped HONLS equation were carried out. We findthat δ

dampedcutoff is generally decreasing as the strength of the

damping increases. The cutoff value inδ for the nonlinearlydamped case is not monotonically decreasing as atypicalcases exist where a small increase in damping may be

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References


Ablowitz, M.J., Hammack, J., Henderson, D., Schober, C.M.(2001). Long time dynamics of the modulational instabilityofdeep water waves. Physica D 152-153:416–433.


Calini, A., Ercolani, N.M., McLaughlin, D.W., Schober, C.M.(1996). Mel’nikov analysis of numerically induced chaos inthenonlinear Schrodinger equation. Physica D 89:227–260.



Fig. 20. The damped HONLS equation (ε = 0.05): δdampedcutoff as a

function of0 (circles) andεβ (boxes).

offset by a coalescence of the modes due to changes in thefocusing times. The overall decay inδdamped

cutoff indicates thatfor rogue waves to occur the JONSWAP initial data must liein a shrinking neighborhood of the homoclinic data. Thusthe proximity to instabilities and homoclinic data is moreessential for the development of rogue waves when dampingis present.

The observations from the nonlinear spectral diagnosticcomplement the earlier results characterizing the nonlineardamped evolution. In Sect. 4, the numerical simulationsstart very close to homolinic data (for initial data (19)the splitting distance is on the order of 10−6, an effectiverogue wave regime). The first pair of rogue waves werenot significantly altered by the damping. However afterpermanent downshifting occured any further rogue waveswere inhibited. Here we are generalizing to JONSWAPinitial data which are not necessarily close to homoclinicdata. We are monitoring the maximum strength in time as afunction ofδ which is a separate issue from the total numberof rogue waves in a given simulation.

Finally we remark that we recently examined rogue wavesand downshifting in the damped Dysthe equation (Islasand Schober, 2010). While the Dysthe equation doesnot conserve momentum, we have obtained qualitativelysimilar results. For the nonlinearly damped Dysthe equation,irreversible downshifting also occurs for all values ofβ,although on a slightly longer timescale. Rogue waves do notoccur in the time series after the downshifting is permanent.Similarly, δ

dampedcutoff generally decreases as the strength of the

damping increases suggesting the heightened importance ofproximity to homoclinic data when obtaining rogue waves inthe damped Dysthe regime.

Acknowledgements.This work was partially supported by NSFGrant # NSF-DMS0608693.



Edited by: C. KharifReviewed by: two anonymous referees

References

Ablowitz, M. J., Hammack, J., Henderson, D., and Schober, C.M.: Modulated Periodic Stokes Waves in Deep Water, Phys. Rev.Lett., 84, 887–890, 2000.

Ablowitz, M. J., Hammack, J., Henderson, D., and Schober, C. M.:Long time dynamics of the modulational instability of deep waterwaves, Physica D, 152–153, 416–433, 2001.

Ablowitz, M. J. and Segur, H.: Solitons and the Inverse ScatteringTransform, SIAM, 1981.

Calini, A., Ercolani, N. M., McLaughlin, D. W., and Schober, C.M.: Mel’nikov analysis of numerically induced chaos in thenonlinear Schrodinger equation, Physica D, 89, 227–260, 1996.

Calini, A. and Schober, C. M.: Homoclinic chaos increases thelikelihood of rogue waves, Phys. Lett. A, 298, 335–349, 2002.

Cox, S. M. and Matthews, P. C.: Exponential time differencing forstiff systems, J. Comput. Phys., 176, 430–455, 2002.

Dysthe, K.: Note on a modification to the nonlinear Schrodingerequation for deep water, Proc. R. Soc. Lon. Ser.-A, 369, 105–114, 1979.

Ercolani, N., Forest, M. G., and McLaughlin, D. W.: Geometry ofthe Modulational Instability Part III: Homoclinic Orbits for thePeriodic Sine-Gordon Equation, Physica D, 43, 349–384, 1990.

Gramstad, O. and Trulsen, K.: Hamiltonian form of the modifiednonlinear Schrodinger equation for gravity waves on arbitrarydepth, J. Fluid Mech., in press, 2011.

Hammack, J. L., Henderson, D. M., and Segur, H.: Deep-waterwaves with persistent, two-dimensional surface patterns, J. FluidMech., 532, 1–51, 2005.

Hara, T. and Mei, C. C.: Frequency down-shift in narrowbandedsurface waves under the influence of wind, J. Fluid Mech., 230,429–477, 1991.

