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Complexiton solutions of the mKdV equation with self-consistent sources
Physics Letters A 374 (2010) 1457–1463
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Physics Letters A www.elsevier.com/locate/pla
Complexiton solutions of the mKdV equation with self-consistent sources Jun Su ∗ , Wei Xu, Liang Gao, Genjiu Xu Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 9 December 2009 Received in revised form 17 January 2010 Accepted 21 January 2010 Available online 28 January 2010 Communicated by A.R. Bishop Keywords: Complexitons Self-consistent sources mKdV equation Darboux transformation
a b s t r a c t A class of complexiton solutions of the mKdV equation with self-consistent sources (mKdVESCSs) are presented by the generalized binary Darboux transformation (GBDT) with N arbitrary t-functions. Taking the special initial seed solution for auxiliary linear problems and the special functions of time t, the real-valued 1-complexiton solution of the mKdVESCSs is considered through the GBDT by selecting the complex spectral parameters in its Lax pair. It is important to point out that the real-valued 1-complexiton solution of the mKdVESCSs is analytical and singular. Moreover, the detailed structures of the 1-complexiton solution are given out analytically and graphically. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The soliton equations with self-consistent sources (SESCSs) have received considerable attention in recent years. It is well known that these equations can exhibit richer nonlinear dynamics than soliton equations and have important physical applications in plasma physics, hydrodynamics, solid state physics, etc. [1–5]. In terms of finding exact solutions, some methods for solving soliton equations have been used to SESCSs as well. For example, through the inverse scattering method, Zeng et al. [6] have obtained the explicit soliton solutions for the KdV, AKNS, mKdV, nonlinear Schrödinger and Kaup–Newell hierarchies with self-consistent sources. In addition to that, the N-soliton solutions for the mKdVESCSs was obtained by Hirota method and Wronskian technique [7]. The Darboux transformation (DT) [8] is one of the most effective tools to construct the exact and explicit solutions of the nonlinear evolution equations. However, the normal DT cannot be directly used to construct the nontrivial solutions for SESCSs. A generalized binary Darboux transformation (GBDT) for the KdV hierarchies and mKdV hierarchy with self-consistent sources have been proposed [9,10]. With an arbitrary function at time t, the GBDT offers a non-auto-Bäcklund transformation between two SESCSs with different degrees of sources. Recently, a kind of classification of exact and explicit solutions of integrable systems was proposed according to the property of associated spectral parameters [11]. In general, positons [12], usually related to the positive spectral parameters, are expressed by the trigonometric functions. Negatons of nonlinear systems [13], usually related to the negative spectral parameters, are expressed by the hyperbolic functions. Solitons are a kind of non-singular negatons. The study of the solitons, negatons, positons and their interaction solutions has been made for the KdV equation in Refs. [11,14]. Moreover, for the SESCSs, the solitons, negatons and positons are also obtained including the KdV, Schrödinger, KP, NLS, mKdV with sources [15–19]. The complexiton solutions [20], which are a novel class of explicit exact solutions to the soliton equations, are related to the complex spectral parameters and are expressed by combinations of the trigonometric functions and the exponential functions. Later, the concepts of complexitons were extended to the Toda lattice equation, the Boussinesq equation, the mKdV equation, etc. [21–24]. Whether the soliton equations with self-consistent sources can have complexiton solutions, Ma provided a positive answer by considering the KdV equation with self-consistent sources [25]. However, not much work has been done on the complexiton solutions of the other SESCSs. In this Letter, we will construct the complexiton solutions of mKdVESCSs. The study of solutions will certainly enrich the theory of the mKdVESCSs. The Letter is organized as follows. In Section 2, we derive the GBDT for the mKdVESCSs. In Section 3, the complexiton solutions of the mKdVESCSs are obtained. Section 4 presents the summary and discussions.
