integrable semi-discrete kundu–eckhaus equation: … semi-discrete kundu–eckhaus equation:...
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J Nonlinear Sci (2018) 28:43–68https://doi.org/10.1007/s00332-017-9399-9
Integrable Semi-discrete Kundu–Eckhaus Equation:Darboux Transformation, Breather, Rogue Waveand Continuous Limit Theory
Hai-qiong Zhao1,2 · Jinyun Yuan2 ·Zuo-nong Zhu3
Received: 5 January 2017 / Accepted: 25 May 2017 / Published online: 3 June 2017© Springer Science+Business Media New York 2017
Abstract To get more insight into the relation between discrete model and continuouscounterpart, a new integrable semi-discrete Kundu–Eckhaus equation is derived fromthe reduction in an extended Ablowitz–Ladik hierarchy. The integrability of the semi-discrete model is confirmed by showing the existence of Lax pair and infinite numberof conservation laws. The dynamic characteristics of the breather and rational solutionshave been analyzed in detail for our semi-discrete Kundu–Eckhaus equation to revealsome new interesting phenomena which was not found in continuous one. It is shownthat the theory of the discrete system including Lax pair, Darboux transformation andexplicit solutions systematically yields their continuous counterparts in the continuouslimit.
Keywords Semi-discrete Kundu–Eckhaus equation · Darboux Transformation ·Breather · Rogue wave · Continuous limit theory
Communicated by Ferdinand Verhulst.
B Hai-qiong [email protected]
Jinyun [email protected]
Zuo-nong [email protected]
1 Department of Applied Mathematics, Shanghai University of International Business andEconomics, 1900 Wenxiang Road, Shanghai 201620, People’s Republic of China
2 Departamento de Matemática, Universidade Federal do Paraná, Centro Politécnico, CP: 19.081,Curitiba, Paraná 81531-980, Brazil
3 School ofMathematical Sciences, Shanghai Jiao TongUniversity, 800Dongchuan Road, Shanghai200240, People’s Republic of China
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44 J Nonlinear Sci (2018) 28:43–68
1 Introduction
There has been a great deal of interest in nonlinear Schrödinger (NLS) family ofequations, which arise in several physical applications including quantum field the-ory, nonlinear optics, water wave, Bose–Einstein condensates. Among such models,Kundu–Eckhaus (KE) equation is one of the simplest integrable extensions of the NLSwith quintic nonlinearity (Kundu 1984; Calogero and Eckhaus 1987)
iχτ + χxx + 2|χ |2χ + 2ia(|χ |2)xχ + a2|χ |4χ = 0, (1.1)
where χ is a complex-valued function of two real variables x and τ , the subscripts xand τ are spatial and temporal partial derivatives, and a is a real constant.Mathematicalstructures have been well studied related to the KE such as explicit form of the Laxpair, Painlevé property, Hamiltonian structures, infinitely many conservation laws,Darboux transformations and Hirota bilinear formula (Clarkson and Cosgrove 1987;Geng 1992; Kakei et al. 1995; Geng and Tam 1999; Qiu et al. 2015). Besides themathematical interests, KE is also used to examine the stability of the Stokes wave(Johnson 1977) and the instabilities of plane solitons associated with the Kadomtsev–Petviashvili equation (Gorshkov and Pelinovsky 1995).
During the last few years, much attention has been paid to discretizations of inte-grable nonlinear partial differential equations (PDEs) due to their wide range ofapplications in diverse number of fields. A typical example is for the quantum fieldtheory, where discretizations provide a powerful tool for the construction of quantumgravity models (Thiemann 2005). Also, numerical analysis, and its applications to allbranches of sciences, relies greatly on discretization of the associated PDEs (Ablowitzet al. 2001). Indeed, discrete models are more effective to access non-perturbationeffects than continuous models. Thus, one of the purposes of discretization is the con-struction of a discrete analogue of the continuous integrable model which preservesas many integrability properties as possible. The motivation of the paper stems froma further question of the effects of discretizationQuestion: how to relate properties from the discrete model to continuous one in thecorresponding continuous limit?The question is handled by considering continuous limit theory of integrable dis-crete model, in which integrability properties computed from the discrete modelwill in general be approximations to their continuous counterparts in the corre-sponding continuous limit. For example, the KdV theory including infinitely manycommuting vector fields, conserved functions and bi-Hamiltonian structure is glob-ally recovered from the continuous limit of the Kac-Moerbeke system, a discrete KdVsystem (Schwarz 1982; Zeng andWojciechowski 1995;Morosi and Pizzocchero 1996;Gieseker 1996; Lin et al. 2002; Zhao and Zhu 2011; Zhu et al. 2011). Discrete NLS-type equation has attracted much study, see Ablowitz and Ladik (1975), Ablowitzand Ladik (1976), Vekslerchik and Konotop (1992), Ablowitz et al. (2004), Flach andGorbach (2008) and Vakhnenko (2017) and references therein. Hence, it is importantand interest to investigate the discretization of the KE equation. The question hasbeen studied since pioneering article of Levi and Scimiterna (2009). They find that theKE and the complex Burgers equation are related by a Miura transformation. Then,
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J Nonlinear Sci (2018) 28:43–68 45
taking into account this relation, they discretize the KE by connecting with discretecomplex Burgers equation. However, to the best of our knowledge, the proceduredoes not clearly guarantee the integrability of the equation due to the fact that it isdifficult to provide Lax pair of the discrete KE from the inversion of discrete Miuratransformation.
In this paper, we introduce a new integrable semi-discrete version of Kundu–Eckhaus (DKE) equation
iqn,t − 2qn − iαqn[qnqn−1e
iα ln(1+|qn−1|2
)− qnqn−1e
−iα ln(1+|qn−1|2
)]
+(1 + |qn|2
) [qn+1e
iα ln(1+|qn |2
)+ qn−1e
−iα ln(1+|qn−1|2
)]= 0, (1.2)
where t ∈ R is a continuous variable, n ∈ N is a discrete variable, overbar ¯denotesthe complex conjugate, and α is a real constant. In Sects. 2 and 3, we show thatthe discrete model comes from an extended Ablowitz–Ladik hierarchy with specialreductions and suitable freely functions. The integrability of the DKE is confirmedby providing a Lax pair, and further, an infinite number of conservation laws. InSect. 4, with the aid of gauge transformations of the Lax pair, we develop an effectivetechnique to construct an explicit N -fold Darboux transformation for the DKE. InSect. 5, by applying the Darboux transformation, we obtain breather and rationalsolutions of the DKE. Analyzing the dynamic behavior of breather solution revealsthat the generation conditions of time-periodic Kuznetsov-Ma breather solution andspace-periodicAkhmediev breather solution are severely constrained bywave number,cubic-quintic nonlinear coefficient, spectral parameter and background constant. Therational solution exhibits not only rogue wave phenomenon but also W-shaped solitoncharacter, which is different from KE that never produces rational soliton solution. InSect. 6, the KE theory including Lax pair, Darboux transformation, breather and roguewave solutions is recovered through the continuous limit of corresponding theory ofour discrete KE. The excitation mechanism of the discrete rational soliton of ourdiscrete KE is discussed. In the last section, the conclusion is given.
