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Negaton, positon and complexiton solutions of the nonisospectral KdV equations with self-consistent sources
Commun Nonlinear Sci Numer Simulat 17 (2012) 110–118
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Negaton, positon and complexiton solutions of the nonisospectral KdV equations with self-consistent sources Jun Su a,⇑, Wei Xu a, Genjiu Xu a, Liang Gao b a b
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an 710072, China Science Research Institute of China – North Group Company, Beijing 100089, China
a r t i c l e
i n f o
Article history: Received 27 August 2010 Received in revised form 6 January 2011 Accepted 10 April 2011 Available online 23 April 2011 Keywords: Nonisospectral KdV equation Negaton Positon Complexiton Darboux transformation Self-consistent sources
a b s t r a c t The negaton, positon, and complexiton solutions of the nonisospectral KdV equations with self-consistent sources (KdVESCSs) are obtained by the generalized binary Darboux transformation (GBDT) with N arbitrary t-functions. Taking the special initial seed solution for auxiliary linear problems, the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs are considered through the GBDT by selecting the negative, positive and complex spectral parameters. It is important to point out that these solutions of the nonisospectral KdVESCSs are analytical and singular. We also show differences between these solutions with singularities. Moreover, the detailed characteristics of these solutions with nonisospectral properties and sources effects are described through some figures. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The nonisospectral evolution equations are used to describe deep water and plasma waves, waves in nonuniform media, in which the spectral parameter depends on the time or space variables [1–11]. In terms of finding exact solutions, some methods have been used to these kind of equations, such as the inverse scattering method [2–5], the Hirota method and the Wronskian technique [1,6–11], and the Darboux transformation [12]. In general, when the spectral parameter k follows kt = kj where j = 0 or 1, the corresponding nonisospectral equations are exactly integrable and mathematically trivial since there exists a gauge transformations between them and their isospectral counterparts; while when j > 1, such transformation cannot be found [12,13]. The soliton equations with self-consistent sources (SESCSs) have also 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. [14–17]. Recently, the SESCSs were investigated by means of the inverse scattering method, the Hirota method and the Wronskian technique, the generalized binary Darboux transformation (GBDT), etc. [18–26]. As is well known, a new classification of exact and explicit solutions of integrable models was proposed according to the property of associated spectral parameters [27–29]. Negaton, related to the negative spectral parameter, was usually a singular or nonsingular reduced two-solitons. Positon, related to the positive spectral parameter, was usually a long-range, slowly decreasing and oscillating singular soliton-like solution. The so-called complexiton, which is related to the complex spectral parameters, was usually expressed by the combinations of trigonometric functions and hyperbolic functions. Some ⇑ Corresponding author. Tel.: +86 29 88460633; fax: +86 29 88495453. E-mail address: [email protected] (J. Su). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.019
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negatons, positons, and complexitons are obtained for the soliton equations with self-consistent sources by the GBDT [22– 26]. In this paper, we investigate the nonisospectral KdV equations with self-consistent sources (KdVESCSs). The N-soliton solutions for the hierarchy of the nonisospectral KdVESCSs has been obtained through the inverse scattering transform [5]. Whether the nonisospectral KdVESCSs have negaton, positon and complexiton solutions or not, like isospectral KdVESCSs? This paper aims to provide a positive answer to this question. Through the GBDT and the gauge transformation, we obtain the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs. We hope that the study of these solutions will enrich the theory of the nonisospectral KdVESCSs. The paper is organized as follows. In Section 2, the nonisospectral KdVESCSs and its gauge transformation are given. In Section 3, the GBDT for the isospectral KdVESCSs is constructed. In Section 4, the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs are obtained. Section 5 presents the conclusions. 2. The nonisospectral KdVESCSs and gauge transformation The nonisospectral KdVESCSs is written as
ut þ 6uux þ uxxx þ 2au þ axux ux gðtÞ þ 4
n X
uj uj;x ¼ 0;
ð2:1aÞ
j¼1
uj;xx þ ðkj þ uÞuj ¼ 0; ðj ¼ 1; 2; . . . ; nÞ;
ð2:1bÞ 2at
