- Email: [email protected]
New positon, negaton and complexiton solutions for the Bogoyavlensky–Konoplechenko equation
Physics Letters A 373 (2009) 1750–1753
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
New positon, negaton and complexiton solutions for the Bogoyavlensky–Konoplechenko equation H.C. Hu ∗ College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China
a r t i c l e
i n f o
Article history: Received 29 January 2009 Received in revised form 3 March 2009 Accepted 9 March 2009 Available online 14 March 2009 Communicated by A.R. Bishop
a b s t r a c t New positon, negaton and complexiton solutions for the Bogoyavlensky–Konoplechenko equation are constructed by means of the Darboux transformation with constant seed solution. The new positon, negaton and complexiton solutions are analytical or singular and given out both analytically and graphically. © 2009 Elsevier B.V. All rights reserved.
PACS: 02.30.Jr 02.30.Ik 05.45.Yv Keywords: Complexiton Positon Negaton Darboux transformation
1. Introduction It is well known that nonlinear phenomena, which are mostly described by the nonlinear evolution equations, play a crucial role in almost all aspects of science, especially in applied mathematics, theoretical physics and optics, etc. Many effective methods have been introduced to construct the exact solutions of the nonlinear evolution equations. For example, the inverse scattering transformation, Bäcklund transformation, Hirota bilinear form, symmetry reduction, multi-linear variable separation approach, Darboux transformation and so on. Among these various methods, Darboux transformation is proved to be a powerful tool to construct the exact and explicit solutions of the given nonlinear evolution equations [1–5]. Recently, by applying the Darboux transformation to the two different types of the coupled KdV system, new positon, negaton, complexiton solutions have been found in [6,7]. The positon, negaton, complexiton solutions are due to a new classification of the exact and explicit solutions of integrable models which were proposed according to the property of associated spectral parameters [8]. The authors of [9–11] provided many interesting positon, negaton, complexiton solutions for the KdV equation, the Toda lattice equation through the Wronskian and Casoratian techniques.
*
Tel.: +86 021 55274316. E-mail address: [email protected].
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.03.022
A new Wronskian formulation called the second Wronskian formulation has been presented in order to construct the rational solution, positon solution and complexiton solutions to the Boussinesq equation in [12]. Negatons, related to the negative spectral parameter, are usually expressed by hyperbolic functions and positons are expressed by means of trigonometric functions related to the positive spectral parameters. For various important integrable systems such as KdV equation, there is no non-singular positon. But negatons can be both singular and non-singular. Especially the non-singular negatons are just the usual solitons which have been studied extensively. The so-called complexiton, which is related to the complex spectral parameters, is expressed by combinations of trigonometric functions and hyperbolic functions. The known complexiton solutions for (1 + 1)-dimensional integrable functions are singular, but some types of analytical complexitons in (2 + 1) and (3 + 1) dimensions can be easily obtained because of the existence of arbitrary functions in the expression of exact solutions [13–15]. In this Letter, we study the positon, negaton and complexiton solutions to the Bogoyavlensky–Konoplechenko equation by its Darboux transformation with constant seed solution. The Bogoyavlensky–Konoplechenko (BK) equation ut + α u xxx + β u xxy + 6α uu x + 4β uu y + 4β u x ∂x−1 u y = 0,
(1.1)
where α and β are two constants, is a two-dimensional version of the KdV equation and a particular case of this equation was
H.C. Hu / Physics Letters A 373 (2009) 1750–1753
1751
introduced in [16,17]. It is pointed out that Eq. (1.1) is a kind of breaking soliton equation by a simple coefficients transformation [18] and has been described as the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis [19]. 2. Darboux transformation for the BK equation It is well known that the Darboux transformation is one of the most powerful tools to construct the exact and explicit solutions for many Lax integrable systems. For a general Lax integrable system in 2n dimensions, the Darboux transformation has been established and a broad class of the exact solutions for the Lax integrable system are given out by means of the corresponding Darboux transformation [20]. In this section, we consider the Darboux transformation for the BK equation and construct its N-step Darboux transformation in the Wronskian determinant. Noting u y = v x , the BK equation (1.1) becomes ut + α u xxx + β u xxy + 6α uu x + 4β uu y + 4β u x v = 0, u y = v x.
(2.1)
The Lax pair for Eq. (2.1) is given by the two operators
L φ = ∂x2 + u − λ φ = 0,
P φ = ∂t + 4α ∂x3 + 4β ∂x2 + u ∂ y + (6α u + 2β v )∂x + 3α u x + 3β u y φ = 0.
