Rogue waves of the Hirota and the Maxwell-Bloch equations

2.4 MB·English

by  Chuanzhong Li

Rogue waves of the Hirota and the Maxwell-Bloch equations
Chuanzhong Lia, Jingsong Hea,∗ and K. Porseizanb
aDepartment of Mathematics, Ningbo University, Ningbo, 315211, China
bDepartment of Physics, Pondicherry University, Puducherry 605014, India.
In this paper, we derive a Darboux transformation of the Hirota and the Maxwell-Bloch(H-
MB) system which is governed by femtosecond pulse propagation through an erbium doped fibre
and further generalize it to the matrix form of the n-fold Darboux transformation of this system.
This n-fold Darboux transformation implies the determinant representation of n-th new solutions
of (E[n],p[n],η[n]) generated from known solution of (E,p,η). The determinant representation of
(E[n],p[n],η[n]) provides soliton solutions, positon solutions, and breather solutions (both bright
3
and dark breathers) of the H-MB system. From the breather solutions, we also construct bright
1
anddarkroguewavesolutionsfortheH-MBsystem,whichiscurrentlyoneofthehottesttopicsin
0
mathematics and physics. Surprisingly, the rogue wave solution for pandη has two peaks because
2
of the order of the numerator and denominator of them. Meanwhile, after fixing time and spatial

n parametersandchangingothertwounknownparametersαandβ, wegeneratearoguewaveshape
a for the first time.
J

5
2 PACS numbers: 42.65.Tg, 42.65.Sf, 05.45.Yv, ters. It has been proved that modulational instability
02.30.Ik. isoneofthemaingeneratingmechanismsfortherogue

] waves[12–17]andcanbewell-describedbytheanalyt-
I
S ical expressions for the spectra of breather solutions
. at the point of extreme compression.
n
i In 1967, McCall and Hahn [18] explored a special
nl I. INTRODUCTION typeoflosslesspulsepropagationintwo-levelresonant
[ media. They have discovered the self-induced trans-
In the past four decades, nonlinear science has ex- parency (SIT) effect which can be explained by using
3
perienced an explosive growth with the invention of theMaxwell-Bloch(MB)system. Ifweconsiderthese
v
several exciting and fascinating new concepts such as effects in erbium doped nonlinear fibre, the system
1
9 solitons, dromions, positons, rogue waves, similari- will be governed by the coupled system of the NLS
1 tons,supercontinuumgeneration,completeintegrabil- and the MB equation (NLS-MB system)[19–23].
1 ity, fractals, chaos etc. Many of the completely inte- Rogue waves have been reported in different
5. grable nonlinear partial differential systems (NPDEs) branches of physics, where the system dynamics is
0 admit one of the most striking aspects of nonlinear governed mostly by a single nonlinear partial differ-
2 phenomena,describedasthesoliton,auniversalchar- ential equation [13, 14]. But our main interest is to
1 acterandisofgreatmathematicalinterest. Thestudy analyzethepossibilityofroguewavesincouplednon-
v: ofthesolitonsandotherrelatedsolutionslikepositons linear systems. The higher-order NLS and Maxwell-
i have become one of the most exciting and extremely Bloch (HNLS-MB) system as a higher-order correc-
X active areas of research in the field of nonlinear sci- tionofNLS-MBsystemwereshowntoadmitLaxpair
r ences. and soliton-type pulse propagation [25–27]. Kodama
a
Among all concepts, in addition to solitons and [24] has shown that with a suitable transformation,
positons [1–4], rogue waves have also been not only higher order NLS equation can be reduced to the Hi-
the subject of intensive research in oceanography [5– rota equation [28] whose rogue wave solution has al-
7] but also they have been studied extensively in sev- readybeenreportedin[29,30]. Inasimilarway,after
eral other areas, such as matter rogue wave [8, 9] in suitablechoiceofself-steepeningandself-frequencyef-
Bose-Einsteincondensates,roguewavesinsurfaceand fects, we obtain the H-MB system in the following
space plasmas [10], financial rogue waves describing form [31]:
thepossiblephysicalmechanismsinfinancialmarkets
and related fields [11]. In some of the above fields,
1
soliton system such as nonlinear Schr¨odinger (NLS) E = iα( E +|E|2E)+β(E +6|E|2E )+2p,
z 2 tt ttt t
equation [12], derivative NLS system [13, 14] and so
(1.1)
on are considered and reported to admit rogue wave
solutions under a certain specific choice of parame- pt = 2iωp+2Eη, (1.2)
η = −(Ep∗+E∗p), (1.3)
t
where E is the normalized slowly varying amplitude
∗Electronicaddress: [email protected] of the complex field envelope, p is the polarization,
η means the population inversion, (ω,α,β) are three be given. Using these Darboux transformations, one
real constants and ∗ represents complex conjugate. β soliton, two soliton and positon solutions are derived
represents the strength of the higher order linear and in section 5 & 6 by assuming trivial seed solutions.
nonlinear effects. In section 7, starting from a periodic seed solution,
The H-MB system has been shown to be integrable breather solution of the H-MB system is provided. A
and also admits a Lax pair and other required prop- Taylor expansion from breather solution will help us
erties of complete integrability [31]. Among many toconstructtheroguewavesolutioninsection8. Sec-
analytical methods, it is well known that the Dar- tion 9 is devoted to conclusion and discussions.
bouxtransformationisoneoftheefficientmethodsto
generate the soliton solutions for integrable systems
[32]. The determinant representation of n-fold Dar-
II. LAX REPRESENTATION OF THE H-MB
boux transformation of the Ablowitz-Kaup-Newell-
SYSTEM
Segur(AKNS)systemwasgivenin[33,34]. Themain
taskofthispaperwillbetoconstructn-foldDarboux
In this section, we will concentrate on the linear
transformation of the H-MB system and find differ-
eigenvalue problem of the Hirota and the Maxwell-
ent kinds of solutions of the H-MB system using the
Bloch(H-MB) system. The linear eigenvalue problem
Darboux transformation.