where a is a real constant and we assume that u tends to 0 as x ? ±1. Taking n = 0, g(t) = c0 + 6be , and c0, b are real constants, the Eq. (2.1) is one of the so-called x-coefficient equations which can describe nonlinear soliton waves in a certain type of non-uniform media with a relaxation effect [1,2]. In [6], the solitons, rational, negatons and positons solutions for this equation were obtained through the Wronskian technique. When the spectral parameter k follows
kt ¼ 2ak:
ð2:2Þ
We can see that the Eq. (2.1) is a nonisospectral soliton equation with self-consistent sources in terms of k. The corresponding Lax pair for the nonisospectral KdVESCSs (2.1) is shown as below
/xx þ ðk þ uÞ/ ¼ 0;
ð2:3aÞ
/t ¼ A/ þ B/x ;
ð2:3bÞ
where
A ¼ ux
n X uj uj;x ; kj k j¼1
B ¼ 4k 2u ax þ gðtÞ þ
n X u2j j¼1
kj k
;
ð2:3cÞ
As kt = 2ak, there exists a gauge transformation connecting the nonisospectral KdVESCSs (2.1) with the isospectral KdVESCSs. The transformation is for both equations and its Lax pairs. We describe it by the following theorem. Theorem 2.1. When kt = 2ak, by the transformation
V ¼ e2at u;
X ¼ eat x þ
Z
t
gðzÞeaz dz;
T¼
1
e3at 1 ; 3a
c ¼ e2at k; w ¼ e2at u;
cj ¼ e2at kj ; wj ¼ e2at uj ; ðj ¼ 1; 2; . . . ; nÞ;
ð2:4aÞ ð2:4bÞ
the nonisospectral KdVESCSs (2.1) can be transformed into the isospectral KdVESCSs
V T þ 6VV X þ V XXX þ 4
n X
wj wj;x ¼ 0;
ð2:5aÞ
j¼1
wj;XX þ ðcj þ VÞwj ¼ 0;
ðj ¼ 1; 2; . . . ; nÞ;
ð2:5bÞ
and the Lax pair (2.3) of the nonisospectral KdVESCSs (2.1) is also transformed to the Lax pair of the isospectral one
UXX þ ðc þ VÞU ¼ 0; UT ¼ C U þ DUx ;
ð2:6aÞ ð2:6bÞ
where
C ¼ Vx
n X wj wj;x ; c j c j¼1
D ¼ 4g 2V þ
n X j¼1
w2j : cj c
ð2:6cÞ
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Proof. Taking Eqs. (2.4) into the nonisospectral KdVESCSs (2.1) leads to the Eq. (2.5). Similarly, we can check that (2.6) holds. This completes the proof. h
3. The GBDT for the isospectral KdVESCSs In this section, we will consider the generalized binary Darboux transformation (GBDT) with an arbitrary function of T for the isospectral KdVESCSs (2.5), which offers a non-auto-Bäcklund transformation between two isospectral KdVSESCSs with different degrees of sources. With the Lax pair (2.6), the GBDT for the isospectral KdVESCSs has been verified in Ref. [22]. Here we only briefly write the final results and construct the negaton, positon and complexiton solutions for the nonisospectral KdVESCSs (2.1) based on the given Darboux transformation. Assume that f and g are two solutions of Eq. (2.6) corresponding to c = n and c = g, respectively. Define
xðf ; gÞ ¼
Wðf ; gÞ ; ng
xðf ; f Þ ¼ lim g!n
Wðf ðnÞ; f ðgÞÞ ; ng
ð3:1Þ
where W(f, g) denotes the Wronskian determinant W(f, g) = fgx fxg. Then, the first step GBDT with an arbitrary function of T for the isospectral KdVESCSs (2.5) is defined by
U½1 ¼ U
f xðf ; UÞ ; eðTÞ þ xðf ; f Þ
ð3:2aÞ
V½1 ¼ V þ 2@ 2x ln½eðTÞ þ xðf ; f Þ; f xðf ; wj Þ wj ½1 ¼ wj ; eðTÞ þ xðf ; f Þ pffiffiffiffiffiffiffiffiffiffi f e0 ðTÞ ; wnþ1 ½1 ¼ eðTÞ þ xðf ; f Þ
ð3:2bÞ
ðj ¼ 1; 2; . . . ; nÞ;
ð3:2cÞ ð3:2dÞ
where e(T) is an arbitrary function of T, and U[1], V[1], w1[1], . . ., wn+1[1] satisfy the Eq. (2.6) with n replaced by n + 1, thus V[1], w1[1], . . ., wn+1[1] is a solution of the equation (2.5) with n replaced by n + 1. For m solutions g1, . . ., gm of Eq. (2.6) and m arbitrary T-function e1(T), . . ., em(T), we define two determinants
W 1 ðg 1 ; . . . ; g m ; e1 ; . . . ; em Þ ¼ det F;
W 2 ðg 1 ; . . . ; g m ; e1 ; . . . ; em1 Þ ¼ det G;
ð3:3Þ
where
F ij ¼ dij ei ðTÞ þ xðg i ; g j Þ;
ði; j ¼ 1; . . . ; mÞ;
Gij ¼ dij ei ðTÞ þ xðg i ; g j Þ;
ði ¼ 1; . . . ; m 1;
Gmj ¼ g j ;
j ¼ 1; . . . ; mÞ;
ðj ¼ 1; . . . ; mÞ:
Then the N-times repeated GBDT with N arbitrary T-functions for the isospectral KdVESCSs (2.5) is given by
W 2 ðf1 ; . . . ; fN ; U; e1 ; . . . ; eN Þ ; W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ
ð3:4aÞ
V½N ¼ V þ 2@ 2x ln W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ;
ð3:4bÞ
W 2 ðf1 ; . . . ; fN ; wj ; e1 ; . . . ; eN Þ ; W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ
ð3:4cÞ
U½N ¼
wj ½N ¼
ðj ¼ 1; 2; . . . ; nÞ;
and
wnþj ½N ¼
qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; fj1 ; fjþ1 ; . . . ; fN ; fj ; e1 ; . . . ; ej1 ; ejþ1 ; . . . ; eN Þ W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ
; ðj ¼ 1; 2; . . . ; NÞ;
ð3:4dÞ
where e1,. . .,eN are N arbitrary function of T, and U[N], V[N], w1[N], . . ., wn+N[N] satisfy the Eq. (2.6) with n replaced by n + N, thus V[N], w1[N], . . ., wn+N[N] is a solution of the Eq. (2.5) with n replaced by n + N. Obviously, the N-times repeated GBDT defined by Eqs. (3.4) contains N arbitrary T-function and provides a non-autoBäcklund transformation between the two isospectral KdVESCSs of degrees n and n + N. The flexibility of the choices of ej(T) and fj, (j = 1, 2, . . ., n) enables us to construct some new solutions, such as negaton, positon and complexiton solutions for the nonisospectral KdVESCSs (2.1).
