Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schrödinger equation with time-dependent nonlinearity

1 MB·English

by  Shou-Fu Tian

Conservation laws, bright matter wave solitons and
modulational instability of nonlinear Schro¨dinger equation
with time-dependent nonlinearity
∗
Shou-Fu Tian1,2 , LiZou3, Qi Ding1 and Hong-QingZhang1.
∗
2 1SchoolofMathematicalSciences,DalianUniversityofTechnology,Dalian116024,
1
0 People’sRepublicofChina
2
2DepartmentofMathematics,UniversityofBritishColumbia,Vancouver,BC,V6T1Z2,Canada

n
a 3SchoolofAeronauticsandAstronautics,DalianUniversityofTechnology,Dalian116024,
J
People’sRepublicofChina
5

]
I
S Abstract: In this paper, we consider a general form of nonlinear Schro¨dinger equation with time-
.
n dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such
i
l
n nonlinear Schro¨dinger equation is identified by admitting an infinite number of conservation laws.
[
UsingtheDarbouxtransformationmethod,weobtainsomeexplicitbrightmulti-solitonsolutionsina

1
recursive manner. Thepropagation characteristic ofsolitons andtheirinteractions undertheperiodic
v
5 planewavebackgroundarediscussed. Finally,themodulationalinstabilityofsolutionsisanalyzedin
4
1 thepresence ofsmallperturbation.
1
PACSnumbers: 02.30.Jr,05.45.Yv,02.30IK.
.
1
0 Keywords: Exact solution, Bright matter wave soliton, Conservation law, Modulational instability,
2
Nonlinear Schro¨dinger equationwithtime-dependent nonlinearity
1
:
v
i
X (Somefiguresinthisarticleareincolouronlyintheelectronic version)
r
a
1. Introduction
It is known that every atom in a Bose-Einstein Condensates (BECs) moves in an effective mean
field due to the other atoms and the mean field equation of motion governing the evolution of the
macroscopic wave function of the Bose-Einstein condensate is the so-called time-dependent Gross-
Pitaevskii(GP)equation [1]
∂Ψ(~r,t) ~2 2
i~ = ∇ +V (~r)+gΨ(~r,t)2 Ψ(~r,t), (1.1)
ext
∂t "− 2m | | #
where Ψ is the BEC order parameter, V is the external trapping potential and the coefficient g =
ext
4π~2a/mcharacterizestheeffectiveinteratomicinteractionsintheBECthroughthes-wavescattering
Correspondingauthor.Tel:[phone]-8136
∗
E-mailaddress:[email protected],[email protected](S.F.Tian)
1
length a. For a parabolic potential and time dependent scattering lengths V = ǫ2x2~ with a(t) =
ext − 4
2π~a, ~ = 2m, the above GP equation in one dimension takes the following nonlinear Schro¨dinger
− m
equation withtime-dependent nonlinearity [2-5]
∂ψ(x,t) ∂2ψ(x,t) 1
i + +2a(t)ψ(x,t)2ψ(x,t)+ ǫ2x2ψ(x,t) = 0. (1.2)
∂t ∂x2 | | 4
HeretheFeshbach-managednonlinearcoefficienta(t)canberedefinedasa(t) = a (t)/a = g exp(ǫt)
s B 0
| |
(a istheBohrradius)[6],whichisalsocalledthetimedependentscatteringlength. InEq.(1.2),timet
B
andcoordinate xaremeasuredinunits2/ω anda ,wherea = (~/mω )1/2anda = (~/mω )1/2are
0 0
⊥ ⊥ ⊥ ⊥
linearoscillatorlengthsinthetransverseandcigar-axisdirections, respectively. ω andω arecorre-
0
⊥
spondingharmonicoscillatorfrequencies,mistheatomicmass,andtheparameterǫ = 2ω /ω 1.
0
| | ⊥ ≪
Investigation of the behaviour of Bose-Einstein Condensates (BECs) requires solving an inho-
mogeneous nonlinear Schro¨dinger equation known as the GP equation [1]. Eventhough numerical
solutionsofGPequationareavailable[7,8],constructionofanalyticsolutionswilloffermoreinsight
into the domain of BECsopening the doors for developing concrete applications of BECsin future.
