High-order nonlinear Schr\"odinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions

0.51 MB·English

by  I. S. Gandzha

High-OrderNonlinear Schro¨dingerEquation
I.S. GANDZHA,1 YU.V.SEDLETSKY,1 D.S.DUTYKH2
1InstituteofPhysics,Nat. Acad. ofSci. ofUkraine
(46, Prosp. Nauky, Kyiv03028, Ukraine; e-mail: [email protected], [email protected])
2Universit´edeSavoieMontBlanc
(CNRS–LAMA UMR 5127, Campus Universitaire, 73376 Le Bourget-du-Lac,
France; e-mail: [email protected])
HIGH-ORDER NONLINEAR
5
1 SCHRO¨DINGER EQUATION FOR THE ENVELOPE
0 OF SLOWLY MODULATED GRAVITY WAVES
2
ON THE SURFACE OF FINITE-DEPTH FLUID

n
AND ITS QUASI-SOLITON SOLUTIONS
a PACS47.35.Bb
J

3
2
We consider the high-order nonlinear Schr¨odinger equation derived earlier by Sedletsky
] [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity
S waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat
P bottom. Thisequationtakesintoaccountthethird-orderdispersionandcubicnonlineardisper-
n. sive terms. We rewrite this equation in dimensionless form featuring only one dimensionless
i parameterkh,wherekisthecarrierwavenumberandhistheundisturbedfluiddepth. Weshow
l
n that one-soliton solutions ofthe classical nonlinearSchr¨odinger equation are transformed into
[ quasi-solitonsolutionswithslowlyvaryingamplitudewhenthehigh-order termsaretakeninto

consideration. These quasi-soliton solutions represent the secondary modulations of gravity
1
waves.
v
Keywords: nonlinear Schr¨odinger equation, gravity waves, finite depth, slow modulations,
3
wave envelope, quasi-soliton, multiple-scale expansions.
3
9
5
1.Introduction neyandNewell[5]. Inthecaseofgravitywavesprop-
0
agating on the surface of infinite-depth irrotational,
. The nonlinear Schr¨odinger equation (NLSE)
1 inviscid,andincompressiblefluid,NLSEwasfirstde-
0 Aτ = a1Aχ ia2Aχχ+ia0,0,0AA2 (1.1) rivedbyZakharov[68]usingtheHamiltonianformal-
5 − − | |
1 arises in describing nonlinear waves in various phys- ism and then by Yuen and Lake [66] using the aver-
: ical contexts, such as nonlinear optics [64], plasma aged Lagrangian method. The finite-depth NLSE of
v
i physics [33], nanosized electronics [12], ferromagnet- form(1.1)wasfirstderivedbyHasimotoandOno[30]
X ics[10],Bose–Einsteincondensates[73],andhydrody- usingthemultiple-scalemethodandthenbyStiassnie
r namics[15,44,66,69]. Here,χisthedirectionofwave and Shemer [57] from Zakharov’s integral equations.
a
propagation, τ is time, A(χ, τ) is the complex first- Noteworthy is also the recent paper by Thomas et
harmonic envelope of the carrier wave, and the sub- al. [61] who derived the finite-depth NLSE for water
scriptsnexttoAdenotethepartialderivatives. NLSE waves on finite depth with constant vorticity.
takes into account the second-order dispersion (term Undercertainrelationshipbetweentheparameters,
with A ) and the phase self-modulation (term with when
χχ
A|A|2). The coefficients a1, a2, and a0,0,0 take vari- a2a0,0,0 < 0, (1.2)
ous values depending on the particular physical con-
text under consideration. NLSE admits exact solutions in the form of solitons,
In the general context of weakly nonlinear disper- whichexistduetothe balanceofdispersionandnon-
sive waves, this equation was first discussed by Ben- linearityandpropagatewithoutchangingtheirshape
and keeping their energy [16]. In this case, the uni-
(cid:13)c I.S.GANDZHA,YU.V.SEDLETSKY, form carrier wave is unstable with respect to long-
D.S.DUTYKH,2014 wave modulations allowing for the formation of en-
ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12 1201
I.S.Gandzha,Yu.V.Sedletsky,D.S.Dutykh
velope solitons. This type of instability is known as part of Dysthe’s equation in new canonical variables
themodulationalorBenjamin–Feirinstability[71](it (the so-called compact Dyachenko–Zakharov equa-
wasdiscoveredforthefirsttimeinopticsbyBespalov tion [21,22]).