is expressed in the form of the Lax pair U and V as
The paper is organized as follows. In section 2, the
Lax representation of H-MB system is introduced. In
section 3, we derived the one-fold Darboux transfor-
mation of the H-MB system. In section 4, the gen- Φt = UΦ, Φz =VΦ, (2.1)
eralization of one-fold Darboux transformation to n-
foldDarbouxtransformationoftheH-MBsystemwill where
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
−i 0 0 E 1 0
U = λ + =−iλσ +U , σ = , (2.2)
0 i −E∗ 0 3 0 3 0 −1
(cid:18)4iβ 0 (cid:19) (cid:18) −αi −4βE(cid:19) (cid:18) −2βi|E|2 αE−2βiE (cid:19)
V = λ3 +λ2 +λ t (2.3)
0 −4iβ 4βE∗ αi −αE∗−2βiE∗ 2βi|E|2
t
(cid:18) αi|E|2−β(EE∗−E E∗) 2β|E|2E+ αiE +βE (cid:19) 1 (cid:18) η −p(cid:19)
+ 2 t t 2 t tt +i
−2β|E|2E∗+ α2iEt∗−βEt∗t −α2i|E|2+β(EEt∗−EtE∗) λ+ω −p∗ −η
1
:= λ3V +λ2V +λV +V +i V , (2.4)
3 2 1 0 λ+ω −1
Φ (λ,t,z)
Φ = Φ(λ)=( 1 ) (2.5)
Φ (λ,t,z)
2
is an eigenfunction associated with eigenvalue param- III. ONE-FOLD DARBOUX
eterλofthelinearEq. (2.1),andV denotesthecoeffi- TRANSFORMATION FOR THE H-MB
i
cientmatrixoftermλi. WeobtaintheclassicalHirota SYSTEM
and the Maxwell-Bloch system when α = 2,β = −1.
BeingdifferentfromAKNSsystem, onlyapartofthe In this section, we construct and prove the one-fold
V matrixispolynomialsintermsofE anditstderiva- Darboux transformation for the H-MB system. First,
tives in this system. Using the above linear system of we consider the transformation about linear function
the H-MB system, one-fold Darboux transformation Φ in the form
will be introduced in the next section.
Φ(cid:48) = TΦ=(λA−S)Φ, (3.1)
where
(cid:18) (cid:19) (cid:18) (cid:19)
a a s s
A= 11 12 , S = 11 12 . (3.2)
a a s s
21 22 21 22
2
The new function Φ(cid:48) satisfies λ2:
Φ(cid:48)t = U(cid:48)Φ(cid:48), (3.3) (ωV1+V0)−S(ωV2+V1) (3.13)
Φ(cid:48) = V(cid:48)Φ(cid:48). (3.4) = (ωV(cid:48)+V(cid:48))−(ωV(cid:48)+V(cid:48))S.
z 1 0 2 1
ThenthematrixT shouldsatisfythefollowingiden- λ3:
tities
(ωV +V )−S(ωV +V ) (3.14)
2 1 3 2
T +TU = U(cid:48)T, (3.5) = (ωV(cid:48)+V(cid:48))−(ωV +V(cid:48))S,
t 2 1 3 2
T +TV = V(cid:48)T. (3.6)
z λ4:
Substituting the matrices A and S into Eq. (3.5) and V(cid:48) =V −[S,V ]. (3.15)
comparing the coefficients of both sides will lead to 2 2 3
the following conditions From the above identities, after simplifications, we
get
a =a =0, (a ) =(a ) =0. (3.7)
12 21 11 t 22 t
E(cid:48) = E−2is , (3.16)
12
For our further discussions, we choose A = I and
T = (λI − S). The relation between old solutions
(E,p,η) and new solutions (E(cid:48),p(cid:48),η(cid:48)), which is called V−(cid:48)1 =(S+ω)V−1(S+ω)−1, (3.17)
Darboux transformation, can be obtained by using
which gives one-fold Darboux transformation of the
Eqs. (3.5) and (3.6).
H-MB system later.
From Eq. (3.5), we have
We suppose
E(cid:48) = E−2is , s =−s∗ (3.8)
12 21 12 S =HΛH−1 (3.18)
(cid:18) (cid:19) (cid:18) (cid:19)
0 E 0 E
S = S−S +i[S,σ ]S.
t −E∗ 0 −E∗ 0 3 (cid:18)λ 0(cid:19)
where Λ= 1 ,
(3.9) 0 λ
2
(cid:18) (cid:19) (cid:18) (cid:19)
Φ (λ ,t,z) Φ (λ ,t,z) Φ Φ
Similarly, using Eq. (3.6), we obtain the following set H = 1 1 1 2 := 1,1 1,2 .
Φ (λ ,t,z) Φ (λ ,t,z) Φ Φ
of relations 2 1 2 2 2,1 2,2
In order to satisfy the constraints of S and V(cid:48)
−1
−S +(λI−S)(λ3V +λ2V +λV +V + iV−1 ) which is similar to V−1, i.e. s21 = −s∗12, following
z 3 2 1 0 λ+ω constraints will be used
iV(cid:48)
=(λ3V +λ2V(cid:48)+λV(cid:48)+V(cid:48)+ −1 )(λI−S). λ = λ∗, s =s∗ , (3.19)
3 2 1 0 λ+ω 2 1 11 22
(cid:18)Φ (λ ,t,z) −Φ∗(λ ,t,z)(cid:19)
(3.10) H = 1 1 2 1 . (3.20)
Φ (λ ,t,z) Φ∗(λ ,t,z)
2 1 1 1
MultiplyingbothsidesofEq.(3.10)byλI−S willlead
After tedious calculations, Eqs. (3.16)-(3.20) and
to
Eq. (2.1) will lead to Eq. (3.5) and Eq. (3.6), i.e.
the transformation Eq. (3.16) and Eq. (3.17) with
−S (λ+ω)+(λI−S)(−4iλ3(λ+ω)σ the conditions Eq. (3.19), Eq. (3.20) is the Darboux
z 3
transformation of the H-MB system.