Aswewellknow,itissignificantlyimportantinmathematicalphysicstosearchforexactsolutionsto
equation (1.2). Exact solutions play avital role inunderstanding various qualitative and quantitative
features ofnonlinear phenomena. Itiswellknown that searching for soliton solutions ofthe nonlin-
earevolution equationsisoneofthemostimportanttopicsinsolitontheory. Darbouxtransformation
(DT) [9, 10] has been proven to be one of the most fruitful algorithmic procedures to obtain exact
solutions ofthenonlinear evolution equations.
Themainaimofthepresentpaperistoconstructsomeinfinitenumberofconservation laws,ex-
plicitbright multi-soliton solutions byusingDTmethod, andinvestigate themodulational instability
of solutions of a general form of nonlinear Schro¨dinger equation with time-dependent nonlinearity
(1.2). In this paper, on the basis of the Lax pair associated with Eq.(1.2), we will derive an infi-
nite number of conservation laws to identify its complete integrability. Furthermore, we will apply
the Darboux transformation method to this integrable model and give the general procedure to re-
cursively generate the bright N-soliton solutions from an initial trivial solution. Moreover, we will
discuss the propagation characteristic and interactions of solitons under periodic plane wave back-
ground andanalyzethelinearstability ofthenonlinear planewaves.
The paper will be organized as follows: In Section 2, based on the linear eigenvalue problem
associated with Eq.(1.2) and an auxiliary 1D NLSE (2.2), we obtain a series of conservation laws.
Then the integrability is identified by admitting an infinite number of conservation laws. In Section
3, we will apply the Darboux transformation method to this integrable model and give the general
procedure to recursively generate the bright N-soliton solutions from an initial trivial solution. The
propagationcharacteristicofsolitonsandtheirinteractionsundertheperiodicplanewavebackground
arediscussedinSection4. InSection5,weanalyzethemodulationalinstabilityofthenonlinearplane
waves. Finally,someconclusions anddiscussions areprovided.
2
2. Lax pair and infinite conservation laws
In this section, using the linear eigenvalue problem associated with 1D NLSE(1.2) and an auxiliary
1D NLSE(2.2), we construct an infinite number of conservation laws for the 1D NLSE(1.2). Then
theintegrability isidentifiedbyadmittinganinfinitenumberofconservation laws.
Underthefollowingtransformation
ψ(x,t) = u(x,t)e−iǫx42−ǫ2t (2.1)
toEq.(1.2),wecanobtainthefollowingnew1DNLSE
∂ ∂2 ∂
i u(x,t)+ u(x,t) iǫx u(x,t)+2a(t)u(x,t)2u(x,t)e ǫt iǫu(x,t) = 0. (2.2)
−
∂t ∂x2 − ∂x | | −
ByvirtueoftheAblowitz-Kaup-Newell-Segurscheme[11,12],theLaxpairassociatedwithequation
(2.2)canbederivedas
Φ = UΦ, Φ = VΦ, (2.3)
x t
where Φ = (φ ,φ )T (the superscript T denotes the vector transpose) is the vector eigenfunction[2],
1 2
andthematricesU andV havethefollowingforms
U = λJ+P, (2.4a)
V = 2iλ2J+ǫλxJP+2iǫP+Q, (2.4b)
with
1 0 0 √g0u(x,t)
J =  , P =  ,
 0 −1  ig0u(x,t)2−√g0u(x,t)∗ √g0ǫx0u(x,t)+ i√g0u(x,t)x
Q =  | | − ,
 √g0ǫxu(x,t)∗ +i√g0u(x,t)∗x −ig0|u(x,t)|2 
where g = a(t)exp( ǫt) is an arbitrary function. Hereafter the asterisk stands for the complex
0
−
conjugate. From the compatibility condition U V + [U,V] = 0, one can derive Eq.(2.2) and
t x
−
Eq.(1.2)inthecaseofψ(x,t) = u(x,t)e−iǫx42−ǫ2t,respectively.