andTalanov[7]). Inthecaseofsurfacegravitywaves, OriginalDysthe’sequationwaswrittenforthefirst-
condition (1.2) holds at kh&1.363, k being the car- harmonicenvelopeofvelocitypotentialratherthanof
rier wavenumber and h being the undisturbed fluid surfaceprofile. InthecaseofstandardNLSE,thisdif-
depth. In addition to theoretical predictions, enve- ferenceinnotessentialbecauseinthatorderthefirst-
lope solitons were observedin numerous experiments harmonic amplitudes of the velocity potential and
performed in water tanks [9,44,50,51,55,60,66,67]. surface displacement differ by a dimensional factor
At the bifurcation point a = 0 (kh 1.363), only,whichisnottrueanymoreintheHONLSEcase,
0,0,0
≈
when the modulational instability changes to stabil- as discussed by Hogan [32]. Keeping this in mind,
ity, NLSE of form (1.1) is not sufficient to describe Trulsenetal. [63]rewroteDysthe’sequationinterms
the wavetrain evolution since the leading nonlinear of the first-harmonicenvelopeof surface profile while
term vanishes. In this case, high-order nonlinear and taking into account the linear dispersion to an arbi-
nonlinear-dispersive terms should be taken into ac- trary order.
count. Inthecaseofinfinite depth, suchahigh-order In the case of finite depth, the effect of induced
NLSE(HONLSE)wasfirstderivedbyDysthe[19]. It mean flow manifests itself in the third order, so that
includes the third-order dispersion (A ) and cubic the NLSE is generally coupled to the equation for
χχχ
nonlinear dispersive terms (A2A , A2A∗, asterisk theinducedmeanflow[6]. However,DaveyandStew-
| | χ χ
denotes the complex conjugate) as well as an addi- artson [13] showed that these coupled equations are
tional nonlinear dispersive term describing the input equivalent to the single NLSE derived by Hasimoto
of the wave-induced mean flow (some of these terms and Ono [30]. On the other hand, such an equiva-
were introduced earlier by Roskes [46] without tak- lence is not preserved for high-order equations. The
ing into consideration the induced mean flow). This first attempt to derive a HONLSE in the case of fi-
equation is usually referred to as the fourth-order nite depth was made by Johnson [35], but only for
HONLSE to emphasize the contrast with the third- kh 1.363,whenthecubicNLSEtermvanishes. The
≈
order NLSE. Janssen [34] re-derived Dysthe’s equa- similarattemptwasmadebyKakutaniandMichihiro
tionandcorrectedthesignatoneofthenonlineardis- [37] (see also a more formal derivation made later by
persiveterms. Hogan[31]followedtheearlierworkby Parkes[45]). Ageneralfourth-orderHONLSEforthe
Stiassnie [56] to derive the similar equation for deep- first-harmonicenvelope of surface profile was derived
water gravity-capillary waves with surface tension by Brinch-Nielsen and Jonsson [8] in coupling with
taken into account. Selezov et al. [49] extended the theintegralequationforthewave-inducedmeanflow.
HONLSE derived by Hogan to the case of nonlinear Gramstadand Trulsen[26,27] deriveda fourth-order
wavetrain propagation on the interface of two semi- HONLSE in terms of canonical variables that pre-
infinitefluidswithouttakingintoaccounttheinduced serves the Hamiltonian structure of the surface wave
mean flow. Worthy of mention is also the paper by problem.
Lukomsky[41]whoderivedDysthe’sequationinadif- Sedletsky [47,48]usedthe multiple-scaletechnique
ferent way. Later, Trulsen and Dysthe [62] extended to derive a single fourth-orderHONLSE for the first-
theequationderivedbyDysthetobroaderbandwidth harmonic envelope of surface profile by introducing
by including the fourth- and fifth-order linear dis- an additional power expansion of the induced mean
persion. Debsarma and Das [14] derived a yet more flow. This equation is the direct counterpart of Dys-
general HONLSE that is one order higher than the the’s equation written in terms of the first-harmonic
equation derived by Trulsen and Dysthe. Gramstad envelope of surface profile [63] but for the case of fi-
and Trulsen [25] derived a set of two coupled fourth- nite depth. Slunyaev [52] confirmed the results ob-
order HONLSEs capable of describing two interact- tained in [48] and extended them to the fifth order.
ing wave systems separated in wavelengths or di- Grimshaw and Annenkov [29] considereda HONLSE
rections of propagation.Zakharov and Dyachenko for water wave packets over variable depth.
[17,18,72] made a conformal mapping of the fluid The deep-water HONLSE in the form of Dysthe’s
domain to the lower half-plane to derive a counter- equation was extensively used in numerical simula-
1202 ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12
High-OrderNonlinear Schro¨dingerEquation
tions of wave evolution [1–3, 11, 20, 23, 40, 53, 54].
However, no such modeling has been performed in
the case of finite depth because of the complexity of
equations as compared to the deep-water limit. The
equation derived in [47,48] can be used as a good
starting point for the simulations of wave envelope
evolutiononfinite depth. The aimofthis paper is(i)
to rewrite this equation in dimensionless form suit-
able for numerical integration and (ii) to observe the Fig. 1. Sketch of the physical domain occupied by an ideal
evolutionofNLSEsolitonstakenasinitialwaveforms incompressiblefluidoffinitedepth
in the case when the HONLSE terms are taken into
considerationforseveralvaluesofintermediatedepth.