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2. The generalized binary Darboux transformation for the mKdVESCSs The negative mKdVESCSs is defined as [19]
ut − 6u 2 u x + u xxx −
n
ϕ12 j + ϕ22 j = 0,
(2.1a)
j =1
ϕ1 j,x = −λ j ϕ1 j + u ϕ2 j ,
ϕ2 j,x = u ϕ1 j + λ j ϕ2 j ,
j = 1, 2, . . . , n ,
(2.1b)
where λ j are distinct constants, and ϕ j = (ϕ1 j , ϕ2 j ) , j = 1, 2, . . . , n. It is associated with the discrete spectrum of the Schrödinger operator and represents the interaction of high-frequency waves ϕ j ( j = 1, 2, . . . , n), with a single, low-frequency wave u. In order to show the existence of complexiton solutions for the mKdVESCSs, we consider the mKdV equation with 2n specific complex self-consistent sources: T
ut − 6u 2 u x + u xxx −
n
ϕ12 j + ϕ22 j + ϕ¯ 12 j + ϕ¯ 22 j = 0,
(2.2a)
j =1
ϕ1 j,x = −λ j ϕ1 j + u ϕ2 j ,
ϕ2 j,x = u ϕ1 j + λ j ϕ2 j ,
j = 1, 2, . . . , n ,
(2.2b)
ϕ¯ 1 j,x = −λ¯ j ϕ¯ 1 j + u ϕ¯ 2 j ,
ϕ¯ 2 j,x = u ϕ¯ 1 j + λ¯ j ϕ¯ 2 j ,
j = 1, 2, . . . , n ,
(2.2c)
¯ j , ϕ¯ 1 j and ϕ¯ 2 j are complex conjugates of λ j , ϕ1 j and ϕ2 j , respectively. Moreover, we will focus on the case of complex eigenvalues where λ λ j and adjoint eigenfunctions ϕ j with only real potential u. The introduction of adjoint eigensystems was proposed for the first time in Ref. [26]. Let
ϕ1 j = ϕ1(1j) + i ϕ1(2j) ,
λ j = α j + iβ j , where
ϕ2 j = ϕ2(1j) + i ϕ2(2j) ,
j = 1, 2, . . . , n ,
(2.3)
α j , β j = 0, ϕ1(1j) , ϕ1(2j) , ϕ2(1j) and ϕ2(2j) are all real. Therefore, the negative mKdVESCSs (2.2) is equivalent to the following real system: ut − 6u 2 u x + u xxx − 2
n
2
ϕ1(1j) + ϕ2(1j)
2
(2) 2 (2) 2 − ϕ1 j + ϕ2 j = 0,
(2.4a)
j =1
ϕ1(1j),x + α j ϕ1(1j) − β j ϕ1(2j) − u ϕ2(1j) = 0,
ϕ1(2j),x + α j ϕ1(2j) + β j ϕ1(1j) − u ϕ2(2j) = 0,
j = 1, 2, . . . , n ,
(2.4b)
ϕ2(1j),x − α j ϕ2(1j) + β j ϕ2(2j) − u ϕ1(1j) = 0,
ϕ2(2j),x − α j ϕ2(2j) − β j ϕ2(1j) − u ϕ1(2j) = 0,
j = 1, 2, . . . , n ,
(2.4c)
where α j , β j = 0 are arbitrary real constants, and u, First, we define a linear map
S:
ϕ1 j ϕ2 j
→
ϕ1(1j) , ϕ1(2j) , ϕ2(1j) and ϕ2(2j) are real independent variables.
ϕ2 j . ϕ1 j
(2.5)
The corresponding Lax pair for the mKdVESCSs (2.4) is shown as below
Φx =
−λ u Φ, u λ
Φt = V (λ, u )Φ +
n H (ϕ j ) j =1
where
V (λ, u ) = H (ϕ j ) =
φ
1 2
(2.6a)
λ − λj
H(Sϕ j)
+
λ + λj
4 λ3 − 2 λ u 2 2 −4λ u − 2λu x − u xx + 2u 3
ϕ12 j −ϕ1 j ϕ2 j 2 −ϕ1 j ϕ1 j ϕ2 j
Φ+
n H (ϕ¯ j ) j =1
λ − λ¯ j
+
H ( S ϕ¯ j )
λ + λ¯ j
−4λ2 u + 2λu x − u xx + 2u 3 −4λ3 + 2λu 2
Φ,
(2.6b)
,
(2.6c)
(2.6d)
,
and Φ = φ 1 is the solution of (2.6) corresponding to the spectral parameter λ. 2 In order to obtain the generalized binary Darboux transformation with an arbitrary function of t for the mKdVESCSs, some symmetric forms are defined. Suppose that f j =
f1 j
f2 j
( j = 1, . . . , N) are different solutions of the system (2.6) with the corresponding spectral
parameters λ = λ j ( j = 1, . . . , N), respectively, φ is a scalar and a group of N arbitrary t-functions e 1 (t ), . . . , e N (t ), h = W 2 are defined as follows:
W 0 {e 1 , f 1 }, . . . , {e N , f N } = det F , (i )
W1
{e 1 , f 1 }, . . . , {e N , f N }; h = det
F
αi
b hi
,
i = 1, 2,
h1 h2
, then W 0 , W 1 ,
J. Su et al. / Physics Letters A 374 (2010) 1457–1463
W 1 {e 1 , f 1 }, . . . , {e N , f N }; h =
(1 )
W 1 ({e 1 , f 1 }, . . . , {e N , f N }; h1 )
(2 )
W 1 ({e 1 , f 1 }, . . . , {e N , f N }; h2 )
W 2 {e 1 , f 1 }, . . . , {e N , f N }; φ = det
F
α1T
α1
φ
1459
,
,
where
F = δi j e i + σ ( f i , f j )
σ ( f i , f j ) := −
N ×N
W ( fi, f j) 2(λi − λ j )
b=
,
,
T
αi = ( f i1 , . . . , f iN ),
σ ( f 1 , h), . . . , σ ( f N , h) , 1
σ ( f i , f i ) := W ( f i , ∂λi f i ), 2
W ( f i , f j ) = f 1i f 2 j − f 2i f 1 j .