UsingLaxpair(2.3)andRefs.[13,14],wecanfurtherderiveaninfinitenumberofconservation
laws of Eqs.(1.2) and (2.2). By introducing the quantity f(x,t) = √g0u(x,t)φφ21, the linear equations
(2.3)canbetransformed intothefollowingRicattiequation
u(x,t) u(x,t)
f(x,t) +i xx f(x,t)+α(x,t) x f(x,t) 2ig f(x,t)2 2ig u(x,t)u(x,t) +β(x,t)f(x,t) = 0,
x 0 0 ∗
u(x,t) u(x,t) − −
(2.5)
with α(x,t) = 2iǫ 2i+ǫλx ǫx+iλ and β(x,t) = ǫλ ǫ+ǫλ2x+2iǫλ ǫλx 4iλ2. Substituting
− − − − −
f(x,t) = fn intoRicattiequation(2.5)andequatingthelikepowersofβ(x,t)tozero,wehave
∞n=1 β(x,t)n
arecursioPn formula
n
u u
f1 = 2ig0uu∗, fn+1 = 2ig0 fifn i fnx i xx fn α x fn = 0, (n = 2,3, ), (2.6)
Xi=1 − − − u − u ···
3
where f (n = 1,2, ) are the functions to be determined. By virtue of the compatibility condition
n
···
lnφ = lnφ ,weobtainthefollowingconservation form
1 xt 1 tx
(cid:0) (cid:1) (cid:0) (cid:1)
∂ ∂
i ρ (x,t)+ I (x,t)= 0, (2.7)
k k
∂t ∂x
whereρ (x,t)andI (x,t)(k = 1,2, )arecalled conserved densities andconserved fluxes, respec-
k k
···
tively. Thefirstthreesignificant physicalconservation lawsarepresented as
ρ (x,t) = 2ig u2, ρ (x,t) = 2ig (1+α)u u 2ig uu +2g u u ,
1 0| | 2 − 0 x ∗− 0 ∗x 0 xx ∗
ρ (x,t) = 8ig3u4 2g u u +2i(1+α)u u +2iǫg (λ 1)u u +2i(1+3α)g u u
3 − 0| | − 0 xx ∗x xx ∗ 0 − x ∗ 0 x ∗x
1
2ig u2 u 2iα(1+α)g u2u +2(1+2α)g u u u ,
− u 0 xx ∗− 0 x ∗ 0 x xx ∗
h i
I (x,t)= ig u2 2g √g u u +2i(ǫλx+2iǫ ǫx)g √g uu ,
1 0 0 0 x ∗ 0 0 ∗
| | − −
u u u u2u
I2(x,t)= ig0|u|2+2g0√g0uxu∗x +2ig0√g0 x uxx ∗ +2(1+α)g0√g0 xu∗ −2ig0√g0(ǫλx+2iǫ
ǫx) uu +iu u +(1+α)u u ,
− ∗x xx ∗ x ∗
(cid:2) u(cid:3)
I (x,t)= ig u2+ √g ǫλx+2iǫ ǫx+i x
3 0 0
| | (cid:18) − u (cid:19)
8ig3u4 2g u u +2i(1+α)u u +2iǫ(λ 1)g u u +2i(1+3α)g u u
× − 0| | − 0 xx ∗x xx ∗ − 0 x ∗ 0 x ∗x
h i
√g0 ǫλx+2iǫ ǫx+iux 2ig u2 u 2iα(1+α)g u2u +2(1+2α)g u u u .
− u (cid:18) − u (cid:19)h 0 xx ∗− 0 x ∗ 0 x xx ∗i
+ +
The conserved quantities J = 2ig0 ∞|u|2dx, H = −2ig0 ∞(1+α)uxu∗ +uu∗x +iuxxu∗dx and
K = 2ig0 ∞ −4g20|u|4+iuxxu∗x+g10(1R+−∞α)uxxu∗+ǫ(λ−1)uxu∗+R(−1∞+3α)uxu∗x−u1 u2xxu∗−α(1+α)u2xu∗
i(1+2α)Ru−∞u u ]dxrepresent theenergy, momentumandHamiltonian, resphectively.
x xx ∗
−
3. Darboux transformation and bright soliton solutions
Inthissection,theDarbouxtransformationmethodisappliedtothisintegrablemodelandthegeneral
procedure is presented to recursively generate the bright N-soliton solutions from an initial trivial
solution. BasedonLaxpair(2.3),theuseoftheDarbouxtransformationtoconstructthebrightsoliton
solution of nonlinear partial differential equations is an optimum choice among various methods in
soliton theory [15-26]. Owing to its purely algebraic algorithm, with the symbolic computation,
the analytical N-soliton solution can be generated through successive application of the Darboux
transformation [31,32].