Thispaperisorganizedasfollows. InSection2,we
write down the fully nonlinear equations of hydro- Φ =0, y = h; (2.4)
y
dynamics used as the starting point in this study. In −
Section 3, we formulate the constraints at which the
whereΦ(x, y, t)isthevelocitypotential(thevelocity
fullynonlinearequationscanbereducedtoHONLSE.
is equal to Φ), g is the acceleration due to gravity,
Then we briefly outline the multiple-scale technique ∇
t is time. Here, (2.1) is the Laplace equation in the
used to derive this equation, which is presented in
fluid domain, (2.2) is the dynamical boundary con-
Section 4. Next, we introduce dimensionless coordi-
dition (the so-called Bernoulli or Cauchy–Lagrange
nate, time, and amplitude to go over to the dimen-
integral), (2.3) and (2.4) are the kinematic boundary
sionless HONLSE. As a result, only one dimension-
conditions (no fluid crosses the free surface and the
less parameter kh appears in the equation. The final
bottom), the indices x, y, and t designate the partial
step is to pass to the reference frame moving with
derivatives overthe correspondingvariables. The po-
the group speed of the carrier wave. In Section 5, we
sition of the zero level y =0 is selected such that the
presenttheresultsofnumericalsimulationsandcom-
Bernoulli constant (the right-hand side of Eq. (2.2))
pare the NLSE and HONLSE solutions. Conclusions
is equal to zero.
are made in Section 6.
Consider a modulated wavetrain with carrier fre-
quency ω and wavenumber k. In this case, a solution
2.Problem Formulation
to Eqs. (2.1)–(2.4) can be looked for in the form of
We consider the dynamics of potential two-dimen-
Fourier series with variable coefficients:
sional waves on the surface of irrotational, inviscid,
and incompressible fluid under the influence of grav- ∞
ity. Waves are assumed to propagate along the hori- Φ(x, y, t) = Φn(x, y, t) ein(kx−ωt),
η(x, t) η (x, t)
zontal x-axis, and the direction of the vertical y-axis (cid:18) (cid:19) n=−∞(cid:18) n (cid:19) (2.5)
X
is selected opposite to the gravity force. The fluid is η−n ≡ηn∗, Φ−n ≡Φ∗n,
assumed to be bounded by a solid flat bed y = h
−
at the bottom and a free surface y = η(x, t) at the where ∗ stands for complex conjugate (here, we as-
top (Fig. 1). The atmospheric pressureis assumedto sumethecarrierwavetobesymmetric),thefunctions
be constant on the free surface. Then the evolution Φ(x, y, t) and η(x, t) are assumed to be real by defi-
ofwavesandassociatedfluidflowsisgovernedbythe nition. Substituting (2.5) in(2.1)–(2.4)andequating
following system of equations [24,58]: the coefficients at the like powersof exp(i(kx ωt)),
−
one can obtain a system of nonlinear partial dif-
Φxx+Φyy =0, <x< , h<y <η(x, t); ferential equations for the functions Φ (x, y, t) and
−∞ ∞ − n
(2.1) η (x, t). Linearization of these equations at n = 1
n
gives the dispersion relation for gravity waves:
1
Φ + Φ2 +Φ2 +gη =0, y =η(x, t); (2.2)
t 2 x y
ω2 =gktanh(kh). (2.6)
η Φ(cid:0)+η Φ =(cid:1) 0, y =η(x, t); (2.3)
t y x x
−
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I.S.Gandzha,Yu.V.Sedletsky,D.S.Dutykh
3.Slowly Modulated Quasi-Harmonic Inpractice,thequasi-harmonicityconditioncanbe
Wavetrains and Multiple-Scale Expansions written as
Generally the system of equations for Φ (x, y, t)
n kη 1, (3.6)
and ηn(x, t) is by no means more simple than orig- | 1|≪
inal equations. It can be simplified when solutions
and the condition of slow modulation (quasi-mono-
are looked for in a class of functions with nar-
chromaticity) can be formalized as
row spectrum, ∆k k. In this case, the problem
| | ≪
has a formal small parameter µ ∆k k (quasi- A
∼ | | x 1, (3.7)
monochromaticity condition), with Φn(x, y, t) and kA ≪
(cid:14) (cid:12) (cid:12)
ηn(x, t)being slow functions ofx andt. Accordingly, (cid:12) (cid:12)
the wave motion can be classified into slow one and w(cid:12)(cid:12) hic(cid:12)(cid:12)h follows from differentiating the function
fast one by introducing different time scales and dif- A(x, t)exp i(kx ωt) withrespecttox. Withthese
−
ferent spatial scales: conditions satisfied, the original system of equations
(cid:0) (cid:1)
(2.1)–(2.4) can be reduced to one evolution equation
Tn µnt, Xn µnx. (3.1) for the first harmonic envelope A(x, t) with the use
≡ ≡
of small-amplitude expansions (3.2) and (3.5).