+λ2(λ+ω)V +λ(λ+ω)V +V (λ+ω)+iV )
2 1 0 −1 The detailed form of one-fold Darboux transforma-
= (−4iλ3(λ+ω)σ3+λ2(λ+ω)V2(cid:48) tion of the H-MB system in terms of eigenfunctions
+λ(λ+ω)V(cid:48)+V(cid:48)(λ+ω)+iV(cid:48) )(λI−S). will be given in the next section.
1 0 −1
Collecting the different powers of λ, we obtain the
following set of identities IV. DETERMINANT REPRESENTATION OF
λ0: n-FOLD DARBOUX TRANSFORMATION
S = (V(cid:48)+iω−1V(cid:48) )S−S(iω−1V +V ),
z 0 −1 −1 0 In this section, we will construct the determinant
(3.11) representation of the n-fold Darboux transformation
of the H-MB system. For this purpose, we introduce
λ:
n eigenfunctions
S = (ωV +iV )−S(ωV +V )+(ωV(cid:48)+V(cid:48))S (cid:18) (cid:19)
z 0 −1 1 0 1 0 Φ1,i =Φ| , i=1,2,...,2n (4.1)
+(−ωV0(cid:48)−iV−(cid:48)1). (3.12) Φ2,i λ=λi
3
with the following constraint on the eigenvalues where
λ = λ∗ and the reduction conditions on eigen-
2n−1 2n
functions as Φ = Φ∗ and Φ = −Φ∗ .
2,2n 1,2n−1 2,2n−1 1,2n
For our further discussions, this reduction condition
has been used. T∆1 = [(ω+λ1)|Φ1,1|2+(ω+λ∗1)|Φ2,1|2]
For completeness, as the simplest Darboux trans- [(ω+λ∗)|Φ |2+(ω+λ )|Φ |2]
1 1,1 1 2,1
formation, the determinant representation of one-fold
−(λ∗−λ )2|Φ |2|Φ |2. (4.5)
Darboux transformation of the H-MB system will be 1 1 1,1 2,1
introduced in the following theorem using identities
(3.16) and (3.17).
It can be easily proved that the new solution η[1]
The one-fold Darboux transformation of the H-MB
is always real. This one-fold transformation will be
system is expressed as
used to generate the one-soliton solution from trivial
seedsolutionsoftheH-MBsystem. Alsothisone-fold
(λ −λ∗)Φ Φ∗ Darbouxtransformationcanbefurthergeneralizedto
E[1] = E−2i 1 1 1,1 2,1, (4.2) construct the n-fold Darboux transformation of the
∆
1 H-MB system which is proposed in the following the-
1
p[1] = (2η[(−ω−λ )|Φ |2+(−ω−λ∗)|Φ |2] orem.
T 1 1,1 1 2,1
∆1
(λ∗−λ )Φ Φ∗ −p∗(λ∗−λ )2Φ2 Φ∗2
1 1 1,1 2,1 1 1 1,1 2,1 Theorem IV.1. The n-fold Darboux transformation
+p[(−ω−λ1)|Φ1,1|2+(−ω−λ∗1)|Φ2,1|2]2), of the H-MB system can be represented as
(4.3)
1
η[1] = [η([(ω+λ )|Φ |2+(ω+λ∗)|Φ |2]
T 1 1,1 1 2,1
∆1 Tn(λ;λ1,λ2,λ3,λ4,...,λ2n) (4.6)
[(ω+λ∗)|Φ |2+(ω+λ )|Φ |2]
1 1,1 1 2,1 = λnI+t[n] λn−1+···+t[n]λ+t[n] (4.7)
+(λ∗1−λ1)2|Φ1,1|2|Φ2,1|2) 1 (cid:18)(Tn−)1 (T ) (cid:19) 1 0
−p∗(λ1−λ∗1)Φ1,1Φ∗2,1[(ω+λ∗1)|Φ1,1|2 = ∆n (Tnn)1211 (Tnn)2212 (4.8)
+(ω+λ )|Φ |2]+p[(ω+λ )|Φ |2
1 2,1 1 1,1
+(ω+λ∗)|Φ |2](λ −λ∗)Φ∗ Φ ], (4.4) where
1 2,1 1 1 1,1 2,1
(cid:12)(cid:12) Φ1,1 Φ2,1 λ1Φ1,1 λ1Φ2,1 ... λn1−1Φ1,1 λn1−1Φ2,1 (cid:12)(cid:12)
∆n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ΦΦΦ111.,,,432 ΦΦΦ222.,,,432 λλλ432ΦΦΦ.111,,,432 λλλ432ΦΦΦ.222,,,432 .......... λλλn4n3n2−−−111.ΦΦΦ111,,,432 λλλ3n24nn−−−111.ΦΦΦ222,,,432 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12) . . . . . . . (cid:12)
(cid:12) . . . . . . . (cid:12)
(cid:12) (cid:12)
(cid:12)Φ Φ λ Φ λ Φ ... λn−1 Φ λn−1 Φ (cid:12)
(cid:12) 1,2n−1 2,2n−1 2n−1 1,2n−1 2n−1 2,2n−1 2n−1 1,2n−1 2n−1 2,2n−1(cid:12)
(cid:12) Φ Φ λ Φ λ Φ ... λn−1Φ λn−1Φ (cid:12)
1,2n 2,2n 2n 1,2n 2n 2,2n 2n 1,2n 2n 2,2n
(Tn)11 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ΦΦ1ΦΦΦΦ,1211111,...n,,,,21234−n1 ΦΦ2ΦΦΦΦ,2222220,...n,,,,21234−n1 λ2nλ−2λλλλn12134ΦΦΦΦΦΦλ...111111,,,,,,221234nn−1 λ2nλ−2λλλλn11234ΦΦΦΦΦΦ0...222222,,,,,,221234nn−1 ........................ λn2λnλλλλ−n2−n1n2n3n4n1−λ−−−−1n1Φ1111...Φ−1ΦΦΦΦ1,111112,,,,,2n1234n−1 λn2λnλλλλ−n2−n1n2n3n4n1−−−−−11Φ11110...Φ2ΦΦΦΦ2,22222,,,,,2n1234n−1 λn2nλ−λλλλn2nn1n2n3n41λΦΦΦΦΦΦ...n111111,,,,,,221234nn−1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