2 Extended Ablowitz–Ladik Hierarchy
Starting from the extended Ablowitz–Ladik discrete iso-spectral problem (Tsuchida2002)
ψn+1 = Nnψn, Nn =(
λwn unvn λ−1sn
), (2.1a)
with the constraint condition wnsn − unvn �= 0, we search for a sequence of timeevolution of the ψn satisfying the differential equation
dψn
dtm= M (m)
n ψn, (2.1b)
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46 J Nonlinear Sci (2018) 28:43–68
such that the corresponding discrete zero-curvature equation
dNn
dtm= M (m)
n+1Nn − NnM(m)n , (2.2)
gives rise to an extended Ablowitz–Ladik hierarchy. In order to do this, we first solvethe stationary discrete zero-curvature equation
�n+1Nn − Nn�n = 0, (2.3a)
where
�n =(An Bn
Cn Dn
). (2.3b)
Through direct calculation, the above equation leads equivalently to
λ(An+1 − An)wn + Bn+1vn − unCn = 0, (2.4a)
un(An+1 − Dn) + λ−1Bn+1sn − λwn Bn = 0, (2.4b)
(Dn+1 − An)vn + λCn+1wn − λ−1snCn = 0, (2.4c)
Cn+1un − vn Bn + λ−1(Dn+1 − Dn)sn = 0. (2.4d)
It follows from
λ × (2.4d) × wn + λ−1 × (2.4a) × sn − (2.4b) × vn − (2.4c) × un= (wnsn − unvn)(Dn+1 − Dn + An+1 − An) = 0,
that for the sake of simplicity, we can take
Dn = −An . (2.4e)
Further, substituting the expansions
An =+∞∑j=0
A( j)n λ−2 j , Bn =
+∞∑j=0
B( j)n λ−2 j+1, Cn =
+∞∑j=0
C ( j)n λ−2 j+1, (2.5)
into (2.4) and comparing each power of λ, we have the following recursion relation:
B(0)n = C (0)
n = 0, A(0)n = γ1 = constant,
B( j+1)n = unw
−1n
(A( j)n+1 + A( j)
n
)+ snw
−1n B( j)
n+1,
C ( j+1)n = vn−1w
−1n−1
(A( j)n + A( j)
n−1
)+ sn−1w
−1n−1C
( j)n−1,
A( j+1)n+1 − A( j+1)
n = unw−1n C ( j+1)
n − vnw−1n B( j+1)
n+1 , j = 0, 1, 2, 3, . . . . (2.6)
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J Nonlinear Sci (2018) 28:43–68 47
Since the last formula in Eq. (2.6) is not local, we can solve the expression of A( j+1)n
with an additive constant. In order to determine the sequence of functions uniquely,we impose the conditions A( j)
n |(un=0,vn=0,sn=0) = 0.On the other hand, if we take �n in (2.3b) as follows,
�n =(An Bn
Cn −An
)=
+∞∑j=1
(A( j)
n B( j)n λ−1
C( j)n λ−1 −A( j)
n
)λ2 j . (2.7)
then, similarly, we can get the following recurrence relation,
B(0)n = C(0)
n = 0, A(0)n = γ2 = constant,
B( j+1)n = wn−1s
−1n−1B
( j)n−1 − un−1s
−1n−1
(A( j)
n + A( j)n−1
),
C( j+1)n = wns
−1n C( j)
n+1 − vns−1n
(A( j)
n+1 + A( j)n
),
A( j+1)n+1 − A( j+1)
n = uns−1n C( j+1)
n+1 − vns−1n B( j+1)
n , j = 0, 1, 2, 3, . . . . (2.8)
Finally, we define
M (m)n =
(�nλ
2m)
+ +(�nλ
−2m)
− + (m)n , (2.9)
where
(�nλ2m)+ =
m∑j=0
(A( j)n B( j)
n λ
C ( j)n λ −A( j)
n
)λ2m−2 j ,
(�nλ−2m)− =
m∑j=0
(A( j)
n B( j)n λ−1
C( j)n λ−1 −A( j)
n
)λ2 j−2m,
(m)n =
(F (m)n − A(m)
n − A(m)n 0
0 G(m)n + A(m)
n + A(m)n
).
We should remark here that the term (m)n and the freely adjustable functions F (m)
n
andG(m)n will play a crucial role in obtaining our desired reductions. Then, the discrete
zero-curvature Eq. (2.2) leads to an extended Ablowitz–Ladik hierarchy
dwn
dtm= wn
(F (m)n+1 − F (m)
n
)+ vn B
(m)n+1 − unC
(m)n , (2.10a)
dundtm
= un(F (m)n+1 − G(m)
n
)+ sn B
(m)n+1 − wnB(m)
n , (2.10b)
dvndtm
= vn
(G(m)
n+1 − F (m)n
)+ wnC(m)
n+1 − snC(m)n , (2.10c)
dsndtm
= sn(G(m)
n+1 − G(m)n
)+ unC(m)
n+1 − vnB(m)n . (2.10d)
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48 J Nonlinear Sci (2018) 28:43–68
3 Semi-discrete Kundu–Eckhaus Equation
By setting vn = −un , sn = wn , G(m)n = F (m)
n , we find
A( j)n = − A( j)
n , B( j)n = −B( j)
n ,
C( j)n = −C ( j)
n , D( j)n = −D( j)
n .
Then, the four-coupled hierarchy (2.10) is reduced to a two-coupled form,
dwn
dtm= wn
(F (m)n+1 − F (m)
n
)− un B
(m)n+1 − unC
(m)n , (3.1a)
dundtm
= un(F (m)n+1 − F (m)
n
)+ wn B
(m)n+1 + wnC
(m)n . (3.1b)
Choosing γ1 = γ2 = i2 , we obtain the first non-trivial member of the hierarchy as
follows,
dwn
dt1= wn
(F (1)n+1 − F (1)
n
)+ i
unun−1
wn−1− i
unun+1
wn+1, (3.2a)
dundt1
= un(F (1)n+1 − F (1)
n
)+ i
wnun−1
wn−1+ i
wnun+1
wn+1, (3.2b)
which is reduced to our discrete KE (1.2) with
un = qnwn,
wn = e−i α2 ln(1+|qn |2),
F (1)n = (α + 2i)unun−1
2wn−1wn− αunun−1
2wn−1wn− i. (3.3)
Thus, the Lax pair of the DKE (1.2) can be written as
ψn+1 = Nnψn,dψn
dt= Mnψn, (3.4a)
where
Nn =(
λwn un−un λ−1wn
), Mn =
(i2 (λ
2 + λ−2) + F (1)n iλ un
wn− iλ−1 un−1
wn−1
−iλ un−1wn−1
+ iλ−1 unwn
− i2 (λ
2 + λ−2) + F (1)n
).
(3.4b)
It is well known that one important characteristic feature of the integrable systemis the existence of an infinite number of conservation laws. A local conservation lawof a differential-difference equation with continuous variable t and discrete variablen is defined by
(Ln)t = Fn+1 − Fn, (3.5)
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J Nonlinear Sci (2018) 28:43–68 49
which is satisfied for all solutions of the system. The functionLn is usually called localconserved density and Fn is the associated flux also known as current density. Withthe aid of Lax pair, we infer that DKE possesses an infinite number of conservationslaws. The conserved densities are expressed as
L(1)n = ln
(1 + |qn|2
),
L(2)n = qnqn+1w
2n,
L(3)n = 1
2qnw
2n
(2w2
n+1|qn+1|2qn+2 + qnq2n+1w
2n + 2w2
n+1qn+2
),
L(k+1)n = tr
{℘(k)(0)
k!
}, k = 3, 4, 5, 6 . . .
where℘(k)(0) denotes the kth derivative of℘(t) = ln(1+qn∑∞
j=1 tj−1g( j)
n ) at t = 0,and the associated fluxes are given by
F (1)n = i
qnqn−1
w2n−1
− iqn−1qnw2n−1,
F (2)n = −iqn−1w
2n−1qn+1w
2n|qn|2 − iqn−1w
2n−1qn+1w
2n + i |qn|2,
F (k)n = iqng
(k−1)n − iqn−1w
2n−1g
(k)n
and
g(1)n = qn, g(k+1)
n = (wn)2g(k)
n+1 + (wn)2qn
k∑j=1
g( j)n+1g
(k+1− j)n , k = 1, 2, . . . .
