Gaussian variational ansatz in the problem of anomalous sea waves: Comparison with direct numerical simulations

0.76 MB·English

by  V. P. Ruban

Gaussian variational ansatz in the problem of anomalous sea waves: Comparison with
direct numerical simulations
∗
V. P. Ruban
Landau Institute for Theoretical Physics RAS, Moscow, Russia
(Dated: February 2, 2015)
The nonlinear dynamics of an obliquely oriented wave packet at sea surface is studied both
analytically and numerically for various initial parameters of the packet, in connection with the
problem of oceanic rogue waves. In the framework of Gaussian variational ansatz applied to the
5 corresponding (1+2D) hyperbolic nonlinear Schr¨odinger equation, a simplified Lagrangian system
1 of differential equations is derived, which determines the evolution of coefficients of the real and
0 imaginary quadratic forms appearing in the Gaussian. This model provides a semi-quantitative
2 descriptionfortheprocessofnonlinearspatio-temporalfocusing, whichisoneofthemostprobable
mechanismsofroguewaveformationinrandomwavefields. Thesystemisintegratedinquadratures,
n
which fact allows ustounderstandqualitativedifferencesbetween thelinear and nonlinearregimes
a
ofthefocusingofwavepacket. Comparison oftheGaussianmodelpredictionswithresultsofdirect
J
numerical simulation of fully nonlinear long-crested water waves is carried out.
0
3 Key words: oceanic rogue waves, nonlinear focusing, variational approximation.

]
PACSnumbers: 47.35.Bb,47.10.Df,02.30.Mv,92.10.Hm
n
y
d
I. INTRODUCTION Here the non-dimensional variables are in use: k A∗
- 0
→
u ψ, ω t t, 2k x x, √2k y y, where A(x,y,t) is
0 0 0
fl Anomalous waves at the ocean surface and in many the com→plex envelo→pe of the ma→in harmonic, k0 is the
. wave number, ω = 2π/T = √gk is the frequency of
s otherphysicalsystems(knownalsoasgiantwaves,rogue 0 0 0
c the carrier wave (the reference frame is moving with the
waves, freak waves) have been a popular subject of
i
s present-day scientific research (see, e.g., reviews [1–3], groupvelocityvgr =(1/2) g/k0alongxaxis). Although
y Eq.(1) has some disadvantages, which are not discussed
andreferencestherein). Themostfrequentlyusedmath-
h p
here, but it is more suitable for approximate analytical
p ematical model for this phenomenon is the (1+1D) non-
studies(becauseinallitsmodifiedvariants,somepseudo-
[ linear Schr¨odinger equation (NLSE) with the focusing
differentialoperatorsappearinevitably,difficulttocalcu-
nonlinearity. In particular, such equation describes the
1 late them “by hands”).
complex amplitude of the main harmonic of a quasi-
v
8 monochromaticweaklynonlinearsurfacewaveoverplane Thus, in the transverse direction the nonlinearity acts
8 potentialflowsofanidealfluid[4]. Duetononlinearself- inthedefocusingmanner,whileinthelongitudinaldirec-
6 focusing, the modulation instability develops in a suffi- tion it works for focusing of a wave packet. In addition,
7 ciently lengthy and high wave group [4, 5]. As the re- the linear dispersion makes its own defocusing contribu-
0 sult, a one-dimensional anomalous wave arises. Such a tion. Owing to the indicated factors, the behavior of a
.
1 scenarioisconfirmedbynumericalandlaboratoryexper- nonlineargroupofwavesonthetwo-dimensionalfreesur-
0 iments [6–11]. The full integrability makes the (1+1D) face of the fluid is more diverse as compared to the one-
5 NLSEattractiveforanalyticalstudiesandprovidesmany dimensional case. Depending on initial conditions, the
1
exactsolutionswhichdescribesomeimportantproperties nonlinearity can in some cases reinforce the linear focus-
:
v of real rogue waves (see, e.g., [12–16]). ing,butaswellitcandestroythefocusinginothercases.