4
(Tn)12 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ΦΦ1ΦΦΦΦ,1211110,...n,,,,21234−n1 ΦΦ2ΦΦΦΦ,2222221,...n,,,,21234−n1 λ2nλ−2λλλλn11234ΦΦΦΦΦΦ0...111111,,,,,,221234nn−1 λ2nλ−2λλλλn11234ΦΦΦΦΦΦλ...222222,,,,,,221234nn−1 ........................ λn2λnλλλλ−n2−n1n2n3n4n1−−−−−11Φ11110...Φ1ΦΦΦΦ1,11112,,,,,2n1234n−1 λn2λnλλλλ−n2−n1n2n3n4n1−λ−−−−1n1Φ1111...Φ−2ΦΦΦΦ2,122222,,,,,2n1234n−1 λn2nλ−λλλλn2nn1n2n3n41ΦΦΦΦΦΦ0...111111,,,,,,221234nn−1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(Tn)21 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ΦΦ1ΦΦΦΦ,1211111,...n,,,,24321−n1 ΦΦ2ΦΦΦΦ,2222220,...n,,,,24321−n1 λ2nλ−2λλλλn14321ΦΦΦΦΦΦλ...111111,,,,,,224321nn−1 λ2nλ−2λλλλn14321ΦΦΦΦΦΦ0...222222,,,,,,224321nn−1 ........................ λn2λnλλλλ−n2−n4n3n2n1n1−λ−−−−1n1Φ1111...Φ−1ΦΦΦΦ1,111112,,,,,2n4321n−1 λ2nλnλλλλ−n2−n4n3n2n1n1−−−−−11Φ11110...Φ2ΦΦΦΦ2,22222,,,,,2n4321n−1 λn2nλ−λλλλn2nn4n3n2n11ΦΦΦΦΦΦ0...222222,,,,,,224321nn−1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(Tn)22 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ΦΦ1ΦΦΦΦ,1211110,...n,,,,21234−n1 ΦΦ2ΦΦΦΦ,2222221,...n,,,,21234−n1 λ2nλ−2λλλλn11234ΦΦΦΦΦΦ0...111111,,,,,,221234nn−1 λ2nλ−2λλλλn11234ΦΦΦΦΦΦλ...222222,,,,,,221234nn−1 ........................ λn2λnλλλλ−n2−n1n2n3n4n1−−−−−11Φ11110...Φ1ΦΦΦΦ1,11112,,,,,2n1234n−1 λn2λnλλλλ−n2−n1n2n3n4n1−λ−−−−1n1Φ1111...Φ−2ΦΦΦΦ2,122222,,,,,2n1234n−1 λn2nλ−λλλλn2nn1n2n3n41λΦΦΦΦΦΦ...n222222,,,,,,221234nn−1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12).
The following identity can be found to prove that these multiplications can have a deter-
minant representation as mentioned above.
(cid:18) (cid:19)
Φ
T (λ;λ ,λ ,λ ,λ ,...,λ )| 1,i =0,(4.9)
n 1 2 3 4 2n λ=λi Φ2,i The n-th new solution after the n-fold Darboux
transformation of the H-MB system will be
wherei=1,2,3,...,2n.Similarly,theDarbouxtrans-
formationfor(E,p,η)canbeconstructedbyusingthe
following identities
E[n] = E+2i(t[n] ) , (4.14)
n−1 12
Tnt+TnU = U[n]Tn, (4.10) p[n] = (2η(T ) (T ) −p∗(T ) (T )
n 11 n 12 n 12 n 12
T +T V = V[n]T , (4.11)
nz n n +p(T ) (T ) ) (4.15)
n 11 n 11
/((T ) (T ) −(T ) (T ) )| ,
which can be further simplified as n 11 n 22 n 12 n 21 λ=−ω
η[n] = (η((T ) (T ) +(T ) (T ) ) (4.16)
n 11 n 22 n 12 n 21
U0[n] =U0+i[σ3,t[nn−]1], (4.12) −p∗(Tn)12(Tn)22+p(Tn)11(Tn)21)
/((T ) (T ) −(T ) (T ) )| ,
n 11 n 22 n 12 n 21 λ=−ω
V[n] =T | V T−1| . (4.13)
−1 n λ=−ω −1 n λ=−ω
The proof for Eq. (4.13) is quite complicated in
where (cid:0)t[n] (cid:1) is the element at the first row and
general when compared to the proof of one and two n−1 12
fold Darboux transformation. However, this will be- second column in the matrix t[n] . So far, we dis-
n−1
come simple if we know the origin of T [33]. Be- cussed about the determinant construction of n-th
n
cause if we treat n-fold Darboux transformation as a Darboux transformation of the H-MB system. As an
generalizationof(n−1)-Darbouxtransformation,the applicationofthesetransformations,solitonandposi-
transformation will be the multiplications of n one- ton solutions of the H-MB system will be constructed
fold Darboux transformations at λ = −ω. It is easy in the next section.