Now, we have the following formula of the GBDT for the mKdVESCSs (2.4). (k)
Theorem 2.1. Let (u , ϕ1 =
(k) ϕ11
(k)
, . . . , ϕn =
(k)
ϕ21
(k) ϕ1n
), k = 1, 2, be a solution of Eq. (2.4), λ = α + i β be a complex number, e (t ) be an arbitrary
(k) ϕ2n
t-function and f be a solution of the system (2.6) with λ = λn+1 , then the GBDT with an arbitrary t-function for the mKdVESCSs (2.4) defined by
Φ˜ =
W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; Φ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
W 2 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; 0)
u˜ = u +
ϕ˜ (jk) =
(2.7a)
,
W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
(2.7b)
,
(k) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; ϕ j )
W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
1 j n , k = 1, 2,
,
ϕ˜n(1+)1
e j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f )
=
e j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
ϕ˜n(2+)1
e j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f )
=
ie j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
(2.7c)
+
e¯ j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f )
−
e¯ j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
e¯ j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f ) i e¯ j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
,
(2.7d)
,
(2.7e)
˜ u˜ , ϕ˜ , . . . , ϕ˜ , k = 1, 2, satisfy Eq. (2.6) with n replaced by n + 1, thus u˜ , ϕ˜ , . . . , ϕ˜ , k = 1, 2, where h¯ is the complex conjugate of h, and Φ, 1 n+1 1 n+1 is a solution of Eq. (2.4) with n replaced by n + 1. (k)
(k)
(k)
(k)
The N-times repeated GBDT with N arbitrary t-functions for the mKdVESCSs (2.4) is given by the following theorem. (k)
Theorem 2.2. Let (u , ϕ1 =
(k) ϕ11 (k)
ϕ21
(k)
, . . . , ϕn =
(k) ϕ1n (k) ϕ2n
), k = 1, 2, be a solution of Eq. (2.4), λ j = α j + i β j , 1 j n + N, be distinct complex numbers,
e j (t ) be an arbitrary t-function and f j be a solution of the system (2.6) with λ = λn+ j , j = 1, . . . , N, then the N-times repeated GBDT with N arbitrary t-functions for the mKdVESCSs (2.4) defined by
Φ˜ =
M1 M0
u˜ = u +
ϕ˜ (jk) = ϕ˜n(1+) j = ϕ˜n(1+) j
=
(2.8a)
, M2
M0 M3 M0
,
(2.8b)
, 1 j n , k = 1, 2,
e j (t ) M 4
e j (t ) M 0
+
e j (t ) M 4
ie j (t ) M 0
(2.8c)
¯4 e¯ j (t ) M
e¯ j (t ) M 0
,
1 j N,
(2.8d)
,
1 j N,
(2.8e)
−
¯4 e¯ j (t ) M
i e¯ j (t ) M 0
where
M 0 = W 0 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N } ,
M 1 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; Φ ,
M 2 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; 0 ,
(k) M 3 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; ϕ j ,
M 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; f j ,
¯ 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; ¯f j , M
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J. Su et al. / Physics Letters A 374 (2010) 1457–1463
˜ u˜ , ϕ˜ , . . . , ϕ˜ ˜ ˜ ˜ h¯ is the complex conjugate of h, and Φ, n+ N , k = 1, 2, satisfy Eq. (2.6) with n replaced by n + N, thus u , ϕ1 , . . . , ϕn+ N , k = 1, 2, is a 1 solution of Eq. (2.4) with n replaced by n + N. (k)
(k)
(k)
(k)
Proof. It can be verified by a similar argument to the one in [9,15,25] that
Φ˜ =
φ˜ 1 φ˜ 2
u˜ ,
,
ϕ˜ j =
ϕ˜ 1 j = ϕ˜ 1(1j) + i ϕ˜ 1(2j) ,
(k)
ϕ˜ j =
ϕ˜ 2 j = ϕ˜ 2(1j) + i ϕ˜ 2(2j) , 1 j n + N ,
(k)
ϕ˜ 1 j
ϕ˜n(k+) j
1 j n,
,
ϕ˜ 2(kj)
ϕ˜ 1 j , ϕ˜ 2 j
=
ϕ˜ 1(k,n) + j ϕ˜ 2(k,n) + j
,
1 j N , k = 1, 2,
present a GBDT for the mKdVESCSs (2.4) from n sources to n + N sources. Therefore, what we need to prove is that (1 )
(2 )
(1 )
(2 )
u˜ , ϕ˜ 1 j , ϕ˜ 1 j , ϕ˜ 2 j , ϕ˜ 2 j ,
ϕ˜ 1(1,n)+ j , ϕ˜ 1(2,n)+ j , ϕ˜ 2(2,n)+ j , ϕ˜ 2(2,n)+ j , 1 j N ,
1 j n, (1)
(2)
(1)
(2)
must be all real, provided that u , ϕ1 j , ϕ1 j , ϕ2 j , ϕ2 j , 1 j n, are real. This can be shown by noting that M 0 is always real,
W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; h
is real for any real function h. These facts are consequences of the basic properties of determinants. This completes the proof of Theorem 2.2. 2 Obviously, the N-times repeated GBDT defined by Eqs. (2.7) contains an arbitrary t-function and provides a non-auto-Bäcklund transformation between the two mKdVESCSs of degrees n and n + N. The flexibility of the choices of e j (t ) and f j , 1 j n, enables us to construct complexiton solutions to the mKdVESCSs (2.1). 3. Complexiton solutions In this section, we study the complexiton solutions for the mKdVESCSs (2.1) by means of GBDT which have been given in Section 2. We take u = 0 as the initial solution of Eq. (2.1) with n = 0. Let α j and β j = 0, 1 j N, be real constants, the spectral parameter λ j , and λ j = α j + i β j , 1 j N, be distinct complex numbers. Suppose that f j ( j = 1, . . . , N) are different solutions of the system (2.6) with the corresponding spectral parameters λ = λ j ( j = 1, . . . , N), respectively. Based on GBDT (2.8), then the mKdVESCSs (2.1) has the solution in the following form with N arbitrary t-function e 1 (t ), . . . , e N (t ),
u=
M2
(3.1a)
,
M0
e j (t ) M 4
ϕ (j1) =
e j (t ) M 0
+
e j (t ) M 4
(2 )
ϕj =
ie j (t ) M 0
¯4 e¯ j (t ) M
e¯ j (t ) M 0
,
1 j N,
(3.1b)
,
1 j N,
(3.1c)
−
¯4 e¯ j (t ) M
i e¯ j (t ) M 0
where
M 0 = W 0 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N } ,
M 2 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; 0 ,
M 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; f j ,
¯ 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; ¯f j , M where h¯ is the complex conjugate of h. We call the above resulting solutions complexiton solutions. More specifically, the solutions defined by (3.1) are called N-complexiton solutions. Now, let us concentrate on the case of N = 1. We construct the real-valued 1-complexiton solution for the mKdVESCSs through the GBDT (2.7), beginning with the zero solution. Taking u = 0 as the initial solution with n = 0 and solving the Lax pair (2.6) with the spectral parameter λ, one can yield
f =
f1 f2
=
c 1 e −λx+4λ t 3 c 2 e λx−4λ t 3
(3.2a)
,
where c 1 and c 2 are two arbitrary constants. And with the spectral parameter −λ,
Sf =
f2 f1
=
c 2 e λx−4λ t 3 c 1 e −λx+4λ t 3
,
we take the simple and special choice of e (t ) as
(3.2b)
J. Su et al. / Physics Letters A 374 (2010) 1457–1463
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Fig. 1. 1-complexiton solution of the mKdVESCSs with c 1 = 2, c 2 = 2, a = 1, b = 0, λ = i (a) t = −0.5; (b) t = 0; (c) t = 0.5.