Xi In the Nature however sea waves are rather far from Inthecaseofreinforcement,aroguewavearises. Besides
the one-dimensional model (see, e.g., [17–23]). When on that, anomalous waves can appear as a result of interac-
r
a thefreefluidsurfacetherearedisturbancesdependingon tion between previously formed coherent structures [20].
bothhorizontalcoordinates,thewavedynamicsbecomes However,fortheusualoceanicwavefieldswherethepres-
more complicated alreadyin the frameworkof the corre- ence of coherent structures is hardly probable, more ac-
sponding NLSE, not to speak about strongly nonlinear tual seems the scenario when anomalous waves rise due
regimes. First, (1+2D) NLSE is not integrable. Second, to occasional spatio-temporal focusing (see [24, 25], and
in the case of deep-water gravity waves, the spatial dif- references therein).
ferential operator in it appears hyperbolic: In general, the question about optimal conditions for
the nonlinearfocusingofwaterwavesis stillfarfrombe-
2iψ +ψ ψ + ψ 2ψ =0. (1) ing clear. It is only obvious that, similarly to the 1D
t xx yy
− | |
case, the initial wave group should be sufficiently high
and/or extensive. But an effect of geometric shape of
the packet,as well as phase modulation, on the develop-
∗Electronicaddress: [email protected] mentofanomalouswaveispoorlystudied. Intheabsence
2
of exact (1+2D) NLSE solutions, which could describe direction. Here it is necessary to note that for applica-
quantitatively all such processes, it seems reasonable to bilityofNLSE,theconditionofnarrowspectralwidthof
carry out approximate consideration of the dynamics of the wave packet should be satisfied. In our case it prac-
an idealized solitary wave packet characterized by a few tically means the following: (X,Y) > 10, (U,V) < 0.1.
time-dependent parameters. Thereby,the abovequalita- Equations of motion determining the∼temporal beh∼avior
tive reasons could be concretized, and a reference point of the unknown quantities X, Y, U, and V, are derived
couldbeputalongthedirectionofdevelopmentoffuture by substitution of the variationalansatz (3) into the La-
more accurate theory of three-dimensional rogue waves. grangianfunctional of the NLSE:
The purpose of this work is to investigate the nonlin-
ear dynamics of a wave packet in the framework of the = (iψ ψ∗ iψψ∗ ψ 2+ ψ 2+ ψ 4/2)dxdy. (4)
fullGaussianvariationalansatz,i.e. inthecasewhenthe L t − t −| x| | y| | |
Z
quadraticformundertheexponentialhasanoff-diagonal
InthiswayweobtainareducedLagrangian ˜depending
part. Such a packet has an elliptical shape, with main
L
axesorientedatsometime-dependent anglewithrespect on N, X(t), Y(t), U(t), and V(t). The action integral
to the coordinate axes. Contrary to the more widely ˜dt should be then variated. The main outcome of
L
knowndiagonalansatz (see, e.g., [26–34]), the full Gaus- thisstandardprocedureisthefollowing. Variationofthe
sian ansatz was previously applied to the elliptic NLSE Rreduced action gives in particular that U = X˙, V = Y˙.
only[35,36]. Thepresentworkfillsthisgapinthetheory. Finally we arrive at the following system of differential
Hereasystemofvariationalequationswillbederivedfor equations of the Newtonian type:
parameters of the ellipse, and its full integrability will
1 N 1 N
be demonstrated. The fact of integrability takes place, X¨ = , Y¨ = + . (5)
X3 − X2Y Y3 Y2X
first, because the third power of the NLSE nonlinearity
is accordedwith the twospatialdimensions, andsecond,
Note that equations (5) constitute a Lagrangian system
because Eq.(1) has a special integral of motion, namely with the Lagrangianfunctional L ˜/N, where
the hyperbolic analog of the angular momentum, ∝L
1 1 2N
i[y(ψxψ∗−ψψx∗)+x(ψyψ∗−ψψy∗)]dxdy =const. (2) 2L=X˙2−Y˙2− X2 + Y2 + XY , (6)
Z
sothekineticenergyisnotpositively-definite(thesecond
The knowledge of general structure of analytical so-
“particle” has a negative mass).
lutions of the variational model will allow us to under-
It should be said that equations of the type (5) and
standqualitativedifferences betweenthe linearandnon-
their generalizationsare activelyused inplasma physics,
linear regimesofthe wavepacketfocusing. Besides that,
in nonlinear optics, and in atomic physics to investigate
with the purpose of immediate testing of the Gaussian
behavior of NLSE and Gross-Pitaevskii equation solu-
model, a comparison between its predictions and the re-
tions in two and three spatial dimensions (see papers
sults of numerical simulations of fully nonlinear long-
[26–34], and references therein). In particular in work
crested waves will be carried out. We shall see that the
[27], analytical and some numerical solutions were pre-
agreementcanberathergoodevenforstronglynonlinear
sentednamelyforthehyperbolic(1+2D)NLSE,asinour
regimes, when the very NLSE is not applicable already.
case,but with suchparametervalues thatare typicalfor
nonlinear optics, not for water waves. In relation to the
problem of anomalous sea wave focusing, this simplified
II. VARIATIONAL ANSATZ
variational ansatz was applied only very recently in the
author’s work [37].