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4. Solutions of the nonisospectral KdVESCSs In this section, we firstly study the solutions of the isospectral KdVESCSs (2.5) by means of the GBDT which have been given in Section 3. By the gauge transformation (2.4), the positon, negaton and complexiton solutions of the nonisospectral KdVESCSs (2.1) are obtained. Moreover, the properties of these solutions are analyzed in detail. 4.1. Negaton solution Hereafter, we always take ej(T) = cjT + dj, (j = 1, 2, . . . , N) and g(t) = c0 + 4be2at, where cj – 0 and dj,c0,b are real constants. We take the solution of the Eq. (2.6) with V = 0, n = 0, and c = j2 < 0, j > 0, Im c = 0, as
f ¼ cosh h;
h ¼ jðX þ x1 4j2 TÞ;
ð4:1Þ
where x1 is an analytic function of j. From (3.2b)–(3.2d) with V = 0, n = 0, and e(T) = cT + d, the one-negaton solution of the isospectral KdVESCSs (2.5) were obtained in [22,27]. Based on the gauge transformations (2.4), we derived the one-negaton solution of the nonisospectral KdVESCSs (2.1) with the spectral parameter k = j2e2at < 0, N = 1 as
u ¼ e2at
8j2 cosh Hðjd sinh H cosh HÞ
ðjd þ sinh H cosh HÞ2 pffiffiffi c cosh H u1 ¼ e2at ; 2jðjd þ sinh H cosh HÞ
;
ð4:2aÞ ð4:2bÞ
where
H ¼ j ðx c0 Þeat þ x1 þ d¼
4j2 ðe3at 1Þ 4b 3at ; e 3a 3a
2c 12j2 3at1 4b Þ þ 2d þ ðx c0 Þeat þ x1 þ j@ j x1 e3at : ðe 3a 3a
Fig. 1. one-negaton solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 1; c ¼ 2; d ¼
ð4:2cÞ ð4:2dÞ
8 (a) t = 0.5; (b) t = 0; (c) t = 0.5. 3
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Here we select the parameters
c0 ¼ 1;
a ¼ 1;
b ¼ 1;
8 3
j ¼ 1; c ¼ 2; d ¼ ;
ð4:3Þ
and x1 ¼ 43 j, the detailed structure of the one-negaton solution is shown in Figs. 1 and 2. 4.2. Positon solution Let f be a solution of the Eq. (2.6) with V = 0, n = 0, and c = j2 > 0, j > 0, Im c = 0. Then it is given by
8 Fig. 2. The shape and motion for the one-negaton solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 1; c ¼ 2; d ¼ . 3
Fig. 3. one-positon solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 2; c ¼ 2; d ¼
4 (d) t = 0.5; (e) t = 0; (f) t = 0.5. 3
J. Su et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 110–118
f ¼ cos h;
h ¼ jðX þ x1 þ 4j2 TÞ;
115
ð4:4Þ
where x1 is an analytic function of j. The generalized binary Darboux transformation for the isospectral KdVESCSs (2.5) with e(T) = cT + d and the gauge transformations (2.4) gives
u ¼ e2at
32j2 cos Hðjr sin H þ cos HÞ
ð2jr þ sin 2HÞ2 pffiffiffi c cos H u1 ¼ e2at ; jð2jr þ sin 2HÞ
ð4:5aÞ
;
ð4:5bÞ
where
H ¼ j ðx c0 Þeat þ x1
4j2 ðe3at 1Þ 4b 3at ; e 3a 3a
ð4:5cÞ
2c þ 12j 3at1 4b Þ þ 2d þ ðx c0 Þeat þ x1 þ j@ j x1 e3at ; ðe 3a 3a 2
r¼
ð4:5dÞ
which presents the one-positon solution of the nonisospectral KdVESCSs (2.1) with the spectral parameter N = 1, k = j2e2at > 0 corresponding to the one-positon solution for the KdV equation in [27,28]. Here we select the parameters
c0 ¼ 1;
a ¼ 1;
b ¼ 2;
4 3
j ¼ 1; c ¼ 2; d ¼ ;
ð4:6Þ
and x1 ¼ 43 j, the detailed structure of the one-positon solution is shown in Figs. 3 and 4. 4.3. Complexiton solution In order to obtain the complexiton solutions of the nonisospectral KdVESCSs (2.1) with the spectral parameter k as complex number, one can construct the complexiton solutions of the isospectral KdVESCSs (2.5) as in [25]. Let cj = aj + Ibj, wj = wj1 + Iwj2, and fj be the solutions of the Eq. (2.6) corresponding to c = cj, (j = 1, 2, . . . , N), where I2 = 1, aj, bj – 0 and wj1, wj2 are all real, then the isospectral KdVESCSs (2.5) has the complexiton solutions as the following form
e ¼ V þ 2@ 2 ln W 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ; V X qffiffiffiffiffiffiffiffiffiffiffi 0 ej ðTÞW 2 ðf1 ; . . . ; ^f j ; . . . ; fN ; f 1 ; . . . ; f N ; fj ; e1 ; . . . ; ^ej ; . . . ; eN ; e1 ; . . . ; eN Þ wj1 ¼ W 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; fN ; f 1 ; . . . ; ^f j ; . . . ; f N ; f j ; e1 ; . . . ; eN ; e1 ; . . . ; ^ej ; . . . ; eN Þ þ ; W 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ
ð4:7aÞ
ðj ¼ 1; 2; . . . ; NÞ qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; ^f j ; . . . ; fN ; f 1 ; . . . ; f N ; fj ; e1 ; . . . ; ^ej ; . . . ; eN ; e1 ; . . . ; eN Þ
ð4:7bÞ
wj2 ¼
IW 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; fN ; f 1 ; . . . ; ^f j ; . . . ; f N ; f j ; e1 ; . . . ; eN ; e1 ; . . . ; ^ej ; . . . ; eN Þ IW 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ
;
ðj ¼ 1; 2; . . . ; NÞ; is the complex conjugate of h and h ^ denotes that h does not appear. where h
4 Fig. 4. The shape and motion for the one-positon solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 2; c ¼ 2; d ¼ . 3
ð4:7cÞ
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Fig. 5. one-complexiton solution of the nonisospectral KdVESCSs with c0 = 0, a = 1, b = 1, c = 1, d = 0, q1 = q2 = 0, c = 2i (g) t = 0.5; (h) t = 0; (k) t = 0.5.