(2.2)
(2.3)
With the Lax pair (2.2) and (2.3), the Darboux transformation for the BK equation has been verified in Ref. [21]. Here we only briefly write the final results and construct the new positon, negaton and complexiton solutions for the BK equation based on the given Darboux transformation. Let φ1 be a nontrivial solution of the Lax pair (2.2) and (2.3) with λ = λ1 , then the new wave funcφ tion φ[1] = φx − φ1x φ and the new potential functions 1
U ≡ u [1] = u + 2(ln φ1 )xx ,
(2.4)
V ≡ v [1] = v + 2(ln φ1 )xy ,
(2.5)
also satisfy the Lax pair (2.2) and (2.3). The map φ → φ[1], u → u [1] and v → v [1] is called the first step Darboux transformation for the BK equation. Using the iteration of the Darboux transformation, one can easily obtain the n step Darboux transformation expressed by the Wronskian determinant. That is to say, if φ1 , φ2 , . . . , φn are solutions of the Lax pair (2.2) and (2.3) at λ = λ1 , . . . , λn , respectively, then
u [ N ] = u + 2 ln W (φ1 , φ2 , . . . , φn ) xx , v [ N ] = v + 2 ln W (φ1 , φ2 , . . . , φn ) xy ,
where W (φ1 , φ2 , . . . , φn ) is the usual Wronskian determinant, are also the solution of the BK equation. 3. Positon, negaton and complexiton solutions for the BK equation In this section, we study the positon, negaton and complexiton solutions of the BK equation in detail by means of the Darboux transformation given in the last section.
Fig. 1. Singular positon solution for the BK equation (1.1) with (3.2) and (3.3) at t = 0.
Substituting the seed solution {u = 0, v = 0} into the Lax pair (2.2) and (2.3), we can easily obtain the wave function in the form of
φ = F1
4βλt − y
e
4βλ
√ λ(α y −β x) β
+ F2
4βλt − y 4βλ
e
√ λ(β x−α y ) β
,
(3.1)
where F 1 and F 2 are two arbitrary functions of the indicated variable. Because of the two arbitrary functions F 1 and F 2 in (3.1), one can select the arbitrary functions properly to construct different singular or analytical positon solutions for the BK equation. Here we give out two kinds of the positon solutions below. Selecting the two arbitrary functions F 1 and F 2 as
F 1 = cn
4βλt − y 4βλ
, 0.99 ,
F 2 = sn
4βλt − y 4βλ
, 0.88
(3.2)
and the parameters
α = 2,
β = 4,
λ = 25,
(3.3)
where “cn” and “sn” are the usual Jacobi elliptic functions. Then we obtain a type of singular positon solution for the BK equation by the first step Darboux transformation and Fig. 1 shows the detail structure. When the two arbitrary functions F 1 and F 2 are chosen to be
F 1 = cn
4βλt − y
4βλ
,
F2 = e
2 sinh(
4βλt − y ) 4βλ
,
(3.4)
and
α = 2,
β = 4,
λ = 9,
(3.5)
an analytical positon solution is obtained for the BK equation (1.1) with the constant selection (3.5) and the detail structure is shown in Fig. 2. 3.2. Negaton solution, λ < 0 In order to obtain the negaton solution for the BK equation, one only need to substitute λ = −k2 into (3.1) and select the arbitrary functions F 1 and F 2 properly. Selecting the two arbitrary functions as
F 1 = F 2 = arctanh
4βλt − y 4βλ
,
(3.6)
and the constants
3.1. Positon solution, λ > 0
α = 4,
Based on the first step Darboux transformation, the positon solution for the BK equation can be constructed directly by taking the seed solution {u = 0, v = 0}.
By substituting (3.6) and (3.7) into (3.1) and (2.4), we have the negaton solution for the BK equation, which was given out graphically in Fig. 3.
β = 4,
λ = −1.
(3.7)
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H.C. Hu / Physics Letters A 373 (2009) 1750–1753
Fig. 5. Singular complexiton solution for the BK equation (1.1) with (3.9), (3.10) and (3.11) at t = 0.
3.3. Complexiton solution
Fig. 2. Analytical positon solution for the BK equation (1.1) with (3.4) and (3.5) at t = 0.
To obtain the complexiton solutions for the BK equation under the seed solution {u = 0, v = 0}, one can select the spectral parameter λ as complex number. When we fix the spectral parameter and constants as
λ = 1 + 2I ,
α = 3,
β = 1,
(3.9)
where I 2 = −1, the corresponding wave function is given by
φ=
C2 6
+
C4 60
x−
C2 2
y−
C4 2
t−
C1 + C3 + C5
2
C2 C4 + − + x + C 2 y + C 4t + C 1 + C 3 + C 5 I , 3
30
(3.10)
and {C 1 , C 2 , C 3 , C 4 , C 5 } are arbitrary constants. Then from the first Darboux transformation (2.4), one can construct the complexiton solution for the BK equation immediately by choosing the five constants. If the five constants in (3.10) are chosen as C 1 = 1, Fig. 3. Negaton solution for the BK equation (1.1) with (3.1), (3.6) and (3.7) at t = 0.