3.1 Darboux transformation
The Darboux transformation is actually a gauge transformation Φ[1] = TΦ of the spectral problem
(2.3)byconsidering thefollowinggaugetransformation
Φ[1] = (λI S)Φ with S = HΛH 1, Λ = diag(λ ,λ ), (3.1)
− − 1 ∗1
where H isanonsingular matrix. Itisrequired thatΦ[1]solvesthesamespectral problems(2.3)
Φ[1] = U[1]Φ[1] and Φ[1] = V[1]Φ[1], (3.2)
x t
4
withU[1] = λJ+P[1],V[1] = 2iλ2J+ǫλxJP[1]+2iǫP[1]+Q[1]and
0 √g0u[1] 1 0
P[1] =  , J =  ,
 −√g0u[1ig]0∗u[1]20  √g00ǫxu[−11]+i√g0u[1]x
Q[1] =  | | − , (3.3)
 √g0ǫxu[1]∗ +i√g0u[1]∗x −ig0|u[1]|2 
whereu[1] = ψ[1]eiǫx42+ǫ2t. Substituting Eq. (3.1)intoEqs. (3.2),wehavethefollowingrelationship
P[1] = P+[J,S], S +[S,JS +P]= 0. (3.4)
x
Nowwediscussaconcretetransformation. ItiseasytoverifythatifΦ = (φ ,φ )T isaneigenfunction
1 2
of Eqs. (2.3) with eigenvalue λ = λ , then (φ , φ )T is also an eigenfunction of Eqs. (2.3) with
1 2 − ∗1
eigenvalue λ= λ . Thuswetakethematrix H intheform
∗1
φ φ
H =  1 ∗2 . (3.5)
 φ2 −φ∗1 
Therefore, bymeansofEqs. (3.4)and(3.5),theonceiterated newpotential ofEqs.(1.2)and(2.2)are
givenby
ψ[1] = u[1]e−iǫx42−ǫ2t,
(λ λ )φ φ
u[1] = u+2 1 − ∗1 1 ∗2. (3.6)
φ φ +φ φ
1 ∗1 2 ∗2
It is straightforward to verify that the Darboux transformations (3.1) and (3.6) can simultaneously
preserve theformoflineareigenvalue problem(2.3).
According totheabovecomputation, wehavethefollowingpropositions.
Proposition 3.1. Whenu(x,t)isgiven, let(φ ,φ )T bethesolution ofEqs.(2.3)withλ = λ ,thenby
1 2 k
usingoftheDarbouxmatrixT (3.1)andDarbouxtransformation (3.6),wehave
U[1] = (T +TU)T 1, with T = λI S. (3.7)
x −
−
Wecanobtainthesameproposition abouttheauxiliary spectral problem.
Proposition 3.2. Whenu(x,t)isgiven, let(φ ,φ )T bethesolution ofEqs.(2.3)withλ = λ ,thenby
1 2 k
usingoftheDarbouxmatrixT (3.1)andDarbouxtransformation (3.6),wehave
V[1]= (T +TV)T 1 with T = λI S. (3.8)
t −
−
Proof. Substituting Eqs. (3.1), (3.2) and (3.6) into Eq. (3.8) by a direct calculation, we can obtain
theconclusion. Theproofiscompleted. 2
Propositions 3.1 and 3.2 show that the transformation (3.1) and (3.6) change the Lax pair (2.3)
intoanotherLaxpairofthetype(3.2)withU[1]andV[1]having thesameformasU andV,respec-
tively. ThereforebothoftheLaxpairsleadtothesameequation (2.2),soEqs. (3.1)and(3.6)arethe
5
Darbouxtransformation ofEq.(2.2). Frompropositions 3.1and3.2,wehavefollowingtheorem.