e (t ) = at + b,
(3.3)
where a = 0 and b are real constants. Fixing the spectral parameter λ as a complex number in (3.2), one can easily obtain the complexiton solutions for the modified KdV equation with self-consistent sources from the generalized binary Darboux transformation. To show a detailed structure of the complexiton, we select the parameters as
c 1 = 2,
c 2 = 2,
a = 1,
b = 0,
λ = i.
(3.4)
Then from the GBDT (2.7) with u = 0 and n = 0, the 1-complexiton solution for the mKdVESCSs with N = 1 is given by
u=
4t cos(2θ) − 4 exp(−θ 2 ) sin(2θ) − 4 sin(θ 3 − θ)
ϕ˜ 1(1) =
exp(θ 4 ) − exp(−2θ 2 )
,
(2 exp(θ 4 ) − 4 exp(−2θ 2 )) cos(θ) + 4t exp(−θ 2 ) sin(θ) + 4 exp(−θ 2 ) cos(θ 3 ) + 4t sin(θ 3 ) − 2 cos(θ 5 ) , t (exp(θ 4 ) − exp(−2θ 2 ))
(3.5)
(3.6a)
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J. Su et al. / Physics Letters A 374 (2010) 1457–1463
Fig. 2. The shape and motion of the 1-complexiton solution for the mKdVESCSs with c 1 = 2, c 2 = 2, a = 1, b = 0, λ = i.
ϕ˜ 1(2) =
4t exp(−θ 2 ) cos(θ) + (4 exp(−2θ 2 ) − 6 exp(θ 4 )) sin(θ) − 4t cos(θ 3 ) − 4 exp(−θ 2 ) sin(θ 3 ) − 2 sin(θ 5 ) t (exp(θ 4 ) − exp(−2θ 2 ))
,
(3.6b)
where θ = x + 4t. Based on those specific forms of the functions f 1 , f 2 , e (t ) and λ, we can see that the resulting 1-complexiton solution of the mKdVESCSs possesses moving singularities like positons and negatons, but different type of singularities. Moreover, they can be described as localized wavepacket whose envelopes and oscillatory components move at different speeds. The graphs of the real-valued 1-complexiton solution of the mKdVESCSs are given as above. Fig. 1 shows the structure plot of 1-complexiton solution for the mKdVESCSs expressed by (3.5) at the fixed time t = −0.5, t = 0 and t = 0.5, respectively. The evolution plot of 1-complexiton solution (3.5) is depicted in Fig. 2. From Figs. 1 and 2, we know that the real-valued 1-complexiton solution of the mKdVESCSs is just a singular solution. In terms of the property of the eigenfunction involved, the solution is different from soliton, positon and negaton solutions, and their interaction solutions. 4. Discussion and conclusion In this Letter, we present the N-times repeated generalized binary Darboux transformation with N arbitrary t-functions which provides non-auto-Bäcklund transformation between two mKdVESCSs with n-degrees of sources and n + N-degrees of sources. This N-times repeated GBDT enables us to construct a class of complexiton solutions of the mKdVESCSs. The resulting complexiton solutions also provide evidence that the mKdVESCSs can have complexiton solutions, in addition to soliton, positon and negaton solutions. Taking the special initial seed solution for auxiliary linear problems and the special functions of time t, the real-valued 1-complexiton solution of the mKdVESCSs is considered through the GBDT by selecting the complex spectral parameters in its Lax pair. It is important to point out that the real-valued 1-complexiton solution for the mKdVESCSs is analytical and singular. The detailed structures of the 1-complexiton solution are given out analytically and graphically. As we know, various other types of exact solutions for the mKdVESCSs, such as N-soliton solution, positon solutions and negaton solutions have been obtained by different approaches in Refs. [7,10,19]. But the complexiton solutions of the mKdVESCSs are firstly given out in this Letter. We believe that the complexiton solutions of the mKdVESCSs would help us learn the internal property of the soliton equations with self-consistent sources deeply. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 10872165) and Scientific and Technological Innovation Foundation of Northwestern Polytechnical University (Grant No. 2008KJ02034).