The derivatives with respect to time and coordinate
are expanded into the following series: 4.High-Order
∞ ∞ Nonlinear Schr¨odinger Equation
∂ ∂ ∂ ∂
= µn , = µn , (3.2) 4.1.Equation derived by Sedletsky
∂t ∂T ∂x ∂X
n n
n=0 n=0
X X Sedletsky [47,48] used the above-described multiple-
the times Tn and coordinates Xn being assumed to scale procedure to derive the following HONLSE for
be independent variables. thefirst-harmonicenvelopeA(x, t)(Eq. (68)in[47]):
When there are no resonancesbetween higher har-
monics, the amplitudes of Fourier coefficients de- i ∂A +V ∂A +
crease with increasing number (quasi-harmonicity ∂(εt) g∂(εx)
(cid:18) (cid:19)
condition): 1 ∂2A
+ε ω′′ +ωk2q A2A +
ηn εnA, n>1, η0 ε2A, ε<1, (3.3) (cid:18)2 ∂(εx)2 3| | (cid:19)
∼ ∼ 1 ∂3A ∂A
+iε2 ω′′′ +ωkQ A2 +
where −6 ∂(εx)3 41| | ∂(εx)
(cid:18)
1 ∂A∗
η εA(x, t). (3.4) +ωkQ A2 =0 [m/s]. (4.1)
1 ≡ 2 42 ∂(εx)
(cid:19)
The parameter ε can be regarded as a formal small
As compared to the standard NLSE, this equation
parameterrelatedtothesmallnessofwaveamplitude
takes into account additional nonlinear and disper-
ascomparedtothecarrierwavelengthλ≡ 2kπ. Inthis sive terms of order Ø(ε2). Equation (4.1) was later
case, the unknown functions Φ (x, y, t) and η (x, t)
n n re-derived by Slunyaev [52], who confirmed the sym-
can be expanded into power series in the formal pa-
boliccomputationspresentedin[47,48]andextended
rameter ε: them to the Ø(ε3) order. Here, we restrictour atten-
Φ (x, y, t) ∞ Φ(m)(x, y, t) tiontotheoriginalequation(4.1). Theparametersof
n = εm n . (3.5) this equation are given by
(cid:18) ηn(x, t) (cid:19) m=1 ηn(m)(x, t) !
X
ω =(gkσ)1/2, σ tanh(kh), (4.2a)
Multiple-scale expansions (3.2) and (3.5) allow the ≡
∂ω ω 2kh
functions Φn(x, y, t) and ηn(x, t) to be expressed in ω′ = Vg = 1+ =
terms of the first harmonic envelope A(x, t), as de- ∂k ≡ 2k sinh(2kh)
(cid:18) (cid:19)
scribed in detail in [47]. Note that in the procedure ω 1 σ2
= 1+ − kh , (4.2b)
described in [47] it is essential to set ε µ. 2k σ
≡ (cid:18) (cid:19)
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High-OrderNonlinear Schro¨dingerEquation
ω′′ = ∂2ω = ω σ2 1 3σ2+1 k2h2 Thefree-surfacedisplacementisexpressedinterms
∂k2 4k2σ2 − − of A as
2σ σ2 1 kh σ(cid:16)(cid:0)2 , (cid:1)(cid:0) (cid:1) (4.2c) η =ε2η +εRe Aei(kx−ωt) +
− − − 0
(cid:0) (cid:1) (cid:17) +ε22Re η e2i((cid:0)kx−ωt) +Ø(cid:1)(ε3), (4.3)
2
∂3ω ω
ω′′′ = = σ2 1 where Re(cid:0) stands fo(cid:1)r the real part of a complex-
∂k3 −8k3σ3 − × {·}
valued function. Here, η and η are defined as
(cid:16)(cid:0) (cid:1) 0 2
15σ4 2σ2+3 k3h3 3σ σ2 1 3σ2+1 k2h2
× − − − − σ+2 1 σ2 kh
η = − k A2, (4.4a)
(cid:0)3σ2 σ2 1 kh(cid:1) 3σ3 , (cid:0) (cid:1)(cid:0) (cid:1) (4.2d) 0 (cid:0) ν (cid:1) | |
− − − 3 σ2
q3 =−(cid:0)16σ14ν(cid:1) σ2−1 2(cid:17)9σ4−10σ2+9 k2h2+ η2 = 8−σ3 kA2. (4.4b)
(cid:16)(cid:0) (cid:1) (cid:0) (cid:1) The corresponding velocity potential is written as
+2σ 3σ6 23σ4+13σ2 9 kh
− − − Φ=εΦ +ε2Re Φ ei(kx−ωt) +
σ2(cid:0)7σ4 38σ2 9 , (cid:1) (4.2e) 0 1
− − − +ε22Re Φ e2i(k(cid:0)x−ωt) +Ø(ε(cid:1)3), (4.5)
Q =(cid:0) 1 σ2 (cid:1)1(cid:17) 5 2
41 32σ5ν2 − × where (cid:0) (cid:1)
3σ6 20σ4(cid:16)(cid:0)21σ2+(cid:1) 54 k5h5 Φ = ω ∂A hσ+ Vg iA cosh k(y+h)
×(cid:0)σ σ2− 1 3 1−1σ8 99σ6 (cid:1)61σ4−+7σ2+270 k4h4+ 1 2kσ (cid:18)∂x(cid:18) ω (cid:19)− (cid:19) co(cid:0)sh(kh) (cid:1)−
− − − − ∂A sinh k(y+h)
(y+h) , (4.6a)
+2σ(cid:0)2 σ2−(cid:1)1(cid:0) 7σ10−58σ8+38σ6+52σ4−(cid:1) − ∂x co(cid:0)sh(kh) (cid:1)!