4 Darboux Transformations
As we know, the Darboux transformation is an important method of constructingsolution to an integrable system, and essentially a special gauge transformation totransform a linear spectral problem to another one of the same type (Matveev andSalle 1991; Gu et al. 2005; Zhu et al. 2010; Zhang et al. 2015). Next, we will constructthe Darboux transformation of our discrete KE (1.2). Let us consider a transformationof eigenfunction
ψ [1]n (λ) = T [1]
n (λ)ψn(λ). (4.1)
We should prove rigorously that ψ [1]n (λ) also satisfies the same spectral problem
ψ[1]n+1(λ) = N [1]
n ψ [1]n (λ),
dψ [1]n (λ)
dt= M[1]
n ψ [1]n (λ), (4.2)
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50 J Nonlinear Sci (2018) 28:43–68
where the matrices N [1]n and M[1]
n have the same structures as Nn and Mn exceptthat qn is replaced by q[1]
n . It is obvious that the Darboux matrix T [1]n (λ) must obey
identities,
N [1]n T [1]
n = T [1]n+1Nn, M[1]
n T [1]n = T [1]
n,t + T [1]n Mn . (4.3)
We should emphasize that the construction of the Darboux matrix T [1]n (λ) is crucial.
4.1 Onefold Darboux Transformation
We begin with a universal Darboux matrix
T [1]n (λ) =
(1λ�
(1)n +
(1)n + λ�
(1)n
1λ�
(2)n +
(2)n + λ�
(2)n
1λ�
(3)n +
(3)n + λ�
(3)n
1λ�
(4)n +
(4)n + λ�
(4)n
), (4.4)
where �( j)n ,
( j)n ,�
( j)n are all functions of the variables n and t . Substituting (3.4) and
(4.4) into the first equation of (4.3) and comparing the different coefficients of λ onboth sides, we have
λ2 :
wn�(1)n+1 − w[1]
n �(1)n = 0, (4.5a)
w[1]n �(2)
n = 0, wn�(3)n+1 = 0, (4.5b)
λ1 :
wn(1)n+1 − un�
(2)n+1 − w[1]
n (1)n − u[1]
n �(3)n = 0, (4.6a)
un�(1)n+1 − w[1]
n (2)n − u[1]
n �(4)n = 0, (4.6b)
u[1]n �(1)
n − un�(4)n+1 + wn
(3)n+1 = 0, (4.6c)
u[1]n �(2)
n + un�(3)n+1 = 0, (4.6d)
λ0 :
wn�(1)n+1 − un
(2)n+1 − w[1]
n �(1)n − u[1]
n (3)n = 0, (4.7a)
wn�(2)n+1 + un
(1)n+1 − w[1]
n �(2)n − u[1]
n (4)n = 0, (4.7b)
u[1]n (1)
n − un(4)n+1 + wn�
(3)n+1 − w[1]
n �(3)n = 0, (4.7c)
u[1]n (2)
n + wn�(4)n+1 + un
(3)n+1 − w[1]
n �(4)n = 0, (4.7d)
λ−1 :
un�(2)n+1 + u[1]
n �(3)n = 0, (4.8a)
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J Nonlinear Sci (2018) 28:43–68 51
un�(1)n+1 − u[1]
n �(4)n + wn
(2)n+1 = 0, (4.8b)
u[1]n �(1)
n − un�(4)n+1 − w[1]
n (3)n = 0, (4.8c)
wn(4)n+1 + un�
(3)n+1 − w[1]
n (4)n + u[1]
n �(2)n = 0 (4.8d)
λ−2 :
wn�(4)n+1 − w[1]
n �(4)n = 0, (4.9a)
w[1]n �(3)
n = 0, wn�(2)n+1 = 0. (4.9b)
Equations (4.5b) and (4.9b) yield �(2)n = �
(3)n = �
(2)n = �
(3)n = 0. In order to
avoid trivial transformations in (4.6a) and (4.8d) (see Remark 1), we choose (1)n = 0
and (4)n = 0. Further, according to the structure of Lax pair and above equations, we
assume that �(1)n = −�
(4)n = T [1]
n,1, �(1)n = −�
(4)n = T [1]
n,3, (2)n = T [1]
n,2, (3)n = T [1]
n,2.
Therefore, the Darboux matrix T [1]n (λ) can be taken as
T [1]n (λ) =
(λT [1]
n,3 + 1λT [1]n,1 T [1]
n,2
T [1]n,2 −λT [1]
n,1 − 1λT [1]n,3
), (4.10)
and the relation between original field function (wn, un) and new one (w[1]n , u[1]
n ) canbe obtained from (4.5a), (4.9a), (4.6c) and (4.8b), namely
w[1]n = wn
T [1]n+1,3
T [1]n,3
, (4.11a)
u[1]n = −un
T [1]n+1,1
T [1]n,3
− wnT [1]n+1,2
T [1]n,3
. (4.11b)
Additionally, from (4.6b), (4.8c),(4.7a) and (4.7d), we find that T [1]n, j , j = 1, 2, 3 must
satisfy
unT[1]n+1,3 − w[1]
n T [1]n,2 + u[1]
n T [1]n,1 = 0, (4.12a)
wnT[1]n+1,1 − unT
[1]n+1,2 − w[1]
n T [1]n,1 − u[1]
n T [1]n,2 = 0. (4.12b)
Remark 1 If selecting w[1]n = wn
(1)n+1
(1)n
from (4.6a) and (4.8d), where (4)n =
(1)n ,
we possess |(1)n | = 1 along with (3.3). Then, from (4.7b) and (4.7c), we have u[1]
n =un
(1)n+1
(1)n
, which is a trivial transformation |u[1]n | = |un|.
The next step is to work out explicit representations of the term T [1]n, j . Let ψn(λ1) =
(ψn,1(λ1), ψn,2(λ1))T be a solution of the linear spectral problem (3.4) with λ = λ1
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52 J Nonlinear Sci (2018) 28:43–68
corresponding to the solution wn and un . By setting(T [1]n (λ)ψn(λ1)
)|λ=λ1 = 0, wehave
T [1]n,1 = −T [1]
n,3λ−11 |ψn,2(λ1)|2 + λ1|ψn,1(λ1)|2
λ−11 |ψn,1(λ1)|2 + λ1|ψn,2(λ1)|2
,
T [1]n,2 = −T [1]
n,3
(|λ1|2 − |λ1|−2)ψn,1(λ1)ψn,2(λ1)
λ−11 |ψn,1(λ1)|2 + λ1|ψn,2(λ1)|2
. (4.13)
Thus, we can check that Eq. (4.12) is verified by using (4.13) and (3.4).Similarly, collecting the coefficients of λ from the second equation of (4.3) and
using the above results, we find that T [1]n,3 must satisfy
dT [1]n,3
dt= −α
2T [1]n,3(|λ1|2 − |λ1|−2)
×[λ1�nunwn − λ1�nunwn
|wn|2(|�n|2 + |λ|21)− λ1�nun−1wn−1 − λ1�nun−1wn−1
|wn−1|2(|λ1�n|2 + 1)
−|�n|2(|λ1|2 − |λ1|−2
) (λ21 − λ21
)(|λ1�n|2 + 1
) (|�n|2 + |λ1|2)
], (4.14)
where �n = ψn,2(λ1)
ψn,1(λ1), and other equations for
dT [1]n, jdt , j = 1, 2 hold.