C 2 = 1,
C 3 = 4,
C 4 = 2,
C 5 = 1,
(3.11)
the singular complexiton solution for the BK equation is arrived and given out in Fig. 5. In this section, with the help of Darboux transformation for the BK equation, we have obtained three types of exact solutions, positon, negaton and complexiton solutions, by selecting the real or complex spectral parameters in the Lax pair from the constant seed solution. Due to the two arbitrary functions F 1 and F 2 in the wave function, we can construct the analytical or singular positon, negaton and complexiton solutions by choosing the two arbitrary functions properly. 4. Summary and discussion
Fig. 4. Second type of the analytical negaton solution for the BK equation (1.1) with (3.1), (3.7) and (3.8) at t = 0.
When we choose the arbitrary functions F 1 and F 2 as
F 1 = arctanh
4βλt − y 4βλ
,
F 2 = arcsinh
4βλt − y 4βλ
,
(3.8)
then another type of the negaton solution is obtained with the constant selection (3.7) and given out directly in Fig. 4.
In this Letter, starting from the trivial constant seed solution, the positon, negaton and complexiton solutions for the BK equation are considered from the Darboux transformation by selecting the real or complex spectral parameters in its Lax pair. It should be pointed out that the obtained positon, negaton and complexiton solutions are analytical or singular due to the proper selection of the two arbitrary functions in the wave function. The detail structures of the positon, negaton and complexiton solutions are given out analytically and graphically. As we know, a family of solutions of the BK equation have been constructed in Ref. [21] which was expressed by exponential function. But the analytical or singular positon, negaton and complexiton solutions for the BK equation are firstly given out to our knowledge. We believe that these new positon, negaton and complexiton solutions for the BK equation will help us to learn the internal property of the nonlinear evolution equation deeply.
H.C. Hu / Physics Letters A 373 (2009) 1750–1753
Acknowledgements The author would like to thank Prof. S.Y. Lou for helpful discussions and the anonymous referees very much for their valuable comments and suggestions. The work was supported by the National Natural Science Foundation of China (No. 10601033). References [1] [2] [3] [4]
V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer, 1991. M. Wadati, H. Sanuki, K. Konno, Prog. Theor. Phys. 53 (1975) 419. K. Konno, M. Wadati, Prog. Theor. Phys. 52 (1975) 1652. C.H. Gu, H.S. Hu, Z.X. Zhou, Darboux Transformation in Soliton Theory and Its Geometric Applications, Shanghai Science and Technology Publishing House, Shanghai, 2005. [5] H.C. Hu, Q.P. Liu, Chaos Solitons Fractals 17 (2003) 921.
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
1753
H.C. Hu, B. Tong, S.Y. Lou, Phys. Lett. A 351 (2006) 403. H.C. Hu, Y. Liu, Phys. Lett. A 372 (2008) 5795. W.X. Ma, Phys. Lett. A 301 (2002) 35. W.X. Ma, K. Maruno, Physica A 343 (2004) 219. W.X. Ma, Y. You, Trans. Am. Math. Soc. 357 (2005) 1753. W.X. Ma, Nonlinear Anal. 63 (2005) e2461. W.X. Ma, C.X. Li, J.S. He, Nonlinear Anal. (2008), doi:10.1016/j.na.2008.09. 010. S.Y. Lou, H.C. Hu, X.Y. Tang, Phys. Rev. E 71 (2005) 036604. X.Y. Tang, S.Y. Lou, Y. Zhang, Phys. Rev. E 66 (2002) 046601. H.C. Hu, S.Y. Lou, Phys. Lett. A 341 (2005) 422. F. Calogero, Lett. Nuovo Cimento 14 (1975) 443. O.I. Bogoyavlenskii, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 243. W.X. Ma, R.G. Zhou, L. Gao, Mod. Phys. Lett. B (2009), in press. B.G. Konopelchenko, Solitons in Multidimensions, World Scientific, Singapore, 1993. W.X. Ma, Lett. Math. Phys. 39 (1997) 33. M.V. Prabhakar, H. Bhate, Lett. Math. Phys. 64 (2003) 1.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
New positon, negaton and complexiton solutions for the Bogoyavlensky–Konoplechenko equation H.C. Hu ∗ College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China
a r t i c l e
i n f o
Article history: Received 29 January 2009 Received in revised form 3 March 2009 Accepted 9 March 2009 Available online 14 March 2009 Communicated by A.R. Bishop
a b s t r a c t New positon, negaton and complexiton solutions for the Bogoyavlensky–Konoplechenko equation are constructed by means of the Darboux transformation with constant seed solution. The new positon, negaton and complexiton solutions are analytical or singular and given out both analytically and graphically. © 2009 Elsevier B.V. All rights reserved.