−181σ(cid:0)2+270(cid:1)k(cid:0)3h3−2σ3 3σ10+18σ8−146σ6− Φ2 =3iω (σ146−σ41) coschos2hk((2yk+h)h) A2. (4.6b)
172σ4+183σ(cid:1)2 270 k2h(cid:0)2 σ4 σ8 109σ6+517σ4+ (cid:0) (cid:1)
− − − −
The term Φ describes the wave-induced mean flow
0
+217σ2+270 kh+σ5(cid:1)σ6 40σ(cid:0)4+193σ2+54 +∆, and is expressed implicitly in terms of its derivatives
−
Q42 = 32σ15ν2(cid:1) − σ2−(cid:0) 1 5× (cid:1)(cid:17)(4.2f) ∂∂Φx0 =εω2kσγν1|A|2+iε8ωσ2γν22 A∂∂Ax∗ −A∗∂∂Ax ,
3σ6+7σ4 (cid:16)11(cid:0)σ2+9(cid:1)k5h5+ (cid:18) (cid:19)(4.7a)
× −
+(cid:0)σ σ2 1 3 11σ8 48σ(cid:1)6+66σ4+8σ2+27 k4h4 ∂Φ0 = V ∂Φ0, (4.7b)
− − − ∂t − g ∂x
2σ(cid:0)2 σ2 (cid:1)1(cid:0) 7σ10 79σ8+282σ6 (cid:1)
− − − − where
154σ(cid:0)4 σ2(cid:1)+(cid:0)9 k3h3+2σ3 3σ10 63σ8+314σ6
− − − − γ = σ2 1 2kh σ σ2 5 , (4.8a)
1
218σ4+19σ2+(cid:1)9 k2h2+σ(cid:0)4 σ8+20σ6 158σ4 − − −
− − − γ =(cid:0)σ2 1(cid:1)5k4h4+(cid:0)4σ σ2 (cid:1) 1 2 13σ2+3 k3h3
28σ2 27 kh σ5 (cid:1)σ6 7σ4+(cid:0)7σ2 9 ∆, (4.2g) 2 − − −
− − − − − −
ν = σ2 1(cid:1) 2k2h2(cid:0) 2σ σ2+1 kh+σ(cid:1)(cid:17)2. (4.2h) −2σ2(cid:0)σ2−1(cid:1) 3σ4+32σ(cid:0)2−3 k(cid:1)2h(cid:0)2+ (cid:1)
− −
+4σ3(cid:0)2σ4 σ(cid:1)(cid:0)2 5 kh 3σ4 (cid:1)σ2 5 . (4.8b)
The(cid:0)quantity(cid:1) V isthewa(cid:0)vegrou(cid:1)pspeed. Theparam- − − − −
g
eter ∆ is the correction introduced by Slunyaev [52] Functi(cid:0)ons (4.3) and(cid:1)(4.5) defi(cid:0)ne an a(cid:1)pproximate so-
to the coefficients derived in [47,48]. This correction lution to the original system of equations (2.1)–(2.4)
is negligible at kh & 1 (see Appendix A), and we in terms of the first-harmonic envelope A, which is
ignore it by keeping ∆=0. found from Eq. (4.1).
ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12 1205
I.S.Gandzha,Yu.V.Sedletsky,D.S.Dutykh
4.2.Dimensionless form Thus, Eq. (4.1) takes the dimensionless form
Introduce the following dimensionless time, coordi-
V 1
nate, and amplitude: i u g u u +q u2u+
τ − ω′′k χ − 2 χχ 3| |
(cid:18) (cid:19)
τ =βt, χ=kx, u=α−1εA, (4.9) 1ω′′′k
+i u +Q u u2+Q u2u∗ =0
6 ω′′ χχχ 41 χ| | 42 χ
αand β being the parametersto be determined. The (cid:18) (cid:19)
relationship between the old and new derivatives is
or, equivalently,
∂ ∂ ∂ ∂
=k , =β . i u +a u a u +a u2u+i a u +
∂x ∂χ ∂t ∂τ τ 1 χ 2 χχ 0,0,0 3 χχχ
− | | −
Then Eq. (4.1) is transformed to +(cid:0)a u u(cid:1)2+a u2u∗ =0, (cid:16) (4.11a)
1,0,0 χ| | 0,0,1 χ
1 (cid:17)
iα(βu +kV u )+α ω′′k2u +ωk2q α2 u2u + which finally yields
τ g χ χχ 3
2 | | | |
(cid:18) (cid:19)
+iα 1ω′′′k3u +ωk2Q α2u u2+ uτ =−a1uχ−ia2uχχ+ia0,0,0|u|2u+
χχχ 41 χ
−6 | | | |
(cid:16) + a u a u u2 a u2u∗ , (4.11b)
+ωk2Q α2u2u∗ =0 [m/s]. 3 χχχ− 1,0,0 χ| | − 0,0,1 χ
42| | χ (cid:16) (cid:17)
(cid:17) where we used the unified notation introduced by
Here,theindicesχandτ designatethepartialderiva- Lukomsky and Gandzha [42]. Here, the coefficients
tiveswithrespecttothe correspondingvariables. Ta-
′′
king into account that ω < 0 at all h > 0 (Fig. 2),
divide this equation by ω′′k2α so that a = Vg = 2 σ2+σ 1 σ2 kh >0,
1 −ω′′k −υ −
i ω′β′k2uτ + ωV′′gkuχ + 12uχχ+ ωω′′|α|2q3|u|2u + a2 = 12, (cid:16) (cid:0) (cid:1) (cid:17)
(cid:18) (cid:19) (cid:18) (cid:19)
+i −16ωω′′′′′kuχχχ+ ωω′′|α|2Q41uχ|u|2+ a3 ≡−16ωω′′′′′k =
ω(cid:16)
+ω′′|α|2Q42u2u∗χ =0 = 1 σ2 1 15σ4 2σ2+3 k3h3 (4.12)
12συ − − −
(cid:17)
and select the values of α and β as 3σ σ2(cid:16)(cid:0) 1 3σ(cid:1)2(cid:0)+1 k2h2 (cid:1)
− − −
α2 = ω′′ >0, β = ω′′k2 >0. (4.10) −3σ2(cid:0) σ2−1(cid:1)(cid:0)kh−3σ(cid:1)3 ,
| | − ω − a0,0,0(cid:0)≡q3, (cid:1)a1,0,0 ≡Q(cid:17)41, a0,0,1 ≡Q42,
υ = σ2 1 3σ2+1 k2h2 2σ σ2 1 kh σ2
0
− − − −
are a(cid:0)ll real (cid:1)a(cid:0)nd depe(cid:1)nd on one (cid:0)dimens(cid:1)ionless pa-
-0.5 rameter kh. Their behavior as functions of kh is
shownin Fig. 3. Itcan be seenthat Eq. (4.1) is valid
at kh&1, where the coefficients a , a , and
0,0,0 1,0,0
Ω¢¢ -1 a do not diverge. At smaller depths, the Korte-
0,0,1
weg–de Vries equation andits generalizations[36,39]
should be used. On the other hand, at large kh,
-1.5
theinfinite-depthlimit(Dysthe’sequation)shouldbe
used. Indeed,thefollowingasymptoticsareeasilyob-
-2 tained at kh :
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 →∞
kh 1 1 3 1
Fig.2. ω′′inm2/sasafunctionofhatk=1andg=9.8m/s2 a3 = 4, a0,0,0 =−2, a1,0,0 = 2, a0,0,1 = 4. (4.13)
1206 ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12
High-OrderNonlinear Schro¨dingerEquation
2
1.5 a3
a0,0,0
1
a1,0,0
0.5
a0,0,1
0
-0.5
-1
-1.5
-2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
kh
Fig. 3. Normalizedcoefficients ofHONLSEasfunctions ofkh
They coincide with the corresponding coefficients of
Dysthe’s equation [63], except for the term includ-
ing the wave-induced mean flow, which cannot be + a3uξξξ−a1,0,0uξ|u|2−a0,0,1u2u∗ξ . (4.16)
explicitly reconstructed from Eq. (4.11) because of (cid:16) (cid:17)
the additional power expansion of the wave-induced This is our target equation for numerical simula-
mean flow made to derive Eq. (4.1). However, this tions. It possesses the integral of motion
term can be reconstructed from the equations gene- ∞
rating Eq. (4.1), at the stage when the wave-induced
I (τ)= u(ξ, τ)2dξ =const, (4.17)
0
mean flow has not been excluded from the equation | |
−Z∞
for A yet [47]. Taking into account these constraints,
we will restrict our attention to the following range whichexpressesthe conservationofwaveaction. The
of intermediate depths: derivation of this conservation law is given in Ap-
1<kh<5. (4.14) pendixC. Itallowsonetotracetherelativenumerical
error of simulations:
4.3.Moving reference frame
I (τ) I (0)
Equation (4.11) can be rewritten in the form with- 0 0
Er(I )= | − |. (4.18)
0
out the uχ term. To this end, let us proceed to the I0(0)
reference frame moving with speed a (dimensionless
1 Of particular interest is to reveal any relationship
group speed):
of Eq. (4.16) to other HONLSEs derived in differ-
ξ =χ a τ, T =τ. (4.15) ent contexts. In Appendix B, we consider one such
1
−
equation(theSasa–Satsumaequation)andprovethat
The relationship between the derivatives in new and
Eq. (4.16) cannot be reduced to it at any kh.