5
V. SOLITON SOLUTIONS OF THE THE one-fold Darboux transformation eq.(4.2), eq.(4.3),
H-MB SYSTEM eq.(4.4) and choosing α = 2, β = −1, then the one-
solition solutions of the classical H-MB system can
In this section, having obtained the Daurboux be obtained whose evolution is shown in Fig.1, which
transformationforoursystem,ournextaimistocon- clearly indicates that E and p are bright solitons be-
structtheonesolitonsolutionoftheH-MBsystemby cause their waves are under the flat non-vanishing
assuming suitable seed solutions. We assume trivial plane whereas η is a dark soliton.
seed solutions as E = 0,p = 0,η = 1, then the linear Now let us discuss the construction of the two-
system becomes soliton solutions of the H-MB system. For this pur-
pose, we have to use two spectral parameters λ =
1
α +iβ and λ = α +iβ . After the second Dar-
1 1 2 2 2
Φt = UΦ, (5.1) boux transformation, we can construct the two so-
Φ = VΦ, (5.2) lition solutions. As the general form of two soliton
z
solution is quite tedious in nature, for simplicity, we
where are giving only the two soliton solution of E with
ω = 1.5,α = 0.5,β = 1,α = 1,β = 1.5,α = 2
1 1 2 2
(cid:18) (cid:19) and β =−1,
Φ
Φ = 1 , (5.3)
Φ
2
(cid:18) (cid:19)
−iλ 0
U = 0 iλ , (5.4) E2−sol =
−i(17e−0.01176470588i(85t−1258z+1815iz+255it)
(cid:18)4iβλ3−αiλ2 0 (cid:19)
V = 0 −4βiλ3+αiλ2 −(36+18i)e−0.01176470588i(170t−4385z+204iz+170it)
i (cid:18)1 0 (cid:19) −7e0.01176470588i(−85t+1258z+1815iz+255it)
+ . (5.5)
λ+ω 0 −1 +(18i−12)e0.01176470588i(−170t+4385z+204iz+170it)
−16ie0.01176470588i(−85t+1258z+1815iz+255it)
From the above system, we construct the explicit
+20ie−0.01176470588i(85t−1258z+1815iz+255it))
eigenfunctions in the form
/(−24cos(0.9999999998t−36.78823529z)
Φ1 = e−iλt+(4βiλ3−αiλ2+λ+iω)z+x0+2iy0, +26cosh(t+18.95294118z)
+2cosh(−5t−23.75294118z).
Φ2 = eiλt+(−4βiλ3+αiλ2−λ+iω)z−x0+2iy0+iθ,
where x , y and θ are all arbitrarily fixed real con- We also constructed the two soliton solution for p
0 0
stants. Substituting these two eigenfunctions into the and η in a similar manner. For completeness, instead
one-foldDarbouxtransformationEq. (4.2),Eq. (4.3), of giving complicated forms of p and η, the graphical
Eq. (4.4) and choosing λ = α +Iβ , x = 0, y = representation of them is shown in Fig.2.
1 1 0 0
0,θ = 0, then the following form of soliton solutions
are obtained:
VI. BRIGHT AND DARK POSITON
E = 2β1e−2iBCsech(2β1BA), (5.6) SOLUTIONS OF THE H-MB SYSTEM
p = iβ1((α1−iβ1+ω)e−2DB +(α1+β1i+ω)e−2BF) In the case of two soliton solution constructed
B above, ifthesecondspectralparameterλ2 isassumed
A to be close to the first spectral parameter λ , and do-
sech2(2β ), (5.7) 1
1B ingtheTaylorexpansionofwavefunctiontofirstorder
2β2 A up to λ1 will lead to a new kind of solution which is
η = 1− B1sech2(2β1B), (5.8) called as degenerate soliton[34] - smooth positon so-
lution. “Positon” was coined by Matvee [1, 2, 4] for
where A,B,C,D,F are explicitly given in Appendix the Korteweg-de Vries (KdV) equation by the same
I. limiting approach. Note that the positon of the KdV
If we choose α = 1, β = 0, the one soliton solution is a singular solution. For the construction of positon
isjustthesolitonsolutionofEq. (15)ofNLS-MBsys- solutions,followingfourlinearfunctionsareneededto
tem mentioned in [22] with α = −ρ , β = ν . Sim- construct the second Darboux transformation and to
1 1 1 1
ilarly, substituting these two eigenfunctions into the generate the positon solutions,
6
Φ1,1 = e−iλ1t+(4βiλ31−αiλ21+λ1i+ω)z+x0+2iy0, φ1(R) := e4(ib(P−12zω+)(Qλ1+tω)), (7.4)
Φ2,1 = eiλ1t+(−4βiλ31+αiλ21−λ1i+ω)z−x0+2iy0+iθ, φ2(R) := 21d(bi+2iλ+R)e4(ib(P−22zω+)(Qλ2+tω)). (7.5)
Φ1,3 = e−iλ3t+(4βiλ33−αiλ23+λ3i+ω)z+x0+2iy0, where P1,Q1,P2,Q2 are polynomials independent of
Φ2,3 = eiλ3t+(−4βiλ33+αiλ23−λ3i+ω)z−x0+2iy0+iθ. tApanpdenzdixanIdI.their complete expressions are given in
If we use the above two wave functions to con-
Now we take λ = λ +(cid:15)(1+i) and use the Tay-
3 1
structthetwonewfunctionsh andh asin[22],then
lor expansion of wave function φ and φ up to first 1 2
3 4
theresultingnewsolutionsobtainedthroughDarboux
order of (cid:15) in terms of λ . For example, choosing
1
transformationarefoundtohavenomeaning. There-
ω = 1.5, α = 0.5, β = 1,α = 2 and β = −1, the
1 1
fore,wewouldliketoconstructmorecomplicatedbut
positon solutions E is constructed in the form
p
physically meaningful solutions in the following part.