We call the above resulting solutions the complexiton solutions. Particularly, the solutions defined by (4.7) are called Ncomplexiton solutions. Now, let us concentrate on the case of N = 1. We take e(T) = cT + d, c = a + Ib and
f ¼ cos n cosh g I sin n sinh g;
ð4:8aÞ
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 a2 þ b2 þ 2a
n¼
g¼
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 4 a2 þ b2 8a T þ q1 ;
ð4:8bÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ b2 2a X þ 4 a2 þ b2 þ 8a T þ q2 ; 2
ð4:8cÞ
and a,b,q1,q2 are arbitrary real constants. Fixing the spectral parameter k of the nonisospectral KdVESCSs (2.1) as a complex number, we can obtain the onecomplexiton solutions for the nonisospectral KdVESCSs (2.1) by the gauge transformations (2.4). To show a detailed structure of the complexiton, we select the parameters
c0 ¼ 0;
a ¼ 1;
b ¼ 1;
c ¼ 1;
d ¼ 0;
q1 ¼ q2 ¼ 0;
c ¼ 2i;
Then the one-complexiton solution for the nonisospectral KdVESCSs (2.1) with u = 0 and n = 0 is given by
ð4:9Þ
J. Su et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 110–118
117
Fig. 6. The shape and motion for one-complexiton solution of the nonisospectral KdVESCSs with c0 = 0, a = 1, b = 1, c = 1, d = 0, q1 = q2 = 0, c = 2i (1) A smooth part of the graph of one-complexiton solution u; (2) A singular part of the graph of one-complexiton solution u.
" u ¼ e2t
2
2
192ðsin 2n þ sinh 2gÞðsin 2n sinh 2g þ 4 cos 2n cosh 2g þ 4Þðe3t 1Þ 2
ðsin 2n þ sinh 2gÞ ð3 sin 2n þ 3 sinh 2g 8e3t þ 8Þ2
# 1024ð1 þ cos 2n cosh 2gÞðe3t 1Þ2 þ 288ðsin 2n þ sinh 2gÞ2 ð1 þ cos 2n cosh 2gÞ ; ðsin 2n þ sinh 2gÞ2 ð3 sin 2n þ 3 sinh 2g 8e3t þ 8Þ2
ð4:10aÞ
and
24 cos n cosh g ; 3 sin 2n þ 3 sinh 2g 8ðe3t 1Þ 8 sin n sinh g ¼ e2t ; ðsin 2n þ sinh 2gÞ
u11 ¼ e2t
ð4:10bÞ
u12
ð4:10cÞ
where
4 n ¼ xet þ ðe3t 1Þ; 3 g ¼ xet 4ðe3t 1Þ:
ð4:10dÞ ð4:10eÞ
Based on those given forms of the functions f, e(t) and k, we can see that the resulting real-valued one-complexiton solution of the nonisospectral KdVESCSs 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 figures of the real-valued one-complexiton solution of the nonisospectral KdVESCSs are given as above. Fig. 5 shows the structure plot of one-complexiton solution for the nonisospectral KdVESCSs expressed by (4.10) at the fixed time t = 0.5, t = 0 and t = 0.5, respectively. The evolution plot of one-complexiton solution (4.10) is depicted in Fig. 6. In terms of the property of the eigenfunction involved, the solution is different from soliton, negaton and positon solutions, and their interaction solutions. 5. Conclusion In this paper, the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs are obtained by the generalized binary Darboux transformation. The nonisospectral KdVESCSs (2.1) can describe soliton waves in a certain type of nonuniform media with a relaxation effect. The negaton, positon, and complexiton solutions correspond, respectively, to negative, positive and complex spectral parameters. The characteristics of these solutions with nonisospectral properties and sources effects have been described through some figures. It is shown that these solutions possess moving singularities
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and the amplitude and velocity of the solitary wave vary with time t. This may be a typical characteristic of ’solitons’ with non-isospectral properties and sources effects. Such singular solutions would help us in recognizing a great diversity of motions of the nonlinear waves described by the nonisospectral SESCSs. To sum up, we have obtained a variety of solutions for the nonisospectral KdVESCSs. We hope that the discussion in this paper would help us learn the internal property of the nonisospectral KdVESCSs deeply and would be useful for the study of other nonisospectral systems. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 10872165, 11002110), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2010JQ1015), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the Scientific and Technological Innovation Foundation of Northwestern Polytechnical University (Grant No. 2008KJ02034). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Hirota R, Satsuma J. N-soliton solution of the KdV equation with loss and nonuniformity terms. J Phys Soc Jpn 1976;41:2141–2. Chen HH, Liu CS. Solitons in nonuniform media. Phys Rev Lett 1976;37:693–7. Gupta MR. Exact inverse scattering solution of a nonlinear evolution equation in a nonuniform media. Phys Lett A 1979;72:420–1. Newell AC. The general structure of integrable evolution equations. Proc R Soc A 1979;365:283–311. Li Q, Zhang DJ, Chen DY. Solving the hierarchy of the nonisospectral KdV equation with self-consistent sources via the inverse scattering transform. J Phys A Math Theor 2008;41:355209. Zhang DJ, Chen DY. Negatons, positons, rational-like solutions and conservation laws of the Korteweg-de Vries equation with loss and non-uniformity terms. J Phys A Math Gen 2004;37:851–65. Zhang Y, Deng SF, Zhang DJ, Chen DY. N-soliton solutions for the non-isospectral mKdV equation. Physica A 2004;399:228–36. Zhang DJ, Bi JB, Hao HH. A modified KdV equation with self-consistent sources in non-uniform media and soliton dynamics. J Phys A Math Gen 2006;39:14627–48. Ning TK, Zhang DJ, Chen DY, Deng SF. Exact solutions and conservation laws for a nonisospectral sine-Gordon equation. Chaos Solitons Fract 2005;25:611–20. Deng SF, Zhang DJ, Chen DY. Exact solutions for the nonisospectral KP equation. J Phys Soc Jpn 2005;74:2383–5. Hao HH, Wang GS, Zhang DJ. KdV equation with self-consistent sources in non-uniform media. Commun Theor Phys 2009;51:989–99. Burtsev SP, Zakharov VE, Mikhailov AV. Inverse scattering method with variable spectral parameter. Theor Math Phys 1987;70:227–40. Ning TK, Zhang WG, Chen DY. Gauge transform between the first order nonisospectral