We can see that the direct calculation of T [1]n,3 from (4.14) is impossible, because
of complicated integral which produces the difficulty to seed solutions with non-vanishing boundary conditions. Moreover, we also fail to achieve N -fold Darbouxtransformation since the iteration of the transformationwill generatemore complicatedintegral. To overcome this problem, the construction of the explicit form of T [1]
n,3 is thecrucial point. Fortunately, by using of the second equation of (3.4), we get
d�n
dt=
[(α − i)unun−1
wn−1wn− (α + i)unun−1
wn−1wn− i(λ1 − λ−1
1 )2]�n
+(
iun−1
λ1wn−1− iλ1un
wn
)�2n − iλ1un−1
wn−1+ i un
λ1wn.
From this equation, the expression of (4.14) can be rewritten as
dT [1]n,3
dt= iαT [1]
n,3(|λ1|4 − 1)
2(|λ1�n|2 + 1
) (|�n|2 + |λ1|2) d
dt|�n|2
= − iα
2T [1]n,3
d
dtln
( |�n|2 + |λ1|2|λ1�n|2 + 1
),
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J Nonlinear Sci (2018) 28:43–68 53
which yields
T [1]n,3 =
(λ−11 |ψn,2(λ1)|2 + λ1|ψn,1(λ1)|2
λ1|ψn,2(λ1)|2 + λ−11 |ψn,1(λ1)|2
)− iα2
. (4.15)
We can summary the above facts as the following theorem:
Theorem 4.1 With Darboux transformation ψ[1]n (λ) = T [1]
n (λ)ψn(λ) defined above,the solution qn of (1.2) is mapped into a new solution q[1]
n under the transformation
q[1]n = u[1]
n
w[1]n
= −T [1]n,3
T [1]n,3
(qn
T [1]n+1,1
T [1]n+1,3
+ eiα ln(1+|qn |2) T[1]n+1,2
T [1]n+1,3
). (4.16)
4.2 Twofold Darboux Transformation
For the once iteration of the Darboux transformation, the twofold transformation ofeigenfunction can be given
ψ [2]n (λ) = T [2]
n (λ)ψ [1]n (λ) = T [2]
n (λ)T [1]n (λ)ψn(λ) � T [2]
n (λ)ψn(λ), (4.17)
where
T [ j]n (λ) =
(λT [ j]
n,3 + 1λT [ j]n,1 T [ j]
n,2
T [ j]n,2 −λT [ j]
n,1 − 1λT [ j]n,3
)
=(T [ j]n,3 0
0 T [ j]n,3
)(λ + 1
λT [ j]n,1 (λ) T [ j]
n,2 (λ)
T [ j]n,2 (λ) −λT [ j]
n,1 (λ) − 1λ
).
By direct calculation, we can rewrite T [2]n (λ) as follows
T [2]n (λ) =
(T [1]n,3T
[2]n,3 0
0 T [1]n,3 T
[2]n,3
) (λ2 + �
(0)n + �
(−2)n λ−2 �
(1)n λ + �
(−1)n λ−1
ϒ(1)n λ + ϒ
(−1)n λ−1 −�
(2)n λ2 − �
(0)n − λ−2
),
(4.18)
where �( j)n , �( j)
n , ϒ( j)n and �
( j)n are determined by
T [2]n (λ j )� j = 0, � j =
(ψn,1(λ j ) ψn,2(λ
−1j )
ψn,2(λ j ) −ψn,1(λ−1j )
), j = 1, 2.
After the action of twofold Darboux matrix T [2]n (λ) on (4.3), the transformation
formula from the original (wn, un) to new one is given as
w[2]n = wn
T [1]n+1,3T
[2]n+1,3
T [1]n,3T
[2]n,3
, (4.19a)
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54 J Nonlinear Sci (2018) 28:43–68
u[2]n = −T [1]
n+1,3T[2]n+1,3
T [1]n,3 T
[2]n,3
(un�
(−2)n+1 + wn�
(−1)n+1
), (4.19b)
where
T [2]n,3 =
(λ−12 |ψ [1]
n,2(λ2)|2 + λ2|ψ [1]n,1(λ2)|2
λ2|ψ [1]n,2(λ2)|2 + λ−1
2 |ψ [1]n,1(λ2)|2
)− iα2
,
�(−2)n = −�
(−2)n
[2]n
, �(−1)n = −�
(−1)n
[2]n
,
with
ψ [1]n (λ2) �
(ψ
[1]n,1(λ2)
ψ[1]n,2(λ2)
)= T [1]
n (λ2)
(ψn,1(λ2)
ψn,2(λ2)
),
�(−1)n =
∣∣∣∣∣∣∣∣∣∣∣
λ−21 ψn,1(λ1) ψn,1 (λ1) λ1ψn,2(λ1) λ21ψn,1(λ1)
λ−22 ψn,1(λ2) ψn,1(λ2) λ2ψn,2(λ2) λ22ψn,1(λ2)
λ−21 ψn,2
(λ−11
)ψn,2
(λ−11
)−λ1ψn,1
(λ−11
)λ21ψn,2
(λ−11
)
λ−22 ψn,2
(λ−12
)ψn,2
(λ−12
)−λ2ψn,1
(λ−12
)λ22ψn,2
(λ−12
)
∣∣∣∣∣∣∣∣∣∣∣
,
�(−2)n =
∣∣∣∣∣∣∣∣∣∣∣
λ21ψn,1(λ1) ψn,1(λ1) λ1ψn,2(λ1) λ−11 ψn,2(λ1)
λ22ψn,1(λ2) ψn,1(λ2) λ2ψn,2(λ2) λ−12 ψn,2(λ2)
λ21ψn,2
(λ−11
)ψn,2
(λ−11
)−λ1ψn,1
(λ−11
)−λ−1
1 ψn,1
(λ−11
)
λ22ψn,2
(λ−12
)ψn,2
(λ−12
)−λ2ψn,1
(λ−12
)−λ−1
2 ψn,1
(λ−12
)
∣∣∣∣∣∣∣∣∣∣∣
,
[2]n =
∣∣∣∣∣∣∣∣∣∣∣
λ−21 ψn,1(λ1) ψn,1(λ1) λ1ψn,2(λ1) λ−1
1 ψn,2(λ1)
λ−22 ψn,1(λ2) ψn,1(λ2) λ2ψn,2(λ2) λ−1
2 ψn,2(λ2)
λ−21 ψn,2(λ
−11 ) ψn,2
(λ−11
)−λ1ψn,1
(λ−11
)−λ−1
1 ψn,1
(λ−11
)
λ−22 ψn,2
(λ−12
)ψn,2
(λ−12
)−λ2ψn,1
(λ−12
)−λ−1
2 ψn,1
(λ−12
)
∣∣∣∣∣∣∣∣∣∣∣
.