Let us first remind that in the simplest — diagonal Let us now turn our attention to the full Gaussian
— variant, the Gaussian ansatz describes an elliptical ansatz, i.e. when off-diagonal elements of the quadratic
wave packet having the symmetry axes coinciding with forms are involved. For further consideration, it is con-
the coordinate axes (see, e.g., [26–31]): venient to introduce linear combinations of the spatial
coordinates,correspondingto a hyperbolic rotationwith
4N x2 y2 Ux2 Vy2 the parameter χ (analogously to the elliptic NLSE case,
ψ = exp +i i +iφ . (3)
XY −2X2−2Y2 2X − 2Y but with the difference that there the rotationwas usual
r
h i trigonometric):
Besides the longitudinal size X(t) and transverse size
Y(t), such a packet is characterized by four more real x¯=xcoshχ+ysinhχ, y¯=xsinhχ+ycoshχ. (7)
quantities: U, V, N, and φ. Parameter N does not de-
pendontimebecause4πN = ψ 2dxdy isanexactinte- Instead of expression (3) we now write
| |
gralofmotionofNLSE,whiletheconjugatevariableφis
cyclic. ParametersU andV deRscribeaphasemodulation, 4N x¯2 y¯2 Ux¯2 Vy¯2
ψ = exp +i i iγx¯y¯+iφ .
withpositive/negativevaluesofU orV correspondingto XY −2X2−2Y2 2X − 2Y −
r
defocusing/focusing configurations respectively in x or y h (8i)
3
Strictly speaking, with χ = 0 the parameters X and III. INTEGRABILITY OF THE MODEL
6
Y are no longer the longitudinal and transverse sizes
of the packet, but we shall continue to call them so for An important advantage of system (11) is an exact
brevity. Parameter χ cannot be too large, otherwise the integrability, because here in analogy with the elliptic
wave packet becomes spectrally wide and goes beyond case, the separation of variables turns out to be pos-
the NLSE applicability domain. Practically,as it will be sible. Integrability of the variational approximation is
seen from the results of direct numerical simulation of typicalnamely for NLSE in 2D space, because in 3D the
nonlinear waveswith realistic parameters,the inequality reduced Lagrangian instead of 2N/(XY) would contain
χ <0.5 should take place. 2N˜/(XYZ),anditwoulddestroythehomogeneityofthe
| |N∼ote that Gaussian form (8) is one of exact solutions
equations. It is well known from the analytical mechan-
of the corresponding linear Schr¨odinger equation, with
ics that for integration of the above system one should
appropriatetemporaldependenciesoftheparametersap-
use the hyperbolic coordinates:
pearing there. Consequently, at small N the approxima-
tion(8)iscertainlyvalid. Inpractice,valuesN 2–4are X =Qsinhξ, Y =Qcoshξ. (14)
≈
of interest. In this case, generally speaking, only quali-
tative agreement of the results of variational model (8) These coordinates describe sector X < Y which is most
with the solutions of the NLSE (and, the more so, with interestinginapplicationtotheproblemofoceanicrogue
the completely nonlinear dynamics of waves on water) waves,because a length of their crest usually 3–10 times
canbe expected. Inmoredetail the comparisonbetween exceeds the wave length λ0 = 2π/k0, while the longitu-
the variationaland numerical solutions will be discussed dinal packet size in the minimum reaches approximately
some later. An undoubted benefit of this approxima- oneλ0 (thatcorrespondstodimensionlessvaluesXmin
∼
tion is that it provides a semiquantitative description of 6–10). The Lagrangian(11) takes the form
the process of spatio-temporal focusing, which is one of
F(ξ)
the mostprobable mechanismsof the formationof rogue 2L =Q2ξ˙2 Q˙2 , (15)
waves under real conditions (see [24, 25], and references M − − Q2
therein).