J. Su et al. / Physics Letters A 374 (2010) 1457–1463
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Physics Letters A www.elsevier.com/locate/pla
Complexiton solutions of the mKdV equation with self-consistent sources Jun Su ∗ , Wei Xu, Liang Gao, Genjiu Xu Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 9 December 2009 Received in revised form 17 January 2010 Accepted 21 January 2010 Available online 28 January 2010 Communicated by A.R. Bishop Keywords: Complexitons Self-consistent sources mKdV equation Darboux transformation
a b s t r a c t A class of complexiton solutions of the mKdV equation with self-consistent sources (mKdVESCSs) are presented by the generalized binary Darboux transformation (GBDT) with N arbitrary t-functions. Taking the special initial seed solution for auxiliary linear problems and the special functions of time t, the real-valued 1-complexiton solution of the mKdVESCSs is considered through the GBDT by selecting the complex spectral parameters in its Lax pair. It is important to point out that the real-valued 1-complexiton solution of the mKdVESCSs is analytical and singular. Moreover, the detailed structures of the 1-complexiton solution are given out analytically and graphically. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The soliton equations with self-consistent sources (SESCSs) have received considerable attention in recent years. It is well known that these equations can exhibit richer nonlinear dynamics than soliton equations and have important physical applications in plasma physics, hydrodynamics, solid state physics, etc. [1–5]. In terms of finding exact solutions, some methods for solving soliton equations have been used to SESCSs as well. For example, through the inverse scattering method, Zeng et al. [6] have obtained the explicit soliton solutions for the KdV, AKNS, mKdV, nonlinear Schrödinger and Kaup–Newell hierarchies with self-consistent sources. In addition to that, the N-soliton solutions for the mKdVESCSs was obtained by Hirota method and Wronskian technique [7]. The Darboux transformation (DT) [8] is one of the most effective tools to construct the exact and explicit solutions of the nonlinear evolution equations. However, the normal DT cannot be directly used to construct the nontrivial solutions for SESCSs. A generalized binary Darboux transformation (GBDT) for the KdV hierarchies and mKdV hierarchy with self-consistent sources have been proposed [9,10]. With an arbitrary function at time t, the GBDT offers a non-auto-Bäcklund transformation between two SESCSs with different degrees of sources. Recently, a kind of classification of exact and explicit solutions of integrable systems was proposed according to the property of associated spectral parameters [11]. In general, positons [12], usually related to the positive spectral parameters, are expressed by the trigonometric functions. Negatons of nonlinear systems [13], usually related to the negative spectral parameters, are expressed by the hyperbolic functions. Solitons are a kind of non-singular negatons. The study of the solitons, negatons, positons and their interaction solutions has been made for the KdV equation in Refs. [11,14]. Moreover, for the SESCSs, the solitons, negatons and positons are also obtained including the KdV, Schrödinger, KP, NLS, mKdV with sources [15–19]. The complexiton solutions [20], which are a novel class of explicit exact solutions to the soliton equations, are related to the complex spectral parameters and are expressed by combinations of the trigonometric functions and the exponential functions. Later, the concepts of complexitons were extended to the Toda lattice equation, the Boussinesq equation, the mKdV equation, etc. [21–24]. Whether the soliton equations with self-consistent sources can have complexiton solutions, Ma provided a positive answer by considering the KdV equation with self-consistent sources [25]. However, not much work has been done on the complexiton solutions of the other SESCSs. In this Letter, we will construct the complexiton solutions of mKdVESCSs. The study of solutions will certainly enrich the theory of the mKdVESCSs. The Letter is organized as follows. In Section 2, we derive the GBDT for the mKdVESCSs. In Section 3, the complexiton solutions of the mKdVESCSs are obtained. Section 4 presents the summary and discussions.
*
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2. The generalized binary Darboux transformation for the mKdVESCSs The negative mKdVESCSs is defined as [19]
ut − 6u 2 u x + u xxx −
n
ϕ12 j + ϕ22 j = 0,
(2.1a)
j =1
ϕ1 j,x = −λ j ϕ1 j + u ϕ2 j ,
ϕ2 j,x = u ϕ1 j + λ j ϕ2 j ,
j = 1, 2, . . . , n ,
(2.1b)
where λ j are distinct constants, and ϕ j = (ϕ1 j , ϕ2 j ) , j = 1, 2, . . . , n. It is associated with the discrete spectrum of the Schrödinger operator and represents the interaction of high-frequency waves ϕ j ( j = 1, 2, . . . , n), with a single, low-frequency wave u. In order to show the existence of complexiton solutions for the mKdVESCSs, we consider the mKdV equation with 2n specific complex self-consistent sources: T
ut − 6u 2 u x + u xxx −
n
ϕ12 j + ϕ22 j + ϕ¯ 12 j + ϕ¯ 22 j = 0,
(2.2a)
j =1
ϕ1 j,x = −λ j ϕ1 j + u ϕ2 j ,
ϕ2 j,x = u ϕ1 j + λ j ϕ2 j ,
j = 1, 2, . . . , n ,
(2.2b)
ϕ¯ 1 j,x = −λ¯ j ϕ¯ 1 j + u ϕ¯ 2 j ,
ϕ¯ 2 j,x = u ϕ¯ 1 j + λ¯ j ϕ¯ 2 j ,
j = 1, 2, . . . , n ,
(2.2c)
¯ j , ϕ¯ 1 j and ϕ¯ 2 j are complex conjugates of λ j , ϕ1 j and ϕ2 j , respectively. Moreover, we will focus on the case of complex eigenvalues where λ λ j and adjoint eigenfunctions ϕ j with only real potential u. The introduction of adjoint eigensystems was proposed for the first time in Ref. [26]. Let
ϕ1 j = ϕ1(1j) + i ϕ1(2j) ,
λ j = α j + iβ j , where
ϕ2 j = ϕ2(1j) + i ϕ2(2j) ,
j = 1, 2, . . . , n ,
(2.3)
α j , β j = 0, ϕ1(1j) , ϕ1(2j) , ϕ2(1j) and ϕ2(2j) are all real. Therefore, the negative mKdVESCSs (2.2) is equivalent to the following real system: ut − 6u 2 u x + u xxx − 2
n
2
ϕ1(1j) + ϕ2(1j)
2
(2) 2 (2) 2 − ϕ1 j + ϕ2 j = 0,
(2.4a)
j =1
ϕ1(1j),x + α j ϕ1(1j) − β j ϕ1(2j) − u ϕ2(1j) = 0,
ϕ1(2j),x + α j ϕ1(2j) + β j ϕ1(1j) − u ϕ2(2j) = 0,
j = 1, 2, . . . , n ,
(2.4b)
ϕ2(1j),x − α j ϕ2(1j) + β j ϕ2(2j) − u ϕ1(1j) = 0,
ϕ2(2j),x − α j ϕ2(2j) − β j ϕ2(1j) − u ϕ1(2j) = 0,
j = 1, 2, . . . , n ,
(2.4c)
where α j , β j = 0 are arbitrary real constants, and u, First, we define a linear map
S:
ϕ1 j ϕ2 j
→
ϕ1(1j) , ϕ1(2j) , ϕ2(1j) and ϕ2(2j) are real independent variables.