Theorem3.3. Assumingthatψ,u,(φ[1,λ ] ,φ[1,λ ] )T,(φ[2,λ ] ,φ[2,λ ] )T, ,(φ[N,λ ] ,φ[N,λ ] )T
1 1 1 2 2 1 2 2 N 1 N 2
···
be the solution of the 1D NLSE (2.2) and N linearly independent solutions of the linear eigenvalue
problem (2.3), respectively, and after iterating the Darboux transformation (3.1) and (3.6) N times
analogoustotheaboveprocedure,wecanfurtherobtaintheNth-iteratedpotentialtransformationas
ψ[N] = u[N]e−iǫx42−ǫ2t,
N (λ λ )φ[i,λ] φ[i,λ]
u[N] = u+2 i− ∗i i 1 i ∗2 , (3.9)
φ[i,λ] φ[i,λ] +φ[i,λ] φ[i,λ]
Xi=1 i 1 i ∗1 i 2 i ∗2
with
A
φ[i+1,λi+1]j = (λi+1 −λ∗i)φ[i,λi+1]j− Bi(λi−λ∗i)φ[i,λi]j,
i
Ai = φ[i,λi]∗1φ[i,λi+1]1+φ[i,λi]∗2φ[i,λi+1]2,
Bi = φ[i,λi+1]1φ[i,λi+1]∗1+φ[i,λi+1]2φ[i,λi+1]∗2, (i= 1,2,··· ,N −1, j = 1,2), (3.10)
where (φ[k,λ ] ,φ[k,λ ] )T (k = 1,2, ,N) is the eigenfunction of Eqs. (2.3) with the eigenvalue
k 1 k ∗2 ···
λ = λ andpotential u[k 1].
k
−
Proof. (By induction) The case N = 1 follows from Darboux transformation (3.1) and (3.6). Now
assumethatequation (3.9)holdsfor N = n. Then
ψ[n] = u[n]e−iǫx42−ǫ2t,
n (λ λ )φ[i,λ] φ[i,λ]
u[n] = u+2 i− ∗i i 1 i ∗2 , (3.11)
φ[i,λ] φ[i,λ] +φ[i,λ] φ[i,λ]
Xi=1 i 1 i ∗1 i 2 i ∗2
where
A
φ[i+1,λi+1]j = (λi+1−λ∗i)φ[i,λi+1]j− Bi(λi −λ∗i)φ[i,λi]j,
i
Ai = φ[i,λi]∗1φ[i,λi+1]1+φ[i,λi]∗2φ[i,λi+1]2,
Bi = φ[i,λi+1]1φ[i,λi+1]∗1+φ[i,λi+1]2φ[i,λi+1]∗2, (i = 1,2,··· ,n−1, j= 1,2). (3.12)
IncaseofN = n+1,usingDarbouxtransformation (3.6),oneobtains
u[n+1] =u[n]+2 (λn+1 −λ∗n+1)φ[n+1,λn+1]1φ[n+1,λn+1]∗2 ,
φ[n+1,λn+1]1φ[n+1,λn+1]∗1+φ[n+1,λn+1]2φ[n+1,λn+1]∗2
n+1 (λ λ )φ[i,λ] φ[i,λ]
=u+2 i− ∗i i 1 i ∗2 . (with Eq.(3.11)) (3.13)
φ[i,λ] φ[i,λ] +φ[i,λ] φ[i,λ]
Xi=1 i 1 i ∗1 i 2 i ∗2
Basedonthefirstequation ofDarbouxtransformation (3.6),itiseasytoobtain
ψ[n+1] = u[n+1]e−iǫx42−ǫ2t. (3.14)
From Eqs.(3.13) and (3.14), the conclusions of Eqs.(3.9) and (3.10) are hold for N = n + 1. This
completestheproof. 2
6
3.2 Bright matter wavesolitonsolution
Brightsolitonhereisdefinedastheemergenceofapositivepulse. Brightsolitonsolutionshavebeen
investigated byoriginal Refs.[11,27,28]. Inthefollowing, theDarbouxtransformation isapplied to
construct the explicit bright soliton solutions of Eq.(1.2). Using the trivial solution ψ = 0 (u=0), we
solvethelinearequations (2.3)withλ= λ = 1(µ +iν )andobtaintheeigenfunction
1 2 1 1
φ1 = e21(℘1+iϑ1), φ2 = e−12(℘1+iϑ1), (3.15)
with
℘ = µ x 2µ ν tξ , ϑ = ν x+ µ2 ν2 t+η , (3.16)
1 1 − 1 1 1 1 1 1− 1 1
(cid:16) (cid:17)
whereξ andη arearbitrary constants.