where function F(ξ) is defined by the formula
Havingsubstitutedthetrialfunction(8)anditspartial
derivatives (expressed it terms of x¯ and y¯) into the La- 4 4N M2
grangian(4),afterstraightforwardcalculationsweobtain F(ξ)= + . (16)
sinh2(2ξ) − sinh(2ξ) cosh2(2ξ)
that variable χ is cyclic, while the corresponding con-
served quantity is (taking into account that N =const)
The corresponding equations of motion are
M =γ(X2+Y2)=const. (9)
F(ξ)
Q¨+Qξ˙2+ = 0, (17)
Herewith, the temporal dependence of parameter χ is Q3
determined by the equation ′
F (ξ)
2QQ˙ξ˙+Q2ξ¨+ = 0. (18)
χ˙ =M(Y2 X2)/(X2+Y2)2. (10) 2Q2
−
The conserved quantity M is a hyperbolic analog of the After multiplying Eq.(18) by 2Q2ξ˙, we obtain in its left
angular momentum. As previously, we have U = X˙,
hand side a total time-derivative, so the integral of mo-
V = Y˙, but the dynamics of unknown functions X(t)
tion follows:
and Y(t) is described now by the following Lagrangian:
1 1 2N M2(X2 Y2) Q4ξ˙2+F(ξ)=C =const. (19)
2L =X˙2 Y˙2 + + + − . (11)
M − −X2 Y2 XY (X2+Y2)2 Looking at the expression (16), it is easy to understand
ThisLagrangianisahyperbolicanalogoftheellipticvari- that at sufficiently small negative values of the integra-
ational problem considered in works [35, 36]. Equations tion constant C, equation F(ξ) = C determining “turn
of motion that follow from here, have the form points”, has two real positive roots (ξ1 and ξ2), with
the motion of variable ξ taking place between them. At
X¨ = X13−XN2Y +(X2M+2XY2)2"1−2((XX22−+YY22))#, (12) CE≥q0.(1th7)ei“stusirmnppliofiinedt”tiosQs¨ol=e. C/Q3. Its general solu-
−
tion is written in explicit form as
Y¨ = 1 + N + M2Y 1 2(Y2−X2) . (13) Q(t)= I(t t∗)2 C/I, (20)
Y3 Y2X (X2+Y2)2" − (X2+Y2)# − −
where t∗ and I are twpo more integration constants. It
The finding solutions of this system and the comparison is easy to check that I = 2E, where E is the energy
of the corresponding evolution of wave packet with a di- integral of the system: −
rectnumericalmodelingofnonlinearwavesistheessence
of the present work. E =(Q2ξ˙2 Q˙2+F(ξ)/Q2)/2. (21)
−
4
WithtakingintoaccountEq.(20),itfollowsfromEq.(19)
80
that X
70 Y
t dt = ξ dξ , (22) 60 XYlliinn
Z0 I(t−t∗)2−C/I ±Zξ0 C−F(ξ) 50
where ξ0 is the fourth — the laspt — integration con- 40
stant. Formulas (20) and (22) provide the full solution
30
of the problem. In particular, with I > 0, C < 0, the
dependence(22)canberepresentedinthefollowingform, 20
10
1 I(t t∗)
sinh2ξ(t)=P a + arctan − , (23)
0 √ C √ C 0
(cid:16) − h − i(cid:17) 0 20 40 60 80 100 120 140 160
t/T
where an even a-periodic function P(a) (having a min- 0
imum at a = 0) is the inverse with respect to elliptic
integral (here the integration variable z =sinh2ξ):
s zdz FIG. 1: The temporal dependencies X(t) and Y(t) for N =
P s, 3.0, M = 0, X0 = 40, Y0 = 50, U0 = −0.05, V0 = −0.10.
(cid:16)±Zs1 2 (1+z2)(Cz2+4Nz−4)−M2z2(cid:17)≡ The solutions for a linear packet with the same initial condi-
(24) tions are also shown for comparison. What is important for
ands1 is thepsmallerrootofthe equation(1+s2)(Cs2+ formation of a rogue wave, the nonlinear focusing along the
4Ns 4) M2s2 = 0. The amplitude of wave packet longitudinal direction goes herewith acceleration for most of
equal−s to −4N/XY, and it is given by the formula the time, while in the linear case it occurs with deceleration
forallthetime. ThevalueofX intheminimumhasbeentoo
A(t)=p√8N/ [I(t t∗)2 C/I]sinh2ξ(t). (25) smallforvalidityoftheNLSE,buttheyareinitialconditions
− − ofthistypewhichinreality result information ofanomalous
p waves of the limiting steepness.