ϕ2 j . ϕ1 j
(2.5)
The corresponding Lax pair for the mKdVESCSs (2.4) is shown as below
Φx =
−λ u Φ, u λ
Φt = V (λ, u )Φ +
n H (ϕ j ) j =1
where
V (λ, u ) = H (ϕ j ) =
φ
1 2
(2.6a)
λ − λj
H(Sϕ j)
+
λ + λj
4 λ3 − 2 λ u 2 2 −4λ u − 2λu x − u xx + 2u 3
ϕ12 j −ϕ1 j ϕ2 j 2 −ϕ1 j ϕ1 j ϕ2 j
Φ+
n H (ϕ¯ j ) j =1
λ − λ¯ j
+
H ( S ϕ¯ j )
λ + λ¯ j
−4λ2 u + 2λu x − u xx + 2u 3 −4λ3 + 2λu 2
Φ,
(2.6b)
,
(2.6c)
(2.6d)
,
and Φ = φ 1 is the solution of (2.6) corresponding to the spectral parameter λ. 2 In order to obtain the generalized binary Darboux transformation with an arbitrary function of t for the mKdVESCSs, some symmetric forms are defined. Suppose that f j =
f1 j
f2 j
( j = 1, . . . , N) are different solutions of the system (2.6) with the corresponding spectral
parameters λ = λ j ( j = 1, . . . , N), respectively, φ is a scalar and a group of N arbitrary t-functions e 1 (t ), . . . , e N (t ), h = W 2 are defined as follows:
W 0 {e 1 , f 1 }, . . . , {e N , f N } = det F , (i )
W1
{e 1 , f 1 }, . . . , {e N , f N }; h = det
F
αi
b hi
,
i = 1, 2,
h1 h2
, then W 0 , W 1 ,
J. Su et al. / Physics Letters A 374 (2010) 1457–1463
W 1 {e 1 , f 1 }, . . . , {e N , f N }; h =
(1 )
W 1 ({e 1 , f 1 }, . . . , {e N , f N }; h1 )
(2 )
W 1 ({e 1 , f 1 }, . . . , {e N , f N }; h2 )
W 2 {e 1 , f 1 }, . . . , {e N , f N }; φ = det
F
α1T
α1
φ
1459
,
,
where
F = δi j e i + σ ( f i , f j )
σ ( f i , f j ) := −
N ×N
W ( fi, f j) 2(λi − λ j )
b=
,
,
T
αi = ( f i1 , . . . , f iN ),
σ ( f 1 , h), . . . , σ ( f N , h) , 1
σ ( f i , f i ) := W ( f i , ∂λi f i ), 2
W ( f i , f j ) = f 1i f 2 j − f 2i f 1 j .