Bycombiningthesetwowavefunctions,wederivethe
Ep =8ie14.8iz−it× new functions h1 and h2 as follows
(1582zcosh(2t+2.40z)−688izsinh(2t+2.4z) h :=g (λ)−g (λ∗)∗,h :=g (λ)+g (λ∗)∗, (7.6)
1 1 1 2 2 1
−50icosh(2t+2.4z)+100itsinh(2t+2.4z))/
where
(400t2+119296z2−5504tz+50+50cosh(4.8z+4t)).
g :=φ (R)+φ (−R),g :=φ (R)+φ (−R).(7.7)
1 1 1 2 2 2
Inthiscase,thepictorialrepresentationoftheposi-
tonsolutions(Ep,pp,ηp)oftheH-MBsystemisshown It can be proved that h1 and h2 are also the solu-
in Fig.3. tions of the Lax equation with λ := α1+iβ1. Using
From the above figures, we observe that two peaks these two wave functions h1 and h2 in the one-fold
of the positon solutions are at same height which is Darboux transformation will lead to the construction
differentfromthetwo-solitonsolution. Meanwhilethe of breather solutions of the H-MB system. To sim-
two waves depart at a relatively less speed after their plify the calculations, we take b = −2α1 and use the
collision. This is also different from the two-solitons, second Darboux transformation discussed in the last
which depart at a fixed speed. Here again, we find section, then the final form of the breather solution
that E and p are bright positon solutions whereas η Eb is obtained in the form
is a dark positon in all three cases discussed above.
Eb = (d2(eiAbY(w) +eAb(Y−w)i)+2dβ1(eiCYb∗ +eCYbi)
VSIIO.LUBTRIIOGNHSTOAFNTDHDEAHR-MKBBRSYEASTTEHMER −dβ1(eDXbi+eDXb∗i)+2β1(w−β1)eFbY(w)i
−2β1(w+β1)eFb(Y−w)i)/ (7.8)
In the last two sections, soliton solutions and posi- G H
(2dcosh(2iwβ z b)−2β cosh(2w b)),
ton solutions have been generated for the H-MB sys- 1 Y 1 Y
tem. In this section, we will now focus on a new and
where
different kind of solution which is also derived from
periodic solutions through Darboux transformation.
Theresultingperiodicsolutionscanbecalledbreather (cid:113)
w = β2−d2, (7.9)
solutions. Now, let us assume the seed solutions as 1
E = deiρ,p = ifE,η = 1,ρ = az+bt, which admits
the constraint in the form X =(α +β i+ω)(−α +β i−ω), (7.10)
1 1 1 1
2a+αb2−2αd2+2βb3−12βd2b−4f =0, (7.1)
Y =(α +ω)(α +β i+ω)(−α +β i−ω),(7.11)
1 1 1 1 1
fb−2ωf +2=0. (7.2)
and A (w),C ,D ,F (w), and G ,H are polynomi-
b b b b b b
Defining als of t,z,ω,w,a,b,d,α,β,α ,β which are defined in
1 1
Appendix-II.
(cid:112)
R:= −b2−4bλ−4λ2−4d2, (7.3) Similarly, the breather form of p and η can be con-
structed. For example, after taking values
(cid:18) (cid:19)
φ
the following wave function Φ= φ12 is obtained in ω =0.5,d=0.5,α1 =−1,β1 =1,α=2,β =−1,
terms of R as (7.12)
7
the breather solution of the H-MB system is plotted
in Fig.4 and Fig.5.
8(1−3iz) 1
E = ( − )e0.5i(−5z+4t), (8.2)
Similarly, for the next choice of parameters ω = r 4+(2t+17z)2+36z2 2
1.5, α = 0.5, β = 1, α = 0, β = 1, the breather
1 1 p = e0.5i(−5z+4t)(1031493073iz4+10312500iz2
solutions of the complex modified Korteweg-de Vries r
−5312500z−625000t−152343760z3
(CMKdV)-MB system are obtained. The picture of
breather solutions is shown in Fig.6. In addition to −1875000t2z−31875000tz2
the above observation, we also find how the values [phone]it2z2+5312500it3z+431640580itz3
of (α,β) changes the direction of breather solution
+1875000itz+156250it4−156250i)/(156250t4
η in the (t,z) plane, see Fig.7. To the best of our
knowledge, we observe this effect for the first time. [phone]z2+312500t2+431640580tz3
[phone]t2z2+5312500t3z
Having constructed bright breathers for E and p
+156250+5312500tz+1031493636z4), (8.3)
andadarkbreatherforη,inthenextsection,ouraim
is to discuss the construction of rogue wave solutions ηr = (16t4+64t2+7224t2z2+105625z4+2992z2
oftheH-MBsystemwhichisinfactonesingleperiod +44200tz3−16+544t3z+896zt)
of breather solutions. /(16t4+32t2+7224t2z2+105625z4
+2600z2+44200tz3+16+544t3z+544zt).
(8.4)
FromEq. (8.2),itisclearlyobservedthattheheight
ofthebackgroundof|E |2 is andtheordersofthenu-
r
meratorsanddenominatorsofp andη arefour. Be-
VIII. BRIGHT AND DARK ROGUE WAVES r r
IN THE H-MB SYSTEM cause of these reasons, the graphs of the rogue waves
for p and η have double peaks which are shown in
r r
Fig.8 and the corresponding density graph is plotted
In this section, using the limit method of the NLS in Fig.9.
equation,weconstructtheroguewavesolutionsofthe For further understanding of our observations, we
H-MBsystem[34]. Thiskindofsolutiononlyappears enlarge the above density in Fig.9, some zoomed por-
in some special regions of time and distance and then tions of the above figures are clearly shown in Fig.10.
will be drowned in one fixed non-vanishing plane. If From the graph of |p|2 shown in Fig.10, we find one
we do the Taylor expansion to the breather solution cave appears on top of the single peak with two caves
(7.8) around β = d, one rogue wave solution of E on both sides of the peak.