IV. SELECTION OF THE SOLUTIONS
momentofmaximalamplitude,atM =0andC =0can
In fact, we shall not use the analytical solutions, ex-
be found fromthe simple conditionsinh2ξ =1/N,as it
1
pressed in terms of special functions, for finding the de-
is clear in view of formula (24).
pendencies X(t)andY(t) withsome giveninitial values.
It is worth noting that at M = 0 the cyclic variable
Practically it is much simpler and faster to carry out
χ remains constant, as it follows from Eq.(10). As we
highly accurate numerical simulation of the variational
shall see soon, the physical wave fields corresponding to
equations (12) and (13) immediately. One should also
different values of χ, strongly differ between each other.
remember thatthe absoluteaccuracyinsolvingapproxi-
In particular, at χ = 0 the wave picture is symmetric
matevariationalproblemishardlynecessary. Theknowl-
with respect to sign change of y coordinate, while at
edge of the analytical structure of the solutions will be
χ = 0 there is no any symmetry (see Figs.2-3). But
needed for a general understanding of properties of the 6
in the framework of NLSE, the envelope of one solution
dynamical system under consideration.
coincides with the envelope of the other solution after
We are interested mainly in those solutions from the
a hyperbolic rotation of the dimensionless coordinates.
whole diversity (20) and (22), which result in essentially
Thepresenceofthis approximatesymmetrysignificantly
higher wave amplitude as compared to the maximally
clarifies the question about optimal conditions for the
possible linear value Amaxlin = 4NU−∞V−∞, where formation of anomalous waves.
U−∞ and V−∞ are the asymptotic values of the focus-
p
ing parameters at large negative times. In essence, only
sucheventsaredeservingtheterribletitle“roguewaves”.
V. COMPARISON WITH NUMERICAL
What is important, the most interesting solutions corre-
EXPERIMENTS
spond to values of the parameters C 0, M 0. The
condition C =0 means that Q2 Y2 ≈X2 =I≈(t t∗)2,
≡ − −
i. e. the plots of functions X(t) and Y(t) touch each It is clear that any simplified model without verifica-
other at t = t∗. It is easy to make sure of the above tionremainsjustatoy. Inordertomakethestatusofthe
assertionbynumericalsimulationsofEqs.(12)-(13)[with variational approximation more distinct, it is necessary
fixed N, U , V , Y , but different X and M], and sub- to compare it with a more accurate model. In our case,
0 0 0 0
sequent comparisonof the maximal amplitudes. Typical numericalexperimentswerecarriedoutintheframework
dependencies X(t) and Y(t) for such a case are shown of the fully nonlinear long-crested wave model which is
in Fig.1. Let us also note that the minimal value of the described in detail in Refs.[38, 39]. The computational
ratio(X/Y) =tanhξ ,whichisreachedcloselytothe domain was a square of the size 5 km, with the periodic
min 1
5
FIG. 2: Wave picture at χ = 0.0 (the other parameters FIG. 3: Wave picture at χ = 0.3 (the other parameters
are the same as in Fig.1): a) wave packet at the initial time are the same as in Fig.1): a) wave packet at the initial time
moment; b) at the moment when the anomalous wave has moment; b) at the moment of the anomalous wave rise. De-
risen. The crest length of the high wave here has been much viations from the Gaussianity are even more prominent here
longerthanitwouldbeinthelinearcase,whichobservationis than at χ=0.0.
inaccordancewiththeGaussianmodelprediction. Deviations
fromtheGaussianityareclearlyseen,basically—intheform
of a small tail consisting of longer waves.