1 1
Thebrightone-soliton solution forEq.(1.2)areobtained as
ψ(x,t) = u(x,t)e−iǫx42−ǫ2t,
u(x,t) = iν eiϑ1sech℘ . (3.17)
1 1
by substitution of the above results into formula (3.6). It implies that the imaginary part ν of the
1
eigenvalueλ determinestheamplitudeofthesolitonsψ(x,t)andu(x,t),whilethevelocityofsolitons
1
arerelatedtobothrealandimaginarypartsoftheeigenvalue λ .
1
Fromabove,wehavethefollowingpropositions.
Proposition 3.4. Assuming that (φ[1,λ ] ,φ[1,λ ] )T and (φ[2,λ ] ,φ[2,λ ] )T betwolinearly in-
1 1 1 2 2 1 2 2
dependentsolutionsofthelineareigenvalueproblem(2.3)correspondingtotwodifferenteigenvalues
λ = 1(µ + iν ) and λ = 1(µ + iν ), respectively, we obtain the bright two-soliton solutions of
1 2 1 1 2 2 2 2
Eq.(1.2)andEq.(2.2)fromformula(3.9)withN = 2
ψ[2] = u[2]e−iǫx42−ǫ2t,
U1eiϑ2cosh℘1+U2eiϑ1cosh℘2+U3 eiϑ1sinh℘2 eiϑ2sinh℘1
u[2] = i − , (3.18)
V cosh(℘ +℘ )+V cosh(℘ ℘(cid:16)) 2ν ν cosh(ϑ ϑ ) (cid:17)
1 1 2 2 1 2 1 2 2 1
− − −
with
℘ = µ x 2µνtξ, ϑ = ν x+ µ2 ν2 t+η,
i i − i i i i i i − i i
(cid:16) (cid:17)
U = ν (µ µ )2 ν2+ν2 , U = ν (µ µ )2+ν2 ν2 ,
1 2 1− 2 − 1 2 2 1 1− 2 1− 2
h i h i
1 1
U = 2iν ν (µ µ ), V = (µ µ )2+ (ν ν )2,
3 1 2 1 2 1 1 2 1 2
− − 2 − 2 −
1 1
V = (µ µ )2+ (ν +ν )2, (i= 1,2), (3.19)
2 1 2 1 2
2 − 2
whereξ,η areallcomplexconstants andtheparameters µ,ν , 0(i = 1,2).
i i i i
Proof. It is straightforward to prove this proposition by using the initial values (3.15), (3.16), and
Theorem3.3foru = 0,N = 2. 2
Proposition3.5. Assumingthat(φ[1,λ ] ,φ[1,λ ] )T,(φ[2,λ ] ,φ[2,λ ] )T, ,(φ[N,λ ] ,φ[N,λ ] )T
1 1 1 2 2 1 2 2 N 1 N 2
···
beN linearlyindependent solutionsofthelineareigenvalue problem(2.3)corresponding toN differ-
enteigenvalues λ = 1(µ +iν ),λ = 1(µ +iν ), ,λ = 1(µ +iν ),respectively, weobtainthe
1 2 1 1 2 2 2 2 ··· N 2 N N
7
bright N-soliton solutions ofEqs.(1.2)and (2.2)fromformula(3.9)withN
ψ[N] = u[N]e−iǫx42−ǫ2t,
N (λ λ )φ[i,λ] φ[i,λ]
u[N] = 2 i − ∗i i 1 i ∗2 , (3.20)
φ[i,λ] φ[i,λ] +φ[i,λ] φ[i,λ]
Xi=1 i 1 i ∗1 i 2 i ∗2
with
℘ = µ x 2µνtξ, ϑ = ν x+ µ2 ν2 t+η, (i = 1,2, ,N), (3.21)
i i − i i i i i i − i i ···
(cid:16) (cid:17)
whereξ,η areallrealconstantsandtheparametersµ,ν , 0(i = 1,2, ,N).(φ[k,λ ] ,φ[k,λ ] )T
i i i i ··· k 1 k ∗2
satisfiesEq.(3.10)andistheeigenfunction ofEqs. (2.3)withtheeigenvalueλ= λ andpotentialu[k]
k
(k = 1,2, ,N).