1
will be obtained and rogue waves for p and η can also To realize the significance of the different parame-
be constructed in a similar way. In the following, for ters α and β, we also consider the case, when α =
brevity, we only report the rogue wave for E in the 0, β = 1, i.e., the CMKdV-MB system, see Fig.11.
r
form From the figures, we also find that the parameters α
and β will change the shape, pulse width, etc., of the
rogue wave. Therefore, in the following, we fix time t
and distance z to see the role of parameters α and β
and their impact on rogue wave dynamics.
A¯(t,z)d i We keep α and β as arbitrary parameters and
E = exp( (8zβα4+8zβα3ω
r B¯(t,z) α1+ω 1 1 choose ω = 0.5, d = 0.5, α1 = −1, β1 = 1, t = 1, z =
1. This is also a rogue wave whose graph is portrayed
−2zαα2ω+2z−2α tω+zαd2ω−12zβd2α ω
1 1 1 in Fig.12. Here we provide only the specific form of
−2zαα13−2α12t+zαd2α1−12zβd2α12)), (8.1) the rogue wave solution Erαβ as
Erαβ = e−4i(8+20β+7α)(40α−17α2+156β−40
where A¯(t,z),B¯(t,z) are polynomials of −585β2−192αβ+i(32+8α+96β))/
t,z,ω,w,a,b,d,α,β,α ,β which are defined in
1 1 (1170β2−80α+34α2−312β+112+384αβ).
Appendix-III.
(8.5)
When we take ω = 0.5, d = 0.5, α = −1, β =
1 1
1, α = 2, β = −1, the final form of the rogue wave This implies that after fixing the values of t and z,
solutions will be the solution depending on parameters α, β is also in
8
theformofaroguewave,whichisobservedforthefirst esting characteristics which might attract physicists
time. Thiswillgiveussomeideaabouthowtomodify to observe them in experiments with higher order op-
the parameters α and β to visualize our theoretical tical effects in the femtosecond regime. The interest-
results in terms of the experimental results in optics. ing characteristics obtained contain the following two
From the above, one can easily conclude that E and sides: (i) The rogue wave solution for p, η is surpris-
p are bright rogue waves and η is a dark rogue wave. inglyfoundbyustohavetwopeaksbecausetheorder
All the solutions mentioned above including posi- of the numerator and denominator of p, η in eq.(8.3)
tons and rogue wave solutions are indeed solutions of and eq.(8.4) is four and (ii) after fixing the time and
theH-MBsystemwhichareverifiedbyusingthesym- spatialparameterandbychangingothertwounknown
bolic computation software MAPLE. parameters α and β, we find a rogue wave shape also
arises out as shown in eq.(8.5). This is the first time
that this phenomenon is obtained to the best of our
IX. CONCLUSION AND DISCUSSIONS knowledge. Still, there are a few interesting questions
which are still unclear. For example, the physical in-
In this paper, after a suitable choice of self- terpretations and observation of higher-order positon
steepening and self-frequency shift effects, we have solutions, the role of higher-order rogue waves solu-
derived the Darboux transformation of the H-MB tions and their applications in physics, in particular,
system which is governed by ultra-short pulse prop- the connection between rogue wave solutions and su-
agation through an erbium doped nonlinear optical percontinuum generation through modulation insta-
waveguide and further generalized it to the matrix bility or soliton fission, etc., with higher-order optical
form of an n-fold Darboux transformation, which im- effects.
pliesthedeterminantrepresentationof(E[n],p[n],η[n])
generated from the known solution (E,p,η). By Acknowledgments This work is supported by the
choosing some special eigenvalues λ = λ∗
2n−1 2n National Natural Science Foundation of China under
and eigenfunctions using the reduction conditions
Grant No.11201251, the Natural Science Foundation of
Φ =Φ∗ , Φ =−Φ∗ ,thedeterminant
1,2n−1 2,2n 2,2n−1 1,2n Zhejiang Province under Grant No. LY12A01007. J.S.He
representation of (E[n],p[n],η[n]) provided some new is supported by the National Natural Science Foundation
solutions of the H-MB system. As examples, soliton of China under Grant No. 10971109 and NO. 11271210,
solutions,breathersolutions,androguewavesolutions andtheK.C.WongMagnaFundatNingboUniversity. K.
of the H-MB system have been constructed explicitly Porseizan wishes to thank the DST, DAE-BRNS, UGC,
byusingtheDarbouxtransformationfromtrivialand and CSIR, Government of India, for financial support
periodic seed solutions. The rogue waves show inter- through major projects.