Gaussianity of the arising rogue waves, Figs.4-5 demon-
strate that there is a qualitative agreement between the
boundaryconditions. The lengthofthe carrierwavewas variational theory and the numerical experiment in the
chosenλ =125m. Thesimulationswerecarriedoutfor importantaspectasthetemporaldependenceofthemax-
0
different sets of initial parameters of the packet. imal amplitude of the wavepacket. It is seen that Gaus-
InFig.2aandFig.3a,examplesareshownofhowGaus- sianmodelsomewhatoverestimatestheheightofanoma-
sianinitialconditionslook. The“portraits”ofanomalous lous wave (especially as parameter χ increases to values
waves developing from those conditions, are presented about unity, when the approximate symmetry of the hy-
in Fig.2b and Fig.3b. In general, deviations from the perbolic rotation becomes invalid), and it moves to the
Gaussian shape are clearly seen there. Despite the non- future the moment of the maximal rise. However, it is
6
0.5 0.5
highest crest highest crest
deepest trough deepest trough
0.4 Gaussian nonlinear 0.4 Gaussian nonlinear
Gaussian linear Gaussian linear
0.3 0.3
A A
0 N=3.0, M=0, χ=0 0 N=4.0, M=0, χ=0
k k
0.2 0.2
0.1 0.1
X =40, Y =50, U =-0.05, V =-0.10 X =47, Y =50, U =-0.05, V =-0.075
0 0 0 0 0 0 0 0
0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
t/T t/T
0 0
0.5 0.5
highest crest highest crest
deepest trough deepest trough
0.4 Gaussian nonlinear 0.4 Gaussian nonlinear
Gaussian linear Gaussian linear
0.3 0.3
A A
0 N=3.0, M=0, χ=0.3 0 N=4.0, M=0, χ=0.3
k k
0.2 0.2
0.1 0.1
X =40, Y =50, U =-0.05, V =-0.10 X =47, Y =50, U =-0.05, V =-0.075
0 0 0 0 0 0 0 0
0 0
0 20 40 60 80 100 120 0 20 40 60 80 100 120
t/T t/T
0 0
0.5
highest crest
deepest trough FIG. 5: Comparison of the maximal amplitudes for different
0.4 Gaussian nonlinear setsofinitialdatathaninFig.4. Itisseenthatwithincrease
Gaussian linear
ofN,thetimedelayofthepredictedpeakincomparisonwith
0.3 thenumerical simulation increases.
A
0 N=3.0, M=0, χ=0.5
k
0.2
hardly reasonable to expect a detailed accordance from
the simplest variational approximation. In any case, the
0.1
time period of existence of anomalous wave is predicted
X =40, Y =50, U =-0.05, V =-0.10
0 0 0 0 by the Gaussian theory rather well. Also another im-
0 portanteffectisconfirmed,the increaseofthetransverse
0 20 40 60 80 100 120
packet size as compared to the linear theory (see Fig.1).
t/T
0 This effect contributes to the subsequent more fast de-
crease of the amplitude in comparison with the linear
case.
FIG.4: Comparisonofthetemporaldependenciesofthemax-
imal amplitudes for wave packets with different χ, the other
parameters being the same as in Fig.1. In each figure shown VI. CONCLUSIONS
are numerically found theheight of thehighest crest and the
depthofthedeepesttrough,andalsopredictionsoftheGaus-
sian models — linear and nonlinear. The numerical curves Thus, based on the variational analysis and on the
haveanoscillatingcharacter,whichpropertyisrelatedtothe comparison with the results of direct numerical simu-
approximately two-fold difference between the phase veloc- lations, it is natural to put forward the following the-
ity of the carrier wave and the group velocity of the packet sis. In the main approximation, a typical long-crested
motion. Heights of crests are typically larger than depths of three-dimensional oceanic rogue wave at the nonlinear
troughs, due to the presence of higher harmonics. Therefore stage is, from the mathematical point of view, a dynam-
thequantitythat should be compared to thetheoretical pre-
ical system with just two degrees of freedom: X(t) and
dictions is a half-sum of envelopes of the numerical curves.
Y(t). This system contains two constant parameters (N
7
and M), which are related to the integrals of motion of casional) focusing at some time moment t∗ the relations
1+2D NLSE and characterize a given group of waves. X∗ Y∗ 20 40, X˙∗ Y˙∗ >0.05shouldtakeplace.
The system is integrable. Real sea states where one can To e≈stima∼te p−robab−ility o≈f −realiz∼ation of such conditions
expect anomalous waves, require values N 2–4. The inrandomweaklynonlinearwavefieldsisthetaskonthe
≈
most favorable focusing parameters for the rogue wave future.
formation are: 1) M 0; 2) on the initial stage of (oc-
≈
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