Hara, T. and Mei, C. C.: Wind effects on the nonlinear evolutionof slowly varying gravity-capillary waves, J. Fluid Mech., 267,221–250, 1994.

Henderson, K. L., Peregrine, D. H., and Dold, J. W.: Unsteady waterwave modulations: fully nonlinear solutions and comparisonwith the nonlinear Schrodinger equation, Wave Motion, 29, 341–361, 1999.

Huang, N., Long, S., and Shen, Z.: The mechanism for frequencydownshift in nonlienar wave evolution, Adv. Appl. Mech., 32,59–117, 1996.

Islas, A. and Schober, C. M.: Predicting rogue waves in randomoceanic sea states, Phys. Fluids, 17, 1–4, 2005.

Islas, A. and Schober, C. M.: Rogue waves, dissipation, anddownshifting, Physica D, submitted, 2010.

Kato, Y. and Oikawa, M.: Wave number down-shift in ModulatedWavetrain through a Nonlinear Damping effect, J. Phys. Soc.Jpn, 64, 4660–4669, 1995.

Khaliq, A. Q. M., Martin-Vaquero, J., Wada, B.A., and Yousuf,M.: Smoothing Schemes for Reaction-Diffusion Systems withNonsmooth Data, J. Comput. Appl. Math. 223, 374–386, 2009.

Kharif, C. and Pelinovsky, E.: Focusing of nonlinear wave groupsin deep water, JETP Lett., 73, 170–175, 2001.

Kharif, C. and Pelinovsky, E.: Physical mechanisms of the roguewave phenomenon, Eur. J. Mech. B-Fluid., 22, 603–634, 2004.

Krasitskii, V.: On reduced equations in the Hamiltonian theoryof weakly nonlinear surface waves, J. Fluid Mech., 272, 1–20,1994.

Lake, B., Yuen, H., Rungaldier, H., and Ferguson, W.: Nonlineardeep-water waves:theory and experiment. Part 2. Evolution of acontinuous wave train, J. Fluid Mech., 83, 49–74, 1977.

Matveev, V. B. and Salle, M. A.: Darboux Transformations andSolitons, Springer-Verlag, 1991.

McLaughlin, D. W. and Schober, C. M.: Chaotic and homo-clinic behavior for numerical discretizations of the nonlinearSchrodinger equation, Physica D, 57, 447–465, 1992.

Onorato, M., Osborne, A., Serio, M., and Bertone, S.: Freak wavein random oceanic sea states, Phys. Rev. Lett., 86, 5831–5834,2001.

Osborne, A., Onorato, M., and Serio, M.: The nonlinear dynamicsof rogue waves and holes in deep-water gravity wave trains,Phys. Lett. A, 275, 386–393, 2000.

Schober, C. M.: Melnikov analysis and inverse spectral analysis ofrogue waves in deep water, Eur. J. Mech. B-Fluids, 25(5), 602–620, doi:10.1016/j.euromechflu.2006.02.005, 2006.

Segur, H., Henderson, D., Carter, J., Hammack, J., Li, C., Pheiff,D., and Socha, K.: Stabilizing the Benjamin-Feir Instability, J.Fluid Mech., 539, 229–271, 2005.

Su, M.: Evolution of groups of gravity waves with moderate to highsteepness, Phys. Fluids, 25, 2167–2174, 1982.

Trulsen, K. and Dysthe, K.: Frequency down-shift through selfmodulation and breaking, in: Water Wave Kinematics, edited by:Torum, A. and Gudmestad, O. T., Kluwer 561–572, 1990.

Trulsen, K. and Dysthe, K.: Frequency down-shift in three-dimensional wave trains in a deep basin, J. Fluid Mech., 352,359–373, 1997a.

Trulsen, K. and Dysthe, K.: Freak waves – a three dimensional wavesimulation, in: Proceedings of the 21st Symposium on NavalHydrodynamics, edited by: Rood, E. P., Nat. Acad. Press, 550–560, 1997b.

Uchiyama, Y. and Kawahara, T.: A possible mechanism forfrequency down-shift in nonlinear wave modulation, WaveMotion, 20, 99–110, 1994.

Zakharov, V. E. and Shabat, A. B. Exact theory of two-dimensionalself-focusing and one-dimensional self-modulation of waves innonlinear media, Sov. Phys. JETP, 34, 62–69, 1972.


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Rogue waves and downshifting in the presence of damping

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