Then, the potential transformation formula for the twofold Darboux transformation isfinally presented in the form
q[2]n = u[2]
n
w[2]n
= −T [1]n,3T
[2]n,3
T [1]n,3 T
[2]n,3
(qn�
(−2)n+1 + eiα ln(1+|qn |2)�(−1)
n+1
). (4.20)
4.3 N-fold Darboux Transformation
Further, let ψn(λk) � (ψn,1(λk), ψn,2(λk))T, k = 1, 2, 3 . . . , N be a column solu-
tions of (3.4) with λ = λk , the similar process yields the so-called N -fold Darbouxtransformation
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J Nonlinear Sci (2018) 28:43–68 55
ψ [N ]n (λ) = T [N ]
n (λ)ψn(λ), (4.21a)
with
T [N ]n (λ) =
⎛⎜⎜⎜⎝
N∏j=1
T [ j]n,3 0
0N∏j=1
T [ j]n,3
⎞⎟⎟⎟⎠
×
⎛⎜⎜⎜⎝
λN +N∑j=1
�(N−2 j)n λN−2 j
N∑j=1
�(N−2 j+1)n λN−2 j+1
N∑j=1
ϒ(N−2 j+1)n λ−N+2 j−1 −λ−N −
N∑j=1
�(N−2 j)n λ−N+2 j
⎞⎟⎟⎟⎠ ,
where �( j)n , �( j)
n , ϒ( j)n and �
( j)n are determined as
T [N ]n (λ j )� j = 0, � j =
(ψn,1(λ j ) ψn,2(λ
−1j )
ψn,2(λ j ) −ψn,1(λ−1j )
), j = 1, 2, . . . , N
and the corresponding N -fold potential transformation formula is given by
q[N ]n = −
N∏j=1
T [ j]n,3
T [ j]n,3
(qn�
(−N )n+1 + eiα ln(1+|qn |2)�(−N+1)
n+1
), (4.21b)
where
T [ j]n,3 =
⎛⎝ λ−1
j |ψ [ j−1]n,2 (λ j )|2 + λ j |ψ [ j−1]
n,1 (λ j )|2λ j |ψ [ j−1]
n,2 (λ j )|2 + λ−1j |ψ [ j−1]
n,1 (λ j )|2
⎞⎠
− iα2
,
�(−N )n = −�
(−N )n
[N ]n
, �(−N+1)n = −�
(−N+1)n
[N ]n
with
(ψ
[k]n,1(λk+1)
ψ[k]n,2(λk+1)
)= T [k]
n (λk+1)
(ψn,1(λk+1)
ψn,2(λk+1)
),
(ψ
[0]n,1(λ1)
ψ[0]n,2(λ1)
)=
(ψn,1(λ1)
ψn,2(λ1)
), k = 1, 2, . . . , N
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56 J Nonlinear Sci (2018) 28:43–68
and
[N ]n =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
λ−N1 ψn,1(λ1) λ−N+2
1 ψn,1(λ1) · · · λN−21 ψn,1(λ1) λN−1
1 ψn,2(λ1) λN−31 ψn,2(λ1) · · · λ−N+1
1 ψn,2(λ1)
.
.
.
.
.
. · · ·...
.
.
.
.
.
. · · ·...
λ−NN ψn,1(λN ) λ−N+2
N ψn,1(λN ) ... λN−2N ψn,1(λN ) λN−1
N ψn,2(λN ) λN−3N ψn,2(λN ) ... λ−N+1
N ψn,2(λN )
λ−N1 ψn,2(λ
−11 ) λ−N+2
1 ψn,2(λ−11 ) · · · λN−2
1 ψn,2(λ−11 ) −λN−1
1 ψn,1(λ−11 ) −λN−3
1 ψn,1(λ−11 ) · · · −λ−N+1
1 ψn,1(λ−11 )
.
.
.
.
.
. · · ·...
.
.
.
.
.
. · · ·...
λ−NN ψn,2(λ
−1N ) λ−N+2
N ψn,2(λ−1N ) · · · λN−2
N ψn,2(λ−1N ) −λN−1
N ψn,1(λ−1N ) −λN−3
N ψn,1(λ−1N ) · · · −λ−N+1
N ψn,1(λ−1N )
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
;
�(−N )n and�
(−N+1)n are described by
[N ]n , but the first column and the last one in
the [N ]n are changed to (λN
1 ψn,1(λ1), . . . , λNNψn,1(λN ), λN
1 ψn,2(λ−11 ), . . . , λN
N ψn,2
(λ−1N ))T, respectively.
5 Breather and Rogue Wave Solutions
In this section, we will construct breather and rogue wave solutions for the DKE (1.2)by using the obtained Darboux transformation. For this purpose, we begin with aplane-wave solution
qn = cei(kn+ωt), (5.1)
where
ω = z2[1 + c2(1 + iα)]e−ik + z−2[1 + c2(1 − iα)]eik − 2, z = (1 + c2)−iα2 ,
and c is a real constant, k is a wave number. Substituting seed solution (5.1) andintroducing a transformation
ψn =(e
i2 (kn+ωt) 0
0 e− i2 (kn+ωt)
)ψn, (5.2)
we can map the variable coefficient differential-difference equation (3.4) to constantcoefficient differential-difference equation,
Eψn = Nnψn,dψn
dt= Mnψn, (5.3)
where
Nn =(
λze−i k2 cze−i k2
−cz−1eik2 (λz)−1ei
k2
),
Mn = i
(λ
zei
k2 − z
λe−i k2
)Nn + i
2
(λ−2 − λ2
)+ i
2(1 − c2)
(z2e−ik − z−2eik
).
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J Nonlinear Sci (2018) 28:43–68 57
To find the fundamental solutions of the last linear system, we should solve theeigenvalues of matrix Nn . This is the solutions of the characteristic equation
det(pI − Nn
) = p2 −(λze−i k2 + (λz)−1 ei
k2
)p + (1 + c2) = 0. (5.4)
(a) If thematrix Nn exists two different eigenvalues p1 and p2, then the correspondingeigenfunction is
ψn =(e
i2 (kn+ωt) 0
0 e− i2 (kn+ωt)
) ⎛⎝
2∑j=1
c j� j pnj e
θ j t
⎞⎠ (5.5)
where � j =(
− p j ze−i k2 −λ−1
c1
), and θ j = i( λ
z ei k2 − z
λe−i k2 )p j + i
2 (λ−2 − λ2) +
i2 (1 − c2)(z2e−ik − z−2eik).
(b) If the matrix Nn has a repeated eigenvalue p = √1 + c2, (λ = p+c
z ek2 i ), then
the corresponding eigenfunction is
ψn = c
(e
i2 (kn+ωt) 0
0 e− i2 (kn+ωt)
)
×[(−ze−i k2
1
) (n + i
(ei
k2λ
z− e−i k2
z
λ
)t
)+ �
]pneθ t , (5.6)
where the vector � =(− z(cs+p)
c e− k2 i
s
)is a solution of the algebraic linear
system (Nn − pI )� = p� with � =(
− pze−i k2 −λ−1
c1
).
At this moment, considering plane-wave solution (5.1) as the seed solution and substi-tuting (5.5) into the onefoldDarboux transformation (4.16),we can construct a breathersolution of (1.2) composed of hyperbolic functions and exponential functions. Sincetheir full expressions are cumbersome, we omit them here. It is shown that time-periodic Kuznetsov-Ma breather and space-periodic Akhmediev breather solution canbe obtained by an appropriate choice of the parameters λ and k. Specially, settingλ = eξ+iη, where ξ and η are real constants, if wave number k, cubic-quintic non-linear coefficient α, η and c satisfy constraint condition k = 2η − α ln(1 + c2), thecharacteristic equation (5.4) reduces to
p2 − (eξ + e−ξ )p + (1 + c2) = 0, (5.7)
where the roots are either real roots or complex conjugation root pairs. To obtainspace-periodic breather, we need to choose the pair of conjugate complex roots, thatis (eξ + e−ξ )2 − 4(1 + c2) < 0. On the other hand, when (eξ + e−ξ )2 − 4(1 +
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58 J Nonlinear Sci (2018) 28:43–68
Fig. 1 Dynamical evolution of breather solution in the DKE. Plotted is the |q[1]n | field
c2) > 0, the solution becomes time-periodic breather. For illustration, we plot thedynamics of the breather solution for parameters c = 0.7, λ = 2, k = 1, α = 1 inFig. 1a. Dynamical evolution of the time-periodic Kuznetsov-Ma breather solutionfor parameters c = 1, λ = e, k = 1, α = − 1
ln(2) is shown in Fig. 1b. The dynamicalevolution of the space-periodic Akhmediev breather solution with specific parametersc = 2, λ = e, k = 1, α = − 1
ln(2) is displayed in Fig. 1c.Furthermore, inserting (5.6) into (4.16) and through a direct but rather tedious
calculation, we obtain an exact rational form solution
q[1]n =
(Bn
Bn−1
)−iα c
z2
(−1 + An
Bn
)ei(kn+ωt+k), (5.8)
where
An = 4 ic2 p2(z2e−ik + z−2eik)t + 2p2,
Bn = 2c2(n + i
(eik z−2 − e−ik z2
)p2t + 2cs + c + p
2c
)2
+ 2c4(eik z−2 + e−ik z2
)2p2t2 + 1
2.