AKNS hierarchy and AKNS hierarchy. Chaos Solitons Fract 2007;34:704–8. Mel’nikov VK. Integration method of the Korteweg-de Vries equation with a self-consistent source. Phy Lett A 1988;133:493–6. Mel’nikov VK. Capture and confinement of solitons in nonlinear integrable systems. Commun Math Phys 1989;120:451–68. Mel’nikov VK. Integration of the Korteweg-de Vries equation with a source. Inverse Problems 1990;6:233–46. Leon J, Latifi A. Solution of an initial-boundary value problem for coupled nonlinear waves. J Phys A Math Gen 1990;23:1385–403. Zeng YB, Ma WX, Lin RL. Integration of the soliton hierarchy with self-consistent sources. J Math Phys 2000;41:5453–89. Zeng YB, Ma WX, Shao YJ. Two binary Darboux transformations for the KdV hierarchy with self-consistent sources. J Math Phys 2001;42:2113–28. Zhang DJ. The N-soliton solutions for the modified KdV equation with self-consistent sources. J Phys Soc Jpn 2002;71:2649–56. Ma WX. Soliton, positon and negaton solutions to a Schrödinger self-consistent source equation. J Phys Soc Jpn 2003;72:3017–9. Zeng YB, Shao YJ, Xue WM. Negaton and positon solutions of the soliton equation with self-consistent sources. J Phys A Math Gen 2003;36:5035–43. Shao YJ, Zeng YB. The solutions of the NLS equations with self-consistent sources. J Phys A Math Gen 2005;38:2441–67. Liu XJ, Zeng YB. On the Toda lattice equation with self-consistent sources. J Phys A Math Gen 2005;38:8951–65. Complexiton solutions of the Korteweg-de Vries equaion with self-consistent sources. Chaos Solitons and Fract 2005;26:1453–1458. Wu HX, Zeng YB, Fan TY. Positon and negaton solutions of the mKdV equation with self-consistent sources. J Phys A Math Theor 2007;40:10505–17. Rasinariu C, Sukhatme U, Khare A. Negaton and positon of the KdV and mKdV hierarchy. J Phys A Math Gen 1996;29:1803–23. Matveev VB. Positons: slowly decreasing analogues of solitons. Theor Math Phys 2002;131:483–97. Ma WX. Complexiton solutions to the Korteweg-de Vries equation. Phy Lett A 2002;301:35–44.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Negaton, positon and complexiton solutions of the nonisospectral KdV equations with self-consistent sources Jun Su a,⇑, Wei Xu a, Genjiu Xu a, Liang Gao b a b
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an 710072, China Science Research Institute of China – North Group Company, Beijing 100089, China
a r t i c l e
i n f o
Article history: Received 27 August 2010 Received in revised form 6 January 2011 Accepted 10 April 2011 Available online 23 April 2011 Keywords: Nonisospectral KdV equation Negaton Positon Complexiton Darboux transformation Self-consistent sources
a b s t r a c t The negaton, positon, and complexiton solutions of the nonisospectral KdV equations with self-consistent sources (KdVESCSs) are obtained by the generalized binary Darboux transformation (GBDT) with N arbitrary t-functions. Taking the special initial seed solution for auxiliary linear problems, the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs are considered through the GBDT by selecting the negative, positive and complex spectral parameters. It is important to point out that these solutions of the nonisospectral KdVESCSs are analytical and singular. We also show differences between these solutions with singularities. Moreover, the detailed characteristics of these solutions with nonisospectral properties and sources effects are described through some figures. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The nonisospectral evolution equations are used to describe deep water and plasma waves, waves in nonuniform media, in which the spectral parameter depends on the time or space variables [1–11]. In terms of finding exact solutions, some methods have been used to these kind of equations, such as the inverse scattering method [2–5], the Hirota method and the Wronskian technique [1,6–11], and the Darboux transformation [12]. In general, when the spectral parameter k follows kt = kj where j = 0 or 1, the corresponding nonisospectral equations are exactly integrable and mathematically trivial since there exists a gauge transformations between them and their isospectral counterparts; while when j > 1, such transformation cannot be found [12,13]. The soliton equations with self-consistent sources (SESCSs) have also 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. [14–17]. Recently, the SESCSs were investigated by means of the inverse scattering method, the Hirota method and the Wronskian technique, the generalized binary Darboux transformation (GBDT), etc. [18–26]. As is well known, a new classification of exact and explicit solutions of integrable models was proposed according to the property of associated spectral parameters [27–29]. Negaton, related to the negative spectral parameter, was usually a singular or nonsingular reduced two-solitons. Positon, related to the positive spectral parameter, was usually a long-range, slowly decreasing and oscillating singular soliton-like solution. The so-called complexiton, which is related to the complex spectral parameters, was usually expressed by the combinations of trigonometric functions and hyperbolic functions. Some ⇑ Corresponding author. Tel.: +86 29 88460633; fax: +86 29 88495453. E-mail address: [email protected] (J. Su). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.019