Now, we have the following formula of the GBDT for the mKdVESCSs (2.4). (k)
Theorem 2.1. Let (u , ϕ1 =
(k) ϕ11
(k)
, . . . , ϕn =
(k)
ϕ21
(k) ϕ1n
), k = 1, 2, be a solution of Eq. (2.4), λ = α + i β be a complex number, e (t ) be an arbitrary
(k) ϕ2n
t-function and f be a solution of the system (2.6) with λ = λn+1 , then the GBDT with an arbitrary t-function for the mKdVESCSs (2.4) defined by
Φ˜ =
W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; Φ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
W 2 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; 0)
u˜ = u +
ϕ˜ (jk) =
(2.7a)
,
W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
(2.7b)
,
(k) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; ϕ j )
W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
1 j n , k = 1, 2,
,
ϕ˜n(1+)1
e j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f )
=
e j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
ϕ˜n(2+)1
e j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f )
=
ie j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
(2.7c)
+
e¯ j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f )
−
e¯ j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
e¯ j (t ) W 1 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f }; f ) i e¯ j (t ) W 0 ({e , f }, {e , S f }, {¯e , ¯f }, {¯e , S ¯f })
,
(2.7d)
,
(2.7e)
˜ u˜ , ϕ˜ , . . . , ϕ˜ , k = 1, 2, satisfy Eq. (2.6) with n replaced by n + 1, thus u˜ , ϕ˜ , . . . , ϕ˜ , k = 1, 2, where h¯ is the complex conjugate of h, and Φ, 1 n+1 1 n+1 is a solution of Eq. (2.4) with n replaced by n + 1. (k)
(k)
(k)
(k)
The N-times repeated GBDT with N arbitrary t-functions for the mKdVESCSs (2.4) is given by the following theorem. (k)
Theorem 2.2. Let (u , ϕ1 =
(k) ϕ11 (k)
ϕ21
(k)
, . . . , ϕn =
(k) ϕ1n (k) ϕ2n
), k = 1, 2, be a solution of Eq. (2.4), λ j = α j + i β j , 1 j n + N, be distinct complex numbers,
e j (t ) be an arbitrary t-function and f j be a solution of the system (2.6) with λ = λn+ j , j = 1, . . . , N, then the N-times repeated GBDT with N arbitrary t-functions for the mKdVESCSs (2.4) defined by
Φ˜ =
M1 M0
u˜ = u +
ϕ˜ (jk) = ϕ˜n(1+) j = ϕ˜n(1+) j
=
(2.8a)
, M2
M0 M3 M0
,
(2.8b)
, 1 j n , k = 1, 2,
e j (t ) M 4
e j (t ) M 0
+
e j (t ) M 4
ie j (t ) M 0
(2.8c)
¯4 e¯ j (t ) M
e¯ j (t ) M 0
,
1 j N,
(2.8d)
,
1 j N,
(2.8e)
−
¯4 e¯ j (t ) M
i e¯ j (t ) M 0
where
M 0 = W 0 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N } ,
M 1 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; Φ ,
M 2 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; 0 ,
(k) M 3 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; ϕ j ,
M 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; f j ,
¯ 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; ¯f j , M
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J. Su et al. / Physics Letters A 374 (2010) 1457–1463
˜ u˜ , ϕ˜ , . . . , ϕ˜ ˜ ˜ ˜ h¯ is the complex conjugate of h, and Φ, n+ N , k = 1, 2, satisfy Eq. (2.6) with n replaced by n + N, thus u , ϕ1 , . . . , ϕn+ N , k = 1, 2, is a 1 solution of Eq. (2.4) with n replaced by n + N. (k)
(k)
(k)
(k)
Proof. It can be verified by a similar argument to the one in [9,15,25] that
Φ˜ =
φ˜ 1 φ˜ 2
u˜ ,
,
ϕ˜ j =
ϕ˜ 1 j = ϕ˜ 1(1j) + i ϕ˜ 1(2j) ,
(k)
ϕ˜ j =
ϕ˜ 2 j = ϕ˜ 2(1j) + i ϕ˜ 2(2j) , 1 j n + N ,
(k)
ϕ˜ 1 j
ϕ˜n(k+) j
1 j n,
,
ϕ˜ 2(kj)
ϕ˜ 1 j , ϕ˜ 2 j
=
ϕ˜ 1(k,n) + j ϕ˜ 2(k,n) + j
,
1 j N , k = 1, 2,
present a GBDT for the mKdVESCSs (2.4) from n sources to n + N sources. Therefore, what we need to prove is that (1 )
(2 )
(1 )
(2 )
u˜ , ϕ˜ 1 j , ϕ˜ 1 j , ϕ˜ 2 j , ϕ˜ 2 j ,
ϕ˜ 1(1,n)+ j , ϕ˜ 1(2,n)+ j , ϕ˜ 2(2,n)+ j , ϕ˜ 2(2,n)+ j , 1 j N ,
1 j n, (1)
(2)
(1)
(2)
must be all real, provided that u , ϕ1 j , ϕ1 j , ϕ2 j , ϕ2 j , 1 j n, are real. This can be shown by noting that M 0 is always real,
W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; h
is real for any real function h. These facts are consequences of the basic properties of determinants. This completes the proof of Theorem 2.2. 2 Obviously, the N-times repeated GBDT defined by Eqs. (2.7) contains an arbitrary t-function and provides a non-auto-Bäcklund transformation between the two mKdVESCSs of degrees n and n + N. The flexibility of the choices of e j (t ) and f j , 1 j n, enables us to construct complexiton solutions to the mKdVESCSs (2.1). 3. Complexiton solutions In this section, we study the complexiton solutions for the mKdVESCSs (2.1) by means of GBDT which have been given in Section 2. We take u = 0 as the initial solution of Eq. (2.1) with n = 0. Let α j and β j = 0, 1 j N, be real constants, the spectral parameter λ j , and λ j = α j + i β j , 1 j N, be distinct complex numbers. Suppose that f j ( j = 1, . . . , N) are different solutions of the system (2.6) with the corresponding spectral parameters λ = λ j ( j = 1, . . . , N), respectively. Based on GBDT (2.8), then the mKdVESCSs (2.1) has the solution in the following form with N arbitrary t-function e 1 (t ), . . . , e N (t ),
u=
M2
(3.1a)
,
M0
e j (t ) M 4
ϕ (j1) =
e j (t ) M 0
+
e j (t ) M 4
(2 )
ϕj =
ie j (t ) M 0
¯4 e¯ j (t ) M
e¯ j (t ) M 0
,
1 j N,
(3.1b)
,
1 j N,
(3.1c)
−
¯4 e¯ j (t ) M
i e¯ j (t ) M 0
where
M 0 = W 0 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N } ,
M 2 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; 0 ,
M 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; f j ,
¯ 4 = W 1 {e 1 , f 1 }, {e 1 , S f 1 }, . . . , {e N , f N }, {e N , S f N }, {¯e 1 , ¯f 1 }, {¯e 1 , S ¯f 1 }, . . . , {¯e N , ¯f N }, {¯e N , S ¯f N }; ¯f j , M where h¯ is the complex conjugate of h. We call the above resulting solutions complexiton solutions. More specifically, the solutions defined by (3.1) are called N-complexiton solutions. Now, let us concentrate on the case of N = 1. We construct the real-valued 1-complexiton solution for the mKdVESCSs through the GBDT (2.7), beginning with the zero solution. Taking u = 0 as the initial solution with n = 0 and solving the Lax pair (2.6) with the spectral parameter λ, one can yield
f =
f1 f2
=
c 1 e −λx+4λ t 3 c 2 e λx−4λ t 3
(3.2a)
,
where c 1 and c 2 are two arbitrary constants. And with the spectral parameter −λ,
Sf =
f2 f1
=
c 2 e λx−4λ t 3 c 1 e −λx+4λ t 3
,
we take the simple and special choice of e (t ) as
(3.2b)
J. Su et al. / Physics Letters A 374 (2010) 1457–1463
1461
Fig. 1. 1-complexiton solution of the mKdVESCSs with c 1 = 2, c 2 = 2, a = 1, b = 0, λ = i (a) t = −0.5; (b) t = 0; (c) t = 0.5.