old variables is given by the formulas
4.4.Dimensionless free surface
∂ ∂ξ ∂ ∂T ∂ ∂
= + = , displacement and velocity potential
∂χ ∂χ∂ξ ∂χ∂T ∂ξ
∂ ∂ξ ∂ ∂T ∂ ∂ ∂ The dimensionless free surface displacement is ex-
= + = a + ,
1 pressed in terms of u as follows:
∂τ ∂τ ∂ξ ∂τ ∂T − ∂ξ ∂T
so that ζ kη =α u2+α Re ueiθ +2α Re u2e2iθ ,
0 1 2
≡ | |
u = ia u +ia u2u+ (4.19)
τ 2 ξξ 0,0,0 (cid:0) (cid:1) (cid:0) (cid:1)
− | |
ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12 1207
I.S.Gandzha,Yu.V.Sedletsky,D.S.Dutykh
cosh(z+kh) sinh(z+kh)
(z+kh) u ,
ξ
2.5 × cosh(kh) − cosh(kh) !
2
c 3i σ4 1 cosh(2(z+kh)
1.5 ϕ = − u2,
a1 2 16σ4 cosh(2kh)
(cid:0) (cid:1) (cid:1)
1
where z ky is the dimensionless vertical coordi-
≡
0.5 nate. The quasi-harmonicity condition is written as
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 u
kh | | 1, (4.24)
√c ≪
Fig. 4. Ratio between the dimensionless phase and group
speedsasafunctionofkh and the quasi-monochromaticitycondition is
u
σ+2 1 σ2 kh 1 3 σ2 ξ 1. (4.25)
α0 = cν− , α1 = √c, α2 = 8−cσ3 , u ≪
(cid:0) (cid:1) (cid:12) (cid:12)
(cid:12) (cid:12)
F(cid:12) ina(cid:12)lly, the original equations of hydrodynamics can
where
be written in the following dimensionless form:
θ =kx ωt=χ cτ =ξ+(a c)τ (4.20)
1
− − − ϕ +ϕ =0, <ξ < , kh<z <ζ(ξ, τ);
ξξ zz
−∞ ∞ −
is the wave phase and (4.26)
1 4σ2 1 c2
c= = (4.21) ϕ + ϕ2+ϕ2 + ζ =0, z =ζ(ξ, τ); (4.27)
α2k2 − υ τ 2 ξ z σ
| |
(cid:0) (cid:1)
ζ ϕ +ζ ϕ =0, z =ζ(ξ, τ); (4.28)
is the dimensionless phase speed. Figure 4 shows the τ z ξ ξ
−
ratio of the phase speed c to the group speed a as a
1
ϕ =0, z = kh. (4.29)
functionofkh. Thisratioisequaltounityatkh 0, z
−
→
anditis twice as largeatkh , infull conformity
→∞ 5.Numerical Simulations
with the classical water wave theory [24]. The wave
envelope is written as In this section, we adopt the split-step Fourier (SSF)
technique described in Appendix D to compute solu-
[ζ]envelope =α1|u|+(α0+2α2)|u|2. (4.22) tions to HONLSE (4.16). To test the accuracyof our
numericalscheme,westartfromclassicalNLSE(1.1)
Thecorrespondingdimensionlessvelocitypotential
written in terms of the coordinate ξ. At a < 0
0,0,0
is expressed as (kh&1.363),ithasanexactone-solitonsolution[69]:
ϕ≡−ω1′′Φ=ϕ0+2Re ϕ1eiθ + u(ξ, τ)= u0exp(iκξ−iΩτ) , (5.1)
cosh K(ξ ξ Vτ)
0
+2Re ϕ2e2iθ , (cid:0) (cid:1) (4.23) Ω= K2 κ2(cid:0)a , V−= −2κa ,(cid:1)K = u a0,0,0,
2 2 0
(ϕ0)ξ =(cid:0) 2γσ1ν|u(cid:1)|2+ 8σiγ2ν22 uu∗ξ −u∗uξ , u0 (cid:0)C, −κ, ξ(cid:1)0 R. − | |r− 2a2
∈ ∈
(cid:0) (cid:1)
(ϕ ) = a (ϕ ) ,
0 τ − 1 0 ξ Here, V is the soliton speed, u0 is the complex am-
plitude, κ and Ω are the soliton’s wavenumber and
√c σ2+1 kh+σ
ϕ = iu+ u frequency, and ξ0 is the soliton’sinitial position. The
1 ξ
2σ (cid:18)− (cid:0) 2(cid:1)σ (cid:19)× amplitude u0 and wavenumber κ should be selected
1208 ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12
High-OrderNonlinear Schro¨dingerEquation
suchthatthequasi-harmonicityandquasi-monochro- 0.01
maticity conditions (4.24), (4.25) hold true. In prac-
0.008
tice,theseconditionsmeanthatthesolitonamplitude
and wavenumber should be small:
0.006
ÈuÈ
u0 1, κ 1. 2 Π c
| |≪ ≪ 0.004
In this study, we restrict our attention by the follow-
0.002
ing choice of parameters:
0
u0 =0.1, κ= K ( Ω=0), ξ0 =0. (5.2) -100 -75 -50 -25 0 25 50 75 100
− ⇒
Ξ
Figures 5 and 6 demonstrate that constraints (4.24)
Fig. 5. Testing the quasi-harmonicity condition (4.24) for
and(4.25)arereadilysatisfiedinthiscase. Notethat kh=3