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negatons, positons, and complexitons are obtained for the soliton equations with self-consistent sources by the GBDT [22– 26]. In this paper, we investigate the nonisospectral KdV equations with self-consistent sources (KdVESCSs). The N-soliton solutions for the hierarchy of the nonisospectral KdVESCSs has been obtained through the inverse scattering transform [5]. Whether the nonisospectral KdVESCSs have negaton, positon and complexiton solutions or not, like isospectral KdVESCSs? This paper aims to provide a positive answer to this question. Through the GBDT and the gauge transformation, we obtain the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs. We hope that the study of these solutions will enrich the theory of the nonisospectral KdVESCSs. The paper is organized as follows. In Section 2, the nonisospectral KdVESCSs and its gauge transformation are given. In Section 3, the GBDT for the isospectral KdVESCSs is constructed. In Section 4, the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs are obtained. Section 5 presents the conclusions. 2. The nonisospectral KdVESCSs and gauge transformation The nonisospectral KdVESCSs is written as
ut þ 6uux þ uxxx þ 2au þ axux ux gðtÞ þ 4
n X
uj uj;x ¼ 0;
ð2:1aÞ
j¼1
uj;xx þ ðkj þ uÞuj ¼ 0; ðj ¼ 1; 2; . . . ; nÞ;
ð2:1bÞ 2at
where a is a real constant and we assume that u tends to 0 as x ? ±1. Taking n = 0, g(t) = c0 + 6be , and c0, b are real constants, the Eq. (2.1) is one of the so-called x-coefficient equations which can describe nonlinear soliton waves in a certain type of non-uniform media with a relaxation effect [1,2]. In [6], the solitons, rational, negatons and positons solutions for this equation were obtained through the Wronskian technique. When the spectral parameter k follows
kt ¼ 2ak:
ð2:2Þ
We can see that the Eq. (2.1) is a nonisospectral soliton equation with self-consistent sources in terms of k. The corresponding Lax pair for the nonisospectral KdVESCSs (2.1) is shown as below
/xx þ ðk þ uÞ/ ¼ 0;
ð2:3aÞ
/t ¼ A/ þ B/x ;
ð2:3bÞ
where
A ¼ ux
n X uj uj;x ; kj k j¼1
B ¼ 4k 2u ax þ gðtÞ þ
n X u2j j¼1
kj k
;
ð2:3cÞ
As kt = 2ak, there exists a gauge transformation connecting the nonisospectral KdVESCSs (2.1) with the isospectral KdVESCSs. The transformation is for both equations and its Lax pairs. We describe it by the following theorem. Theorem 2.1. When kt = 2ak, by the transformation
V ¼ e2at u;
X ¼ eat x þ
Z
t
gðzÞeaz dz;
T¼
1
e3at 1 ; 3a
c ¼ e2at k; w ¼ e2at u;
cj ¼ e2at kj ; wj ¼ e2at uj ; ðj ¼ 1; 2; . . . ; nÞ;
ð2:4aÞ ð2:4bÞ
the nonisospectral KdVESCSs (2.1) can be transformed into the isospectral KdVESCSs
V T þ 6VV X þ V XXX þ 4
n X
wj wj;x ¼ 0;
ð2:5aÞ
j¼1
wj;XX þ ðcj þ VÞwj ¼ 0;
ðj ¼ 1; 2; . . . ; nÞ;
ð2:5bÞ
and the Lax pair (2.3) of the nonisospectral KdVESCSs (2.1) is also transformed to the Lax pair of the isospectral one
UXX þ ðc þ VÞU ¼ 0; UT ¼ C U þ DUx ;
ð2:6aÞ ð2:6bÞ
where
C ¼ Vx
n X wj wj;x ; c j c j¼1
D ¼ 4g 2V þ
n X j¼1
w2j : cj c
ð2:6cÞ
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Proof. Taking Eqs. (2.4) into the nonisospectral KdVESCSs (2.1) leads to the Eq. (2.5). Similarly, we can check that (2.6) holds. This completes the proof. h
3. The GBDT for the isospectral KdVESCSs In this section, we will consider the generalized binary Darboux transformation (GBDT) with an arbitrary function of T for the isospectral KdVESCSs (2.5), which offers a non-auto-Bäcklund transformation between two isospectral KdVSESCSs with different degrees of sources. With the Lax pair (2.6), the GBDT for the isospectral KdVESCSs has been verified in Ref. [22]. Here we only briefly write the final results and construct the negaton, positon and complexiton solutions for the nonisospectral KdVESCSs (2.1) based on the given Darboux transformation. Assume that f and g are two solutions of Eq. (2.6) corresponding to c = n and c = g, respectively. Define
xðf ; gÞ ¼
Wðf ; gÞ ; ng
xðf ; f Þ ¼ lim g!n
Wðf ðnÞ; f ðgÞÞ ; ng
ð3:1Þ
where W(f, g) denotes the Wronskian determinant W(f, g) = fgx fxg. Then, the first step GBDT with an arbitrary function of T for the isospectral KdVESCSs (2.5) is defined by
U½1 ¼ U
f xðf ; UÞ ; eðTÞ þ xðf ; f Þ
ð3:2aÞ
V½1 ¼ V þ 2@ 2x ln½eðTÞ þ xðf ; f Þ; f xðf ; wj Þ wj ½1 ¼ wj ; eðTÞ þ xðf ; f Þ pffiffiffiffiffiffiffiffiffiffi f e0 ðTÞ ; wnþ1 ½1 ¼ eðTÞ þ xðf ; f Þ
ð3:2bÞ
ðj ¼ 1; 2; . . . ; nÞ;
ð3:2cÞ ð3:2dÞ
where e(T) is an arbitrary function of T, and U[1], V[1], w1[1], . . ., wn+1[1] satisfy the Eq. (2.6) with n replaced by n + 1, thus V[1], w1[1], . . ., wn+1[1] is a solution of the equation (2.5) with n replaced by n + 1. For m solutions g1, . . ., gm of Eq. (2.6) and m arbitrary T-function e1(T), . . ., em(T), we define two determinants
W 1 ðg 1 ; . . . ; g m ; e1 ; . . . ; em Þ ¼ det F;
W 2 ðg 1 ; . . . ; g m ; e1 ; . . . ; em1 Þ ¼ det G;
ð3:3Þ
where
F ij ¼ dij ei ðTÞ þ xðg i ; g j Þ;
ði; j ¼ 1; . . . ; mÞ;
Gij ¼ dij ei ðTÞ þ xðg i ; g j Þ;
ði ¼ 1; . . . ; m 1;
Gmj ¼ g j ;
j ¼ 1; . . . ; mÞ;
ðj ¼ 1; . . . ; mÞ:
Then the N-times repeated GBDT with N arbitrary T-functions for the isospectral KdVESCSs (2.5) is given by
W 2 ðf1 ; . . . ; fN ; U; e1 ; . . . ; eN Þ ; W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ
ð3:4aÞ
V½N ¼ V þ 2@ 2x ln W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ;
ð3:4bÞ
W 2 ðf1 ; . . . ; fN ; wj ; e1 ; . . . ; eN Þ ; W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ
ð3:4cÞ
U½N ¼
wj ½N ¼
ðj ¼ 1; 2; . . . ; nÞ;
and
wnþj ½N ¼
qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; fj1 ; fjþ1 ; . . . ; fN ; fj ; e1 ; . . . ; ej1 ; ejþ1 ; . . . ; eN Þ W 1 ðf1 ; . . . ; fN ; e1 ; . . . ; eN Þ
; ðj ¼ 1; 2; . . . ; NÞ;
ð3:4dÞ
where e1,. . .,eN are N arbitrary function of T, and U[N], V[N], w1[N], . . ., wn+N[N] satisfy the Eq. (2.6) with n replaced by n + N, thus V[N], w1[N], . . ., wn+N[N] is a solution of the Eq. (2.5) with n replaced by n + N. Obviously, the N-times repeated GBDT defined by Eqs. (3.4) contains N arbitrary T-function and provides a non-autoBäcklund transformation between the two isospectral KdVESCSs of degrees n and n + N. The flexibility of the choices of ej(T) and fj, (j = 1, 2, . . ., n) enables us to construct some new solutions, such as negaton, positon and complexiton solutions for the nonisospectral KdVESCSs (2.1).