e (t ) = at + b,
(3.3)
where a = 0 and b are real constants. Fixing the spectral parameter λ as a complex number in (3.2), one can easily obtain the complexiton solutions for the modified KdV equation with self-consistent sources from the generalized binary Darboux transformation. To show a detailed structure of the complexiton, we select the parameters as
c 1 = 2,
c 2 = 2,
a = 1,
b = 0,
λ = i.
(3.4)
Then from the GBDT (2.7) with u = 0 and n = 0, the 1-complexiton solution for the mKdVESCSs with N = 1 is given by
u=
4t cos(2θ) − 4 exp(−θ 2 ) sin(2θ) − 4 sin(θ 3 − θ)
ϕ˜ 1(1) =
exp(θ 4 ) − exp(−2θ 2 )
,
(2 exp(θ 4 ) − 4 exp(−2θ 2 )) cos(θ) + 4t exp(−θ 2 ) sin(θ) + 4 exp(−θ 2 ) cos(θ 3 ) + 4t sin(θ 3 ) − 2 cos(θ 5 ) , t (exp(θ 4 ) − exp(−2θ 2 ))
(3.5)
(3.6a)
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J. Su et al. / Physics Letters A 374 (2010) 1457–1463
Fig. 2. The shape and motion of the 1-complexiton solution for the mKdVESCSs with c 1 = 2, c 2 = 2, a = 1, b = 0, λ = i.
ϕ˜ 1(2) =
4t exp(−θ 2 ) cos(θ) + (4 exp(−2θ 2 ) − 6 exp(θ 4 )) sin(θ) − 4t cos(θ 3 ) − 4 exp(−θ 2 ) sin(θ 3 ) − 2 sin(θ 5 ) t (exp(θ 4 ) − exp(−2θ 2 ))
,
(3.6b)
where θ = x + 4t. Based on those specific forms of the functions f 1 , f 2 , e (t ) and λ, we can see that the resulting 1-complexiton solution of the mKdVESCSs possesses moving singularities like positons and negatons, but different type of singularities. Moreover, they can be described as localized wavepacket whose envelopes and oscillatory components move at different speeds. The graphs of the real-valued 1-complexiton solution of the mKdVESCSs are given as above. Fig. 1 shows the structure plot of 1-complexiton solution for the mKdVESCSs expressed by (3.5) at the fixed time t = −0.5, t = 0 and t = 0.5, respectively. The evolution plot of 1-complexiton solution (3.5) is depicted in Fig. 2. From Figs. 1 and 2, we know that the real-valued 1-complexiton solution of the mKdVESCSs is just a singular solution. In terms of the property of the eigenfunction involved, the solution is different from soliton, positon and negaton solutions, and their interaction solutions. 4. Discussion and conclusion In this Letter, we present the N-times repeated generalized binary Darboux transformation with N arbitrary t-functions which provides non-auto-Bäcklund transformation between two mKdVESCSs with n-degrees of sources and n + N-degrees of sources. This N-times repeated GBDT enables us to construct a class of complexiton solutions of the mKdVESCSs. The resulting complexiton solutions also provide evidence that the mKdVESCSs can have complexiton solutions, in addition to soliton, positon and negaton solutions. Taking the special initial seed solution for auxiliary linear problems and the special functions of time t, the real-valued 1-complexiton solution of the mKdVESCSs is considered through the GBDT by selecting the complex spectral parameters in its Lax pair. It is important to point out that the real-valued 1-complexiton solution for the mKdVESCSs is analytical and singular. The detailed structures of the 1-complexiton solution are given out analytically and graphically. As we know, various other types of exact solutions for the mKdVESCSs, such as N-soliton solution, positon solutions and negaton solutions have been obtained by different approaches in Refs. [7,10,19]. But the complexiton solutions of the mKdVESCSs are firstly given out in this Letter. We believe that the complexiton solutions of the mKdVESCSs would help us learn the internal property of the soliton equations with self-consistent sources deeply. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 10872165) and Scientific and Technological Innovation Foundation of Northwestern Polytechnical University (Grant No. 2008KJ02034).
J. Su et al. / Physics Letters A 374 (2010) 1457–1463
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