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J Nonlinear Sci (2018) 28:43–68 59
Fig. 2 The W-shaped rational soliton solution |q[1]n | with specific parameters α = 1, c = 0.5, s = −1,
k = −α ln(1 + c2) + π2 . a Profile plot, b intensity contour plot, c evolution plot
The above expression exists three free parameters c, s and k, whose role will beexplained in detail next.
If we restrict the condition z2e−ik + z−2eik = 2 cos(k + α ln(1 + c2)) = 0, i.e.,k = −α ln(1+ c2) + π
2 , the solution becomes a W-shaped rational soliton moving on
a constant background. By letting n → ∞ and t → ∞, so |q[1]n | → c. The maximum
peak of |q[1]n | occurs at A � {(n, t)|n + i(eik z−2 − e−ik z2)p2t + 2cs+c+p
2c = 0}and it is equal to c(3 + 4c2). Similarly, the minimum peak of |q[1]
n | occurs at A �{(n, t)|n + i(eik z−2 − e−ik z2)p2t + 2cs+c+p
2c = ±√3+4c22c } and it is equal to 0. These
phenomena are shown in Fig. 2.To describe the rogue wave phenomenon, we will require z2e−ik + z−2eik �= 0
in the latter. If wave number and cubic-quintic nonlinear coefficient satisfy constraintcondition k = −α ln(1+ c2), we obtain the rogue wave solution described in Fig. 3a.The effect of the free parameter s in the solution controls the position of the roguewave (see Fig. 4), and the free parameter c determines the background amplitude. Themaximum amplitude of |q[1]
n | occurs at (n, t) = (s + 1 + p2c , 0), and it is equal to
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60 J Nonlinear Sci (2018) 28:43–68
Fig. 3 The evolution plot of the roguewave solution |q[1]n |with specific parametersα = 1, c = 0.5, s = −1
and a k = −α ln(1 + c2), b k = 0.5, c k = 1
Fig. 4 Density plots of rogue wave solution |q[1]n | with α = 1, σ = −1, c = 0.5, k = −α ln(1 + c2) and
a s = −15, b s = −1, c s = 10
c(3+4c2) (see Fig. 5), which reveals a large amplitudes rogue wave. For comparison,we note that their continuous counterparts are exactly three times the background level(see Eq. (4.7) of Ref. Qiu et al. (2015)). If k �= −α ln(1 + c2), the rogue wave whichrises from the constant background traverses across the lattice and then disappears intothe background again which named as traveling rogue wave (Ohta and Yang 2014).The increase in |k| slows down the speed of ‘rise’ and ‘disappearance’ (see Fig. 3b,
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J Nonlinear Sci (2018) 28:43–68 61
Fig. 5 Plot of the amplitude of the roguewave solution |q[1]n |withα = 1, s = −1, k = −α ln(1+c2), t = 0
and a c = 0.1, b c = 0.5, c c = 1, which shows the large amplitude of the DKE
Fig. 6 Density plots of roguewave solution |q[1]n |with c = 0.5, s = −1, k = −α ln(1+c2) and a α = 1.5,
b α = 2, c α = 3, which exhibit the prolongation of the existing state of the traveling rogue wave
c), and the cubic-quintic nonlinear term exhibits accelerating (α > 0) or decelerating(α < 0) of the process (see Fig. 6).
6 Continuous Limit Theory
In this section, we prove that the theory of the DKE including Lax pair, Darboux trans-formation and exact solutions yields the corresponding results of KE in the continuouslimit. To this end, we set
qn = δχ(nδ, δ2t) + o(δ3)= δχ(x, τ ) + o(δ3), α = a
δ(6.1a)
andλ = 1 + δμ + o(δ), μ = μ1, μ2, . . . , μN , (6.1b)
then
ψn, j (λ) = δφ j (δn, δ2t, μ) + o(δ3)= δφ j (x, τ, μ) + o(δ3) = δφ j (μ) + o(δ3), j = 1, 2. (6.1c)
where μ is a constant spectral parameter. It is shown that, in the limit δ −→ 0, thetheory of our discrete KE (1.2) yields the theory of KE (1.1) with the transformation(6.1).
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62 J Nonlinear Sci (2018) 28:43–68
6.1 Continuum Limit for the Lax Pair
Under the transformation, we see that, from (3.4), the discrete linear eigenvalue prob-lem leads to
ψn+1 − Nnψn = (φx − Sφ)δ2 + o(δ2), (6.2a)dψn
dt− Mnψn = (φτ − Xφ)δ3 + o(δ3), (6.2b)
where
S =(
μ − ia2 |χ |2 χ
−χ −μ + ia2 |χ |2
),
X =(2iμ2 + i |χ |2 + ia2|χ |4 − a
2 (χχx − χx χ ) 2iμχ + iχx − a|χ |2χ−2iμχ + i χx + a|χ |2χ −2iμ2 − i |χ |2 − ia2|χ |4 + a
2 (χχx − χx χ )
).
One can check that the zero-curvature equation Sτ − Xx + SX − XS = 0 gives riseto the KE.
6.2 Continuum Limit for the Darboux Transformation
Notice that (4.13) and (4.15) have the expansions
T [1]n,1
T [1]n,3
= − λ−11 |ψn,2(λ1)|2 + λ1|ψn,1(λ1)|2
λ−11 |ψn,1(λ1)|2 + λ1|ψn,2(λ1)|2
= −1 − 2D[1]1 δ + o(δ),
T [1]n,2
T [1]n,3
= − (|λ1|2 − |λ1|−2)ψn,1(λ1)ψn,2(λ1)
λ−11 |ψn,1(λ1)|2 + λ1|ψn,2(λ1)|2
= 2D[1]2 δ + o(δ),
T [1]n,3 =
(1 + (μ1 + μ1)
|φ1|2 − |φ2|2|φ1|2 + |φ2|2 δ
)− ia2δ
= D[1]3 + O(δ), (6.3)
where
D[1]1 = μ1|φ1|2 − μ1|φ2|2
|φ1|2 + |φ2|2 ,
D[1]2 = −(μ1 + μ1)
φ1φ2
|φ1|2 + |φ2|2 ,
D[1]3 = e
−i a2 (μ1+μ1)|φ1|2−|φ2 |2|φ1|2+|φ2 |2 ,
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J Nonlinear Sci (2018) 28:43–68 63
then the expansion of the Darboux matrix T [1]n (λ) is
T [1]n (λ) =
(T [1]n,3 0
0 T [1]n,3
)⎛⎜⎜⎝
λ + 1λ
T [1]n,1
T [1]n,3
T [1]n,2
T [1]n,3
T [1]n,2
T [1]n,3
−λT [1]n,1
T [1]n,3
− 1λ
⎞⎟⎟⎠
=(D[1]3 + O(δ) 0
0 D[1]3 + O(δ)
)
×(1 + δμ − (1 − δμ)(1 + 2D[1]
1 δ) + o(δ) 2D[1]2 δ + o(δ)
2D[1]2 δ + o(δ) (1 + δμ)(1 + 2D[1]
1 δ) − 1 + δμ + o(δ)
)
= 2δD[1](μ) + o(δ),
where
D[1](μ) =(D[1]3 0
0 D[1]3
) (μ − D[1]
1 D[1]2
D[1]2 μ + D[1]
1
).