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4. Solutions of the nonisospectral KdVESCSs In this section, we firstly study the solutions of the isospectral KdVESCSs (2.5) by means of the GBDT which have been given in Section 3. By the gauge transformation (2.4), the positon, negaton and complexiton solutions of the nonisospectral KdVESCSs (2.1) are obtained. Moreover, the properties of these solutions are analyzed in detail. 4.1. Negaton solution Hereafter, we always take ej(T) = cjT + dj, (j = 1, 2, . . . , N) and g(t) = c0 + 4be2at, where cj – 0 and dj,c0,b are real constants. We take the solution of the Eq. (2.6) with V = 0, n = 0, and c = j2 < 0, j > 0, Im c = 0, as
f ¼ cosh h;
h ¼ jðX þ x1 4j2 TÞ;
ð4:1Þ
where x1 is an analytic function of j. From (3.2b)–(3.2d) with V = 0, n = 0, and e(T) = cT + d, the one-negaton solution of the isospectral KdVESCSs (2.5) were obtained in [22,27]. Based on the gauge transformations (2.4), we derived the one-negaton solution of the nonisospectral KdVESCSs (2.1) with the spectral parameter k = j2e2at < 0, N = 1 as
u ¼ e2at
8j2 cosh Hðjd sinh H cosh HÞ
ðjd þ sinh H cosh HÞ2 pffiffiffi c cosh H u1 ¼ e2at ; 2jðjd þ sinh H cosh HÞ
;
ð4:2aÞ ð4:2bÞ
where
H ¼ j ðx c0 Þeat þ x1 þ d¼
4j2 ðe3at 1Þ 4b 3at ; e 3a 3a
2c 12j2 3at1 4b Þ þ 2d þ ðx c0 Þeat þ x1 þ j@ j x1 e3at : ðe 3a 3a
Fig. 1. one-negaton solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 1; c ¼ 2; d ¼
ð4:2cÞ ð4:2dÞ
8 (a) t = 0.5; (b) t = 0; (c) t = 0.5. 3
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Here we select the parameters
c0 ¼ 1;
a ¼ 1;
b ¼ 1;
8 3
j ¼ 1; c ¼ 2; d ¼ ;
ð4:3Þ
and x1 ¼ 43 j, the detailed structure of the one-negaton solution is shown in Figs. 1 and 2. 4.2. Positon solution Let f be a solution of the Eq. (2.6) with V = 0, n = 0, and c = j2 > 0, j > 0, Im c = 0. Then it is given by
8 Fig. 2. The shape and motion for the one-negaton solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 1; c ¼ 2; d ¼ . 3
Fig. 3. one-positon solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 2; c ¼ 2; d ¼
4 (d) t = 0.5; (e) t = 0; (f) t = 0.5. 3
J. Su et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 110–118
f ¼ cos h;
h ¼ jðX þ x1 þ 4j2 TÞ;
115
ð4:4Þ
where x1 is an analytic function of j. The generalized binary Darboux transformation for the isospectral KdVESCSs (2.5) with e(T) = cT + d and the gauge transformations (2.4) gives
u ¼ e2at
32j2 cos Hðjr sin H þ cos HÞ
ð2jr þ sin 2HÞ2 pffiffiffi c cos H u1 ¼ e2at ; jð2jr þ sin 2HÞ
ð4:5aÞ
;
ð4:5bÞ
where
H ¼ j ðx c0 Þeat þ x1
4j2 ðe3at 1Þ 4b 3at ; e 3a 3a
ð4:5cÞ
2c þ 12j 3at1 4b Þ þ 2d þ ðx c0 Þeat þ x1 þ j@ j x1 e3at ; ðe 3a 3a 2
r¼
ð4:5dÞ
which presents the one-positon solution of the nonisospectral KdVESCSs (2.1) with the spectral parameter N = 1, k = j2e2at > 0 corresponding to the one-positon solution for the KdV equation in [27,28]. Here we select the parameters
c0 ¼ 1;
a ¼ 1;
b ¼ 2;
4 3
j ¼ 1; c ¼ 2; d ¼ ;
ð4:6Þ
and x1 ¼ 43 j, the detailed structure of the one-positon solution is shown in Figs. 3 and 4. 4.3. Complexiton solution In order to obtain the complexiton solutions of the nonisospectral KdVESCSs (2.1) with the spectral parameter k as complex number, one can construct the complexiton solutions of the isospectral KdVESCSs (2.5) as in [25]. Let cj = aj + Ibj, wj = wj1 + Iwj2, and fj be the solutions of the Eq. (2.6) corresponding to c = cj, (j = 1, 2, . . . , N), where I2 = 1, aj, bj – 0 and wj1, wj2 are all real, then the isospectral KdVESCSs (2.5) has the complexiton solutions as the following form
e ¼ V þ 2@ 2 ln W 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ; V X qffiffiffiffiffiffiffiffiffiffiffi 0 ej ðTÞW 2 ðf1 ; . . . ; ^f j ; . . . ; fN ; f 1 ; . . . ; f N ; fj ; e1 ; . . . ; ^ej ; . . . ; eN ; e1 ; . . . ; eN Þ wj1 ¼ W 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; fN ; f 1 ; . . . ; ^f j ; . . . ; f N ; f j ; e1 ; . . . ; eN ; e1 ; . . . ; ^ej ; . . . ; eN Þ þ ; W 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ
ð4:7aÞ
ðj ¼ 1; 2; . . . ; NÞ qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; ^f j ; . . . ; fN ; f 1 ; . . . ; f N ; fj ; e1 ; . . . ; ^ej ; . . . ; eN ; e1 ; . . . ; eN Þ
ð4:7bÞ
wj2 ¼
IW 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ qffiffiffiffiffiffiffiffiffiffiffi e0j ðTÞW 2 ðf1 ; . . . ; fN ; f 1 ; . . . ; ^f j ; . . . ; f N ; f j ; e1 ; . . . ; eN ; e1 ; . . . ; ^ej ; . . . ; eN Þ IW 1 ðf1 ; . . . ; fN ; f 1 ; . . . ; f N ; e1 ; . . . ; eN ; e1 ; . . . ; eN Þ
;
ðj ¼ 1; 2; . . . ; NÞ; is the complex conjugate of h and h ^ denotes that h does not appear. where h
4 Fig. 4. The shape and motion for the one-positon solution of the nonisospectral KdVESCSs with c0 ¼ 1; j ¼ 1; a ¼ 1; b ¼ 2; c ¼ 2; d ¼ . 3
ð4:7cÞ
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Fig. 5. one-complexiton solution of the nonisospectral KdVESCSs with c0 = 0, a = 1, b = 1, c = 1, d = 0, q1 = q2 = 0, c = 2i (g) t = 0.5; (h) t = 0; (k) t = 0.5.