···
Proof. It is straightforward to prove this proposition by using the initial values (3.15), (3.16), and
Theorem3.3foru = 0. 2
Thegraphsofthebrightone-solitonperiodicwavesolution(3.17)andtwo-solitonperiodicwave
solution (3.18)areplottedinFig. 1andFig. 2,respectively.
–0.06
–0.04 –0.06
–0.02 –0.04
0 –0.02
0
0.02
0.02
0.04
0.04
0.06
0.06
–10 –10 –10 –10
–5 –5 –5 –5
x 0 5 5 0 t x 0 5 5 0 t
10 10 10 10
(a) (b)
0.08
0.06
0.05 0.04
0.02
–80 –60 –40 –20 0 20 40 –80 –60 –40 –20 0 20 40
x x
–0.02
–0.05 –0.04
–0.06
–0.1 –0.08
(c) (d)
Fig. 1. (Coloronline)Asymmetricbrightone-solitonsolution ψ(x,t)2 ofEq.(1.2)withparameters: µ = 1,
1
| |
ν =0.1,ξ =1+i,η =3andǫ=0.01.Thisfigureshowsthatthesymmetricbrightone-solitonperiodicwave
1 1 1
isspatiallyperiodicintwodirections,butitneednotbeperiodicineitherthe xortdirections. (a)Perspective
viewfortherealpartofwave. (b)Perspectiveviewoftheimaginarypartwave. (c)Wavepropagationpattern
of the real partwave alongthe x axis. (d) Wave propagationpatternof the imaginarypart wave along the x
axis.
8
4 6
2 4
0 2
0
–2
–2
–4 –4
6 6 6 6
4 4 4 4
2 2 2 2
0 0 0 0
t –2 –2 x t –2 –2 x
–4 –4 –4 –4
–6 –6 –6 –6
(a) (b)
5
4 4
3
2
2
1
–10 –8 –6 –4 –2 0 2 4 6 8 10
t
–10 –8 –6 –4 –2 0 2 4 6 8 10
–2
–1 t
–2 –4
(c) (d)
Fig. 2. (Coloronline)Asymmetricbrighttwo-solitonsolution ψ(x,t)2ofEq.(1.2)withparameters:µ =0.1,
1
| |
ν =0.1,ξ =i,η =0.1i,µ =1,ν =1,ξ =i,η =0.1i,andǫ=0.01.Thisfigureshowsthatthesymmetric
1 1 1 2 1 2 2
bright two-soliton periodic wave is spatially periodic in two directions, but it is periodic in either the x or t
directions. (a)Perspectiveviewfortherealpartofwave. (b)Perspectiveviewoftheimaginarypartwave. (c)
Wavepropagationpatternoftherealpartwavealongthetaxis. (d)Wavepropagationpatternoftheimaginary
partwavealongthetaxis.
4. Bright soliton interactions on the periodic background
Inthissection,weinvestigatesomebrightsolitoninteractionsontheperiodicbackgroundbyusingEqs.(3.6)
and(3.9). ThesimpleexactsolutionofEq.(1.2)istheplanewave
ψ(x,t)0 =u(x,t)0e−iǫx42−ǫ2t,
u(x,t) =γei(kx+ωt), (4.1)
0
whereγandkarerealconstants,andthefrequencyωsolvesthenonlineardispersionrelation
ǫ2
ω= k2+ǫk+2γ2 iǫ+ . (4.2)
− − 4
Consideringsolution(4.1)astheinitialseedsolutionofEq.(1.2),weobtainthelinearequations(2.3)andhave
theeigenfunctioncorrespondingtotheeigenvalueλ intheform
1
φ[1,λ1]1 =d1eD1 +d2eD2, φ[1,λ1]2 =d3e−D1 +d2e−D2, (4.3)
with
1 1
D1 = i(kx+ωt)+Γ(x+∆t), D2 = i(kx+ωt) Γ(x+∆t),
2 2 −
d ik k 2 d ik
d = 2 Γ λ , Γ2 = λ + γ2, d = 1 +Γ λ ,
3 γ√g0 2 − − 1! 1 2! − 4 γ√g0 2 − 1!