at κ < 0 we have V > 0. In this case, solitons move
from left to right with speed exceeding the carrier
group speed. 0.08
Figure7showsasolitoncomputedforkh=3using
analytical formula (5.1) for the initial moment τ =0 0.06
and moment τ = 10000.The same soliton was taken ÉuΞÉ
ÈuÈ
as the initial condition for the simulation with the 0.04
SSFtechnique. Thedeviationfromtheexactsolution
is seen to be negligible. Indeed, the numerical error 0.02
estimated with formula (4.18) is
0
S(2) : Er(I )=1.3 10−10%, -100 -75 -50 -25 0 25 50 75 100
τ=10000 0 Ξ
| ×
∆ (u , u )=1.0 10−4%, Fig. 6. Testing the quasi-monochromaticity condition (4.25)
rms exact comp
× forkh=3
S(4) : Er(I )=2.3 10−10%,
|τ=10000 0 × oscillations decreases with time (Fig. 9). Such a so-
∆ (u , u )=3.5 10−9%, lution does not fall under the definition of soliton
rms exact comp
× because it does not preserve the constant amplitude
where S(2) and S(4) designate the order of the SSF and shape during the evolution. On the other hand,
technique adopted for calculation (see Appendix D) itmoveswithnearlytheconstantspeed(Fig. 10)and
and stillpossessesthe unique propertyofsolitonsto exist
over long periods of time without breaking. In view
∞ (u(ξ, τ) g(ξ, τ))2dξ of this unique property, we call such solutions quasi-
∆rms(u, g)(τ)= −∞ | |−| | solitons. The term quasi-soliton was introduced ear-
s I
R 0
lierbyZakharovandKuznetsov[70],butinsomewhat
is the relative r.m.s. deviation between two func- different context; then Karpmanet al. [38] and Slun-
tions. Thus,ournumericalschemereproducestheex- yaev[53]useditinthesamecontextasinthepresent
act one-solitonsolution to NLSE with high accuracy. study.
Figure 8 shows the evolution of the same one- Such a behavior of NLSE solitons in the HONLSE
soliton waveform taken as the initial condition in case was first described by Akylas [3] in the context
HONLSE(4.16). AscomparedtotheNLSEcase,the of asymptotic modeling andnumericalsimulations of
wave amplitude is smaller, the pulse width is larger, Dysthe’sequationinthe infinite-depthlimit. Growth
and the wave speed is higher. The wave amplitude in the soliton speed corresponds to the well-known
does not remain constant and exhibits slow oscilla- carrier frequency downshift observed in deep-water
tions that can be interpreted as the secondary mod- experimentsbySu[60]andinsimulationsofDysthe’s
ulation of the carrier wave. The amplitude of these equationbyLoandMei[40]. Dysthe [19]pointedout
ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12 1209
I.S.Gandzha,Yu.V.Sedletsky,D.S.Dutykh
0.1
NLSE,kh=3
0.075
Τ=10000
ÈuÈ computations
0.05 NLSEsoliton
Κ=-K
0.025
0
-100 0 100 200 300 400 500 600 700 800
Ξ
Fig.7. Evolutionofone-solitonsolution(5.1)toNLSE(1.1)atkh=3.SSFparameters:
∆τ =1, ∆ξ=2, ξ∈[−1000,1000); V ≈0.0566
0.1
HONLSE,kh=3
0.075
Τ=10000n,n=0,5
ÈuÈ computations
0.05 NLSEsoliton
Κ=-K
0.025
0
0 500 1000 1500 2000 2500 3000 3500
Ξ
Fig.8. Evolutionofone-solitonwaveform(5.1)takenastheinitialconditioninHONLSE
(4.16) at kh = 3.SSF parameters: ∆τ = 0.5, ∆ξ = 2, ξ ∈ [−4000,4000). Accuracy:
S(2)|τ=50000: Er(I0)=0.050%
ÈuÈmax V
0.095
0.07
0.09
0.065
0.085
0.06
0.08
0.055
0.075
0.07 0 500 1000 1500 2000 2500 3000
0 500 1000 1500 2000 2500 3000 Ξ
Ξ
Fig.10. Meanwavespeedasafunctionofdistanceatkh=3:
Fig. 9. Variations inthe amplitudeof the quasi-solitonsolu- solidcurve —quasi-soliton,dashedline—NLSE soliton(V ≈
tionwithdistanceatkh=3 ≈0.0566)
1210 ISSN 2071-0186. Ukr. J. Phys. 2014. Vol. 59, No. 12