We call the above resulting solutions the complexiton solutions. Particularly, the solutions defined by (4.7) are called Ncomplexiton solutions. Now, let us concentrate on the case of N = 1. We take e(T) = cT + d, c = a + Ib and
f ¼ cos n cosh g I sin n sinh g;
ð4:8aÞ
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 a2 þ b2 þ 2a
n¼
g¼
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 4 a2 þ b2 8a T þ q1 ;
ð4:8bÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ b2 2a X þ 4 a2 þ b2 þ 8a T þ q2 ; 2
ð4:8cÞ
and a,b,q1,q2 are arbitrary real constants. Fixing the spectral parameter k of the nonisospectral KdVESCSs (2.1) as a complex number, we can obtain the onecomplexiton solutions for the nonisospectral KdVESCSs (2.1) by the gauge transformations (2.4). To show a detailed structure of the complexiton, we select the parameters
c0 ¼ 0;
a ¼ 1;
b ¼ 1;
c ¼ 1;
d ¼ 0;
q1 ¼ q2 ¼ 0;
c ¼ 2i;
Then the one-complexiton solution for the nonisospectral KdVESCSs (2.1) with u = 0 and n = 0 is given by
ð4:9Þ
J. Su et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 110–118
117
Fig. 6. The shape and motion for one-complexiton solution of the nonisospectral KdVESCSs with c0 = 0, a = 1, b = 1, c = 1, d = 0, q1 = q2 = 0, c = 2i (1) A smooth part of the graph of one-complexiton solution u; (2) A singular part of the graph of one-complexiton solution u.
" u ¼ e2t
2
2
192ðsin 2n þ sinh 2gÞðsin 2n sinh 2g þ 4 cos 2n cosh 2g þ 4Þðe3t 1Þ 2
ðsin 2n þ sinh 2gÞ ð3 sin 2n þ 3 sinh 2g 8e3t þ 8Þ2
# 1024ð1 þ cos 2n cosh 2gÞðe3t 1Þ2 þ 288ðsin 2n þ sinh 2gÞ2 ð1 þ cos 2n cosh 2gÞ ; ðsin 2n þ sinh 2gÞ2 ð3 sin 2n þ 3 sinh 2g 8e3t þ 8Þ2
ð4:10aÞ
and
24 cos n cosh g ; 3 sin 2n þ 3 sinh 2g 8ðe3t 1Þ 8 sin n sinh g ¼ e2t ; ðsin 2n þ sinh 2gÞ
u11 ¼ e2t
ð4:10bÞ
u12
ð4:10cÞ
where
4 n ¼ xet þ ðe3t 1Þ; 3 g ¼ xet 4ðe3t 1Þ:
ð4:10dÞ ð4:10eÞ
Based on those given forms of the functions f, e(t) and k, we can see that the resulting real-valued one-complexiton solution of the nonisospectral KdVESCSs 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 figures of the real-valued one-complexiton solution of the nonisospectral KdVESCSs are given as above. Fig. 5 shows the structure plot of one-complexiton solution for the nonisospectral KdVESCSs expressed by (4.10) at the fixed time t = 0.5, t = 0 and t = 0.5, respectively. The evolution plot of one-complexiton solution (4.10) is depicted in Fig. 6. In terms of the property of the eigenfunction involved, the solution is different from soliton, negaton and positon solutions, and their interaction solutions. 5. Conclusion In this paper, the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs are obtained by the generalized binary Darboux transformation. The nonisospectral KdVESCSs (2.1) can describe soliton waves in a certain type of nonuniform media with a relaxation effect. The negaton, positon, and complexiton solutions correspond, respectively, to negative, positive and complex spectral parameters. The characteristics of these solutions with nonisospectral properties and sources effects have been described through some figures. It is shown that these solutions possess moving singularities
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and the amplitude and velocity of the solitary wave vary with time t. This may be a typical characteristic of ’solitons’ with non-isospectral properties and sources effects. Such singular solutions would help us in recognizing a great diversity of motions of the nonlinear waves described by the nonisospectral SESCSs. To sum up, we have obtained a variety of solutions for the nonisospectral KdVESCSs. We hope that the discussion in this paper would help us learn the internal property of the nonisospectral KdVESCSs deeply and would be useful for the study of other nonisospectral systems. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 10872165, 11002110), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2010JQ1015), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the Scientific and Technological Innovation Foundation of Northwestern Polytechnical University (Grant No. 2008KJ02034). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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