ǫk ǫ ǫ2 k 2
∆=kλ ǫk 2iǫλ i +iγ2 i +(2iǫ k) λ + γ2 , (4.4)
1− − 1− 2 − 2 − 4 −  1 2! − 
 
9
whered andd aretwoarbitrarycomplexconstants.
1 2
Substituting expressions(4.3) into formula (3.9), the one-soliton and multi-soliton solutionson the pe-
riodic background can be obtained with the iterative algorithm of the Darboux transformation. The bright
one-solitonsolutionsareplottedinFigure3,whichshowsfourkindsofsolitonprofilestructureswithdifferent
wavenumbers.AlthoughsolitonsinFigures3(c)and3(d)holdlargerwavenumbersthanthoseofFigures3(a)
and3(b),theybothcanpropagatestablyforlongdistancesbytheresultsofnumericalsimulationsofnonlinear
pulses.
Proposition4.1. Assuming that(φ[1,λ ] ,φ[1,λ ] )T be the solution ofthe lineareigenvalueproblem (2.3)
1 1 1 2
witheigenvaluesλ ,weobtainthebrighttwo-solitonsolutionsofEqs.(1.2)and(2.2)correspondingtoeigen-
1
valuesλ fromformula(3.9)withN =2
2
ψ[2]=u[2]e−iǫx42−ǫ2t,
2 (λ λ )φ[i,λ] φ[i,λ]
u[2]=u(x,y) +2 i− ∗i i 1 i ∗2 . (4.5)
0 φ[i,λ] φ[i,λ] +φ[i,λ] φ[i,λ]
Xi=1 i 1 i ∗1 i 2 i ∗2
Thesolutionofthelineareigenvalueproblem(2.3)witheigenvaluesλ is
2
(λ +λ )φ[1,λ ] φ[1,λ ] 2 λ φ[1,λ ] φ[1,λ ] λ φ[1,λ ] φ[1,λ ]2
φ[2,λ ] = 1 ∗1 1 1| 1 2| − ∗1 1 1 1 ∗2− 1 1 1 1 2,
2 1 φ[1,λ ] φ[1,λ ] +φ[1,λ ] φ[1,λ ]
1 1 1 ∗1 1 2 1 ∗2
(λ λ )φ[1,λ ] 2φ[1,λ ] +λ φ[1,λ ] 2φ[1,λ ] λ φ[1,λ ]3
φ[2,λ ] = 1− ∗1 | 1 1| 1 2 ∗1| 1 1| 1 ∗2− 1 1 2. (4.6)
2 2 φ[1,λ ] φ[1,λ ] +φ[1,λ ] φ[1,λ ]
1 1 1 ∗1 1 2 1 ∗2
Thebrighttwo-solitonsolutionsareplottedinFigures4. Figure4depictsthebrighttwo-solitonperiodic
wave solutionsin the three-dimensionalspace. From the interactionprocessin two sets of figures, it can be
clearlyseenthattheinteractingsolitonslikeparticlescrosseachotherunaffectedlyonlybyaphaseshift,and
their respectiveamplitudesand velocitiesare the same as those beforecollision. Furthertheoreticalanalysis
for soliton solutions shows that the sign and value of real part of the eigenvaluedetermines the propagation
directionofsoltionandtheamplitude,respectively.
30 25
25
20
20
15
15
10
10
5 5
0 0
–8 –6 –4 –2 0 2 4 6 8 10 –10 –6 –4 –2 0 2 4 6 8 10
x x
(a) (b)
30
25 20
20
15
15
10
10
5 5
–8 –6 –4 –2 0 2 4 6 8 10 –8 –6 –4 –2 0 2 4 6 8 10
x x
(c) (d)
Fig. 3. (Color online) Four kinds of bright one-soliton solutions ψ(x,t)2 of Eq.(1.2) with different wave
| |
10