Thus, the passage to the limit δ −→ 0 for the Darboux transformation gives
limδ→0
ψ[1]n (λ)
2δ2= lim
δ→0
T [1]n (λ)ψn(λ)
2δ2= φ[1](μ), (6.4a)
limδ→0
q[1]n (λ)
δ= lim
δ→0
T [1]n,3
( − qnT[1]n+1,1 − eiα ln(1+|qn |2)T [1]
n+1,2
)
δT [1]n,3T
[1]n+1,3
= limδ→0
[(D[1]3
)2 + O(δ)][χδ(1 + 2D[1]1 δ) − 2δD[1]
2 eia|χ |2δ + o(δ)]
δ
= χ [1], (6.4b)
where
φ[1](μ) = D[1](μ)φ(μ), (6.5a)
χ [1] = (D[1]3
)2(χ − 2D[1]
2
). (6.5b)
One can examine that Eq. (6.5) satisfies the linear equation
φ[1]x = S[1]φ[1], φ[1]
τ = X [1]φ[1],
whereS[1] andX [1] are defined byS andX , respectively, but the potentialχ is replacedby χ [1].
Further, from (4.19), we find
�(−1)n = −8δ7♥[2] + o(δ7),
�(−2)n = −4δ6♣[2] + o(δ6),
[2]n = 4δ6♣[2] + o(δ6),
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64 J Nonlinear Sci (2018) 28:43–68
where
♥[2] =
∣∣∣∣∣∣∣∣∣∣
μ1φ1(μ1) φ1(μ1) φ2(μ1) μ21φ1(μ1)
μ2φ1(μ2) φ1(μ2) φ2(μ2) μ22φ1(μ2)
μ1φ2(−μ1) φ2(−μ1) −φ1(−μ1) μ21φ2(−μ1)
μ2φ2(−μ2) φ2(−μ2) −φ1(−μ2) μ22φ2(−μ2)
∣∣∣∣∣∣∣∣∣∣,
♣[2] =
∣∣∣∣∣∣∣∣∣∣
μ1φ1(μ1) φ1(μ1) φ2(μ1) μ1φ2(μ1)
μ2φ1(μ2) φ1(μ2) φ2(μ2) μ2φ2(μ2)
μ1φ2(−μ1) φ2(−μ1) −φ1(−μ1) −μ1φ1(−μ1)
μ2φ2(−μ2) φ2(−μ2) −φ1(−μ2) −μ2φ1(−μ2)
∣∣∣∣∣∣∣∣∣∣.
Hence, the continuous limit of twofold potential transformation formula (4.20) givesrise to continuous one
q[2]n (λ) = δ
(D[1]3 D[2]
3
)2(χ − 2
♥[2]
♣[2]) + o(δ). (6.6)
Moreover, the transformation formula of the N -fold Darboux transformation (4.21b)converges to counterparts of KE
limδ→0
q[N ]n (λ)
δ=
⎛⎝
N∏j=1
D[ j]3
⎞⎠
2 (χ − 2
♥[N ]
♣[N ]
)= χ [N ], (6.7)
where
♣[N ] =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
φ1(μ1) μ1φ1(μ1) · · · μN−11 φ1(μ1) φ2(μ1) μ1φ2(μ1) · · · μN−1
1 φ2(μ1)
.
.
.... · · ·
.
.
....
.
.
. · · ·...
φ1(μN ) μNφ1(μN ) ... μN−1N φ1(μN ) φ2(μN ) μNφ2(μN ) ... μN−1
N φ2(μN )
φ2(−μ1) μ1φ2(−μ1) · · · μN−11 φ2(−μ1) −φ1(−μ1) −μ1φ1(−μ1) · · · −μN−1
1 φ1(−μ1)
.
.
.... · · ·
.
.
....
.
.
. · · ·...
φ2(−μN ) μN φ2(−μN ) · · · μN−1N φ2(−μN ) −φ1(−μN ) −μN φ1(−μN ) · · · −μN−1
N φ1(−μN )
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
,
and ♥[N ] are defined by♣[N ] where the last column is replaced by (μN1 φ1(μ1), μ
N2 φ1
(μ2), . . . , μNNφ1(μN ) , μN
1 φ2(−μ1), μN2 φ2(−μ2), . . . , μ
NN φ2(−μN ))T.
6.3 Continuum Limit for the Breather and Rogue Wave Solutions
In order to show that the solutions of discrete KE (1.2) obtained by the Darbouxtransformation converge to the corresponding solutions of the KE in the continuouslimit, we set
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c = δc, k = δb, s = s
δ. (6.8)
From this transformation, seed solution (5.1) is presented as
qn = cei(kn+ωt) = δχ + o(δ), (6.9)
where χ is a plane-wave solution of (1.1)
χ = cei(bx+ωτ ), ω = a2c4 + 2c2 − b2. (6.10)
Breather Solution Introducing p = eδp, we have λ = 1+μδ+(
μ2
2 − ipc2
ac2+2iμ+b
)δ2+
o(δ2), which is consistent with the definition of (6.1b). The characteristic equation of(5.4) yields continuous one
limδ→0
det(pI − Nn)
δ2= p2 + b2
4+ ibμ + ac2b
2− μ2 + iμac2 + c2 + a2c4
4= 0.
(6.11)
Then, the eigenfunction of (5.5) becomes
ψn =(e
i2 (δbn+δ2ωt)+o(δ2) 0
0 e− i2 (δbn+δ2ωt)+o(δ2)
)
×⎛⎝
2∑j=1
δc j (ϒ j + O(δ))ep j (δn + θ jδ2t) + o(δ2)
⎞⎠
= δφ + O(δ), (6.12a)
where
φ =(e
i2 (bx+ωτ ) 0
0 e− i2 (bx+ωτ )
) ⎛⎝
2∑j=1
c jϒ jep j (x+θ j τ)
⎞⎠ , (6.12b)
with ϒ j =(
−p j+μ− i2 ac
2− i2 b
c1
), θ j = 2iμ−ac2 −b and p j is the solution of (6.11).
We see that (6.12b) can be solved by Lax pair (6.2) with solution (6.10). Therefore,from (6.4), the breather solution of DKE (1.2) converges to the breather solution ofKE (1.1) as δ → 0.Rogue Wave Solution It is noted that repeated eigenvalue p = √
1 + c2 = 1 +c2
2 δ2 + o(δ2) and λ = p+cz ei
k2 = 1 + μδ + μ2
2 δ2 + o(δ2) are also consistent with
the definition of (6.1b), where μ = 2c+ib+iac22 is also the repeated condition of p in
(6.11). Analogously, the expression of (5.6) leads to
ψn = δφ + o(δ), (6.13a)
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where
φ = c
(e
i2 (bx+ωτ ) 0
0 e− i2 (bx+ωτ )
)
×[(−1
1
)(x + (2ic − 2ac2 − 2b)τ ) +
(− cs+1cs
)]eθ τ . (6.13b)
We remark here that (6.13b) is just the eigenfunction of (6.2) corresponding to (6.10)andμ = 2c+ib+iac2
2 . Consequently, from (6.4), the expansion of rational form solution(5.8) achieves to
q[1]n = δχ [1] + o(δ). (6.14)
Here χ [1] is a rogue wave solution of (1.1) expressed by
χ [1] = cei(bx + ωτ ) + 2iac
C
B
(−1 + A
B
), (6.15)
where
A = 8i c2τ + 2,
C = 2cx − (4ac3 + 4bc)τ + 2cs + 1,
B = 2c2(x − 2(ac2 + b)τ + 2cs + 1
2c
)2
+ 8c4τ 2 + 1
2.