X. APPENDICES
Appendix I: In this appendix, we are providing explicit expression for A, B, C, D, and F
A = −24zβα3ω+4zαα2ω−8zβα2β2+2zαα β2+4β2zβω2+2zαα ω2
1 1 1 1 1 1 1 1
−12zβα2ω2+z+8zβα β2ω+tα2+tβ2+tω2+2tα ω−12zβα4+2zαα3+4zββ4,
1 1 1 1 1 1 1 1 1
B = α2+2α ω+ω2+β2,
1 1 1
C = 2tα2ω+tα ω2+tβ2α −4zβα5+zαα4−zαβ4−zα +tα3
1 1 1 1 1 1 1 1 1
+24zβα2β2ω+12zβα β2ω2−2zαα β2ω−8zβα4ω−4zβα3ω2
1 1 1 1 1 1 1 1
+8zβα3β2+12zββ4α +2zαα3ω+zαα2ω2−zαβ2ω2−zω,
1 1 1 1 1 1 1
D := −24β zβα3ω+4β zαα2ω+2β zαα ω2−12β zβα2ω2+8β3zβα ω+β z+β3t+4β5zβ
1 1 1 1 1 1 1 1 1 1 1 1 1
+β tω2+β tα2−8β3zβα2+2β3zαα +4β3zβω2+2β tα ω−12β zβα4+2β zαα3
1 1 1 1 1 1 1 1 1 1 1 1 1 1
+i(tβ2α +zαα4−zα −zω+tα3+tα ω2+2tα2ω−zαβ2ω2
1 1 1 1 1 1 1 1
−4zβα3ω2−8zβα4ω−4zβα5−zαβ4−2zαα β2ω
1 1 1 1 1 1
+8zβα3β2+12zββ4α +2zαα3ω+24zβα2β2ω+12zβα β2ω2+zαα2ω2),
1 1 1 1 1 1 1 1 1 1
9
F := 24β zβα3ω−4β zαα2ω−2β zαα ω2+12β zβα2ω2−8β3zβα ω−β z−β3t−4β5zβ
1 1 1 1 1 1 1 1 1 1 1 1 1
−β tω2−β tα2+8β3zβα2−2β3zαα −4β3zβω2−2β tα ω+12β zβα4−2β zαα3
1 1 1 1 1 1 1 1 1 1 1 1 1 1
+i(tβ2α +zαα4−zα −zω+tα3+tα ω2+2tα2ω−zαβ2ω2
1 1 1 1 1 1 1 1
−4zβα3ω2−8zβα4ω−4zβα5−zαβ4−2zαα β2ω
1 1 1 1 1 1
+8zβα3β2+12zββ4α +2zαα3ω+24zβα2β2ω+12zβα β2ω2+zαα2ω2),
1 1 1 1 1 1 1 1 1 1
Appendix II: In this appendix, we are furnishing the expression for P ,Q ,P ,Q
1 1 2 2
P = 12βd2b2λ−4αd2λω+4βb3λω+2αd2bω+2αd2bλ
1
+2αb2λω−24βd2bω2+12βd2b2ω−8λ−8ω−4αd2ω2−2βb4λ
−2βb4ω+2αb2ω2+4βb3ω2−αb3λ−αb3ω−24βd2bλω+2iωβRb3
+2iλβRb3+λαRb2i+ωαRb2i+16iλ2ωβRb+4iR+8iβλbω2R
−8iβλb2ωR−4id2βωRb−4id2βλRb−4iαbλRω+8id2βλRω
−4iβb2ω2R−16iβλ3Rω−2iαλ2Rb−2iαbω2R−16iβλ2ω2R
−4iβλ2b2R+4iαλ2Rω+8iβλ3Rb+8id2βω2R+4iαλω2R,
Q = 2b2ω−4bλω−4bω2+2b2λ−2iRbω+4iRωλ+4iRω2−2iRbλ,
1
P = −12βd2b2λ+4αd2λω−4βb3λω−2αd2bω−2αd2bλ−2αb2λω+24βd2bω2
2
−12βd2b2ω+8λ+8ω+4αd2ω2+2βb4λ+2βb4ω−2αb2ω2−4βb3ω2
+αb3λ+αb3ω+24βd2bλω+2iωβRb3+2iλβRb3+λαRb2i+ωαRb2i
+16iλ2ωβRb+4iR+8iβλbω2R−8iβλb2ωR−4id2βωRb−4id2βλRb
−4iαbλRω+8id2βλRω−4iβb2ω2R−16iβλ3Rω−2iαλ2Rb−2iαbω2R
−16iβλ2ω2R−4iβλ2b2R+4iαλ2Rω+8iβλ3Rb+8id2βω2R+4iαλω2R,
Q = 4bλω+4bω2−2b2ω−2b2λ−2iRbω+4iRωλ+4iRω2−2iRbλ,
2
Appendix III: In this appendix, we are providing the expression for
A (w), C , D , F (w), G H ,A¯(t,z)andB¯(t,z)
b b b b b b
A = −2wβ z−72zβα3ωwβ −72zβα2ω2wβ +6zαα2ωwβ +6zαα ω2wβ
b 1 1 1 1 1 1 1 1 1
−24zβωwα β3−24zβα ω3wβ +2zαα3wβ −24zβα4wβ +2zαω3wβ −24zβwα2β3
1 1 1 1 1 1 1 1 1 1 1
+2zαwα β3+2zαwωβ3−(azα3+azω3+btα3+btω3+3azα2ω
1 1 1 1 1 1
+3azα ω2+3btα2ω+3btα ω2+azα β2+azωβ2+btα β2+btωβ2),
1 1 1 1 1 1 1 1 1
C = −2zω2−2zβ2+2α4t−2zα2−4zωα +6α3tω+2zαα5−8zβα6
b 1 1 1 1 1 1 1
+6α2tω2+2α tω3−zαd2ω3−zαd2α3+6zαα4ω+6zαα3ω2
1 1 1 1 1
+2zαα2ω3−24zβα5ω−24zβα4ω2−8zβα3ω3+12zβd2α4
1 1 1 1 1
−3zαd2ω2α −3zαd2α2ω+36zβd2α3ω+36zβd2α2ω2
1 1 1 1
+12zβd2α ω3−8zβα4β2+2zαα3β2+2α tωβ2+2α2tβ2
1 1 1 1 1 1 1 1 1
+4izd2βwωβ2+4izαα wωβ2+4izd2βwα β2+24izβα ω2wβ2
1 1 1 1 1 1 1
+12izd2βω2wα +12izd2βwα2ω+2itwωβ2+2itwα β2
1 1 1 1 1
+6itwω2α +6itwα2ω+4izαα4w−24izβα5w+12zβd2α2
1 1 1 1 1
β2+2zαα2ωβ2−zαd2α β2−zαd2ωβ2−8zβα3ωβ2
1 1 1 1 1 1 1 1
+12zβd2α ωβ2−72izβα4ωw−72izβα3ω2w−24izβα2ω3w
1 1 1 1 1
+4izαwα2β2+12izαα3ωw+12izαα2ω2w+4izαα ω3w
1 1 1 1 1
+4izd2βω3w+4izd2βwα3−16izβα3wβ2+8izβω3wβ2+8izβwα
1 1 1 1 1
β4+8izβwωβ4+2izwα +2izwω+2itwω3+2itwα3,
1 1 1 1
10