This implies that the rogue wave feature of the rational solution coincides with thecorresponding results of KE. However, it is surprising to note that the characteristicof the rational soliton is decaying and vanishing when the step size δ −→ 0. Theexcitationmechanism of the discrete rational soliton is clearly controlled by restrictioncondition k = −α ln(1+ c2) + (2m + 1)π
2 , m ∈ N. But the restriction condition doesnot hold in continuous case. So, there is no rational soliton solution to KE. Moreover,the expansion for the maximum amplitude is c(3+ 4c2) = 3cδ + o(δ), where leadingterm 3c is exactly corresponding result of (6.15), from which we see that infinitesimalof higher order does not affect continuous equation.
7 Conclusion
Starting from the extended Ablowitz–Ladik discrete eigenvalue problem, a four-component generalization of Ablowitz–Ladik lattice hierarchy with two arbitraryadjustable functions is constructed. With special reductions and suitable freelyfunctions, a new integrable semi-discrete Kundu–Eckhaus equation is derived. Theintegrability of the new discrete model is confirmed by providing the Lax pair andan infinite number of conservation laws. Further, the Darboux transformation is con-structed based on the gauge transformation of the discrete eigenvalue problem. Byusing the Darboux transformation, we obtain breather and rational wave solutions to
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J Nonlinear Sci (2018) 28:43–68 67
our discrete Kundu–Eckhaus equation. Their dynamical properties are also discussedin detail to reveal some novel interesting phenomena. Finally, we investigate the topicof the continuous limits of the discrete model, as well as its Lax formulation, Darbouxtransformation, breather and rogue solutions.
It is known that the discussion of continuous limits theory for the whole discretehierarchy is extremely difficult topic. It deserves a further investigation to developa systematic limit procedure not only on the level of semi-discrete Kundu–Eckhausequation hierarchy but also on the level of solutions and the corresponding integrabil-ities.
Acknowledgements The work of HQZ is supported by National Natural Science Foundation of ChinaunderGrant 11301331, Natural Science Foundation of Shanghai underGrant 17ZR1411600, and InnovationProgram of Shanghai Municipal Education Commission under Grant 14YZ135, that of JYY by CAPES andCNPq of Brazil, that of ZNZ by National Natural Science Foundation of China under Grants 11271254,11428102 and 11671255, and by the Ministry of Economy and Competitiveness of Spain under grandsMTM2012-37070 and MTM2016-80276-P(AEI/FEDER,EU). We sincerely thank the referees for theirvery useful and constructive comments.
References
Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations. J. Math. Phys. 16, 598–603 (1975)Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys.
17, 1011–1018 (1976)Ablowitz, M.J., Herbst, B.M., Schober, C.M.: Discretizations, integrable systems and computation. J. Phys.
A Math. Gen. 34, 10671–10693 (2001)Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schrödinger Systems.
Cambridge University Press, New York (2004)Calogero, F., Eckhaus, W.: Nonlinear evolution equations, rescalings, model PDEs and their integrability:
I. Inverse Probl. 3, 229–262 (1987)Clarkson, P.A., Cosgrove, C.M.: Painleve analysis of the nonlinear Schrödinger family of equations. J. Phys.
A Math. Gen. 20, 2003–2024 (1987)Flach, S., Gorbach, A.V.: Discrete breathers—advances in theory and applications. Phys. Rep. 467, 1–116
(2008)Geng, X.G.: A hierarchy of non-linear evolution equations, its Hamiltonian structure and classical integrable
system. Physica A 80, 241–251 (1992)Geng,X.G., Tam,H.W.:Darboux transformation and soliton solutions for generalized nonlinear Schrödinger
equations. J. Phys. Soc. Jpn. 68, 1508–1512 (1999)Gieseker, D.: The Toda hierarchy and the KdV hierarchy. Commun. Math. Phys. 181, 587–603 (1996)Gorshkov, K.A., Pelinovsky, D.E.: Asymptotic theory of plane soliton self-focusing in two-dimensional
wave media. Physica D 85, 468–484 (1995)Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformations in Integrable Systems: Theory and Their Appli-
cations to Geometry. Springer, Berlin (2005)Johnson, R.S.: On the modulation of water waves in the neighbourhood of kh ≈ 1.363. Proc. R. Soc. Lond.
A 357, 131–141 (1977)Kakei, S., Sasa, N., Satsuma, J.: Bilinearization of a generalized derivative nonlinear Schrödinger equation.
J. Phys. Soc. Jpn. 64, 1519–1523 (1995)Kundu, A.: Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear
Schrödinger-type equations. J. Math. Phys. 25, 3433–3438 (1984)Levi, D., Scimiterna, C.: The Kundu–Eckhaus equation and its discretizations. J. Phys. A Math. Theor. 42,
465203 (2009)Lin, R.L., Ma, W.X., Zeng, Y.B.: Higher order potential expansion for the continuous limits of the Toda
hierarchy. J. Phys. A Math. Gen. 35, 4915–4928 (2002)Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)
123
68 J Nonlinear Sci (2018) 28:43–68
Morosi, C., Pizzocchero, L.: On the continuous limit of integrable lattices I. The Kac-Moerbeke system andKdV theory. Commun. Math. Phys. 180, 505–528 (1996)
Ohta, Y., Yang, J.K.: General rogue waves in the focusing and defocusing Ablowitz–Ladik equations. J.Phys. A Math. Theor. 47, 255201 (2014)
Qiu, D.Q., He, J.S., Zhang, Y.S., Porsezian, K.: The Darboux transformation of the Kundu–Eckhaus equa-tion. Proc. R. Soc. A 471, 20150236 (2015)
Schwarz,M.: Korteweg-deVries and nonlinear equations related to the Toda lattice. Adv.Math. 44, 132–154(1982)
Thiemann, T.: Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge(2005)
Tsuchida, T.: Integrable discretizations of derivative nonlinear Schrödinger equations. J. Phys. A Math.Gen. 35, 7827–7847 (2002)
Vakhnenko,O.O.: Semi-discrete integrable nonlinear Schrödinger systemwith background-controlled inter-site resonant coupling. J. Nonlinear Math. Phys. 24, 250–302 (2017)
Vekslerchik, V.E., Konotop, V.V.: Discrete nonlinear Schrödinger equation under non-vanishing boundaryconditions. Inverse Probl. 8, 889–909 (1992)
Zeng, Y.B., Wojciechowski, S.R.: Continuous limits for the Kac-Van Moerbeke hierarchy and for theirrestricted flows. J. Phys. A Math. Gen. 28, 3825–3840 (1995)
Zhang, Y.S., Guo, L.J., Zhou, Z.X., He, J.S.: Darboux transformation of the second-type derivative nonlinearSchrödinger equation. Lett. Math. Phys. 105, 853–891 (2015)
Zhao, H.Q., Zhu, Z.N.: Multisoliton, multipositon, multinegaton and multiperiodic solutions of a coupledVolterra lattice system and their continuous limits. J. Math. Phys. 52, 023512 (2011)
Zhu, Z.N., Zhao, H.Q., Zhang, F.F.: Explicit solutions to an integrable lattice. Stud. Appl. Math. 125, 55–67(2010)
Zhu, Z.N., Zhao, H.Q., Wu, X.N.: On the continuous limits and integrability of a new coupled semidiscretemKdV system. J. Math. Phys. 52, 043508 (2011)
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