PACS: 02.30.Jr 02.30.Ik 05.45.Yv Keywords: Complexiton Positon Negaton Darboux transformation
1. Introduction It is well known that nonlinear phenomena, which are mostly described by the nonlinear evolution equations, play a crucial role in almost all aspects of science, especially in applied mathematics, theoretical physics and optics, etc. Many effective methods have been introduced to construct the exact solutions of the nonlinear evolution equations. For example, the inverse scattering transformation, Bäcklund transformation, Hirota bilinear form, symmetry reduction, multi-linear variable separation approach, Darboux transformation and so on. Among these various methods, Darboux transformation is proved to be a powerful tool to construct the exact and explicit solutions of the given nonlinear evolution equations [1–5]. Recently, by applying the Darboux transformation to the two different types of the coupled KdV system, new positon, negaton, complexiton solutions have been found in [6,7]. The positon, negaton, complexiton solutions are due to a new classification of the exact and explicit solutions of integrable models which were proposed according to the property of associated spectral parameters [8]. The authors of [9–11] provided many interesting positon, negaton, complexiton solutions for the KdV equation, the Toda lattice equation through the Wronskian and Casoratian techniques.
*
Tel.: +86 021 55274316. E-mail address: [email protected].
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.03.022
A new Wronskian formulation called the second Wronskian formulation has been presented in order to construct the rational solution, positon solution and complexiton solutions to the Boussinesq equation in [12]. Negatons, related to the negative spectral parameter, are usually expressed by hyperbolic functions and positons are expressed by means of trigonometric functions related to the positive spectral parameters. For various important integrable systems such as KdV equation, there is no non-singular positon. But negatons can be both singular and non-singular. Especially the non-singular negatons are just the usual solitons which have been studied extensively. The so-called complexiton, which is related to the complex spectral parameters, is expressed by combinations of trigonometric functions and hyperbolic functions. The known complexiton solutions for (1 + 1)-dimensional integrable functions are singular, but some types of analytical complexitons in (2 + 1) and (3 + 1) dimensions can be easily obtained because of the existence of arbitrary functions in the expression of exact solutions [13–15]. In this Letter, we study the positon, negaton and complexiton solutions to the Bogoyavlensky–Konoplechenko equation by its Darboux transformation with constant seed solution. The Bogoyavlensky–Konoplechenko (BK) equation ut + α u xxx + β u xxy + 6α uu x + 4β uu y + 4β u x ∂x−1 u y = 0,
(1.1)
where α and β are two constants, is a two-dimensional version of the KdV equation and a particular case of this equation was
H.C. Hu / Physics Letters A 373 (2009) 1750–1753
1751
introduced in [16,17]. It is pointed out that Eq. (1.1) is a kind of breaking soliton equation by a simple coefficients transformation [18] and has been described as the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis [19]. 2. Darboux transformation for the BK equation It is well known that the Darboux transformation is one of the most powerful tools to construct the exact and explicit solutions for many Lax integrable systems. For a general Lax integrable system in 2n dimensions, the Darboux transformation has been established and a broad class of the exact solutions for the Lax integrable system are given out by means of the corresponding Darboux transformation [20]. In this section, we consider the Darboux transformation for the BK equation and construct its N-step Darboux transformation in the Wronskian determinant. Noting u y = v x , the BK equation (1.1) becomes ut + α u xxx + β u xxy + 6α uu x + 4β uu y + 4β u x v = 0, u y = v x.
(2.1)
The Lax pair for Eq. (2.1) is given by the two operators
L φ = ∂x2 + u − λ φ = 0,
P φ = ∂t + 4α ∂x3 + 4β ∂x2 + u ∂ y + (6α u + 2β v )∂x + 3α u x + 3β u y φ = 0.
(2.2)
(2.3)
With the Lax pair (2.2) and (2.3), the Darboux transformation for the BK equation has been verified in Ref. [21]. Here we only briefly write the final results and construct the new positon, negaton and complexiton solutions for the BK equation based on the given Darboux transformation. Let φ1 be a nontrivial solution of the Lax pair (2.2) and (2.3) with λ = λ1 , then the new wave funcφ tion φ[1] = φx − φ1x φ and the new potential functions 1
U ≡ u [1] = u + 2(ln φ1 )xx ,
(2.4)
V ≡ v [1] = v + 2(ln φ1 )xy ,
(2.5)
also satisfy the Lax pair (2.2) and (2.3). The map φ → φ[1], u → u [1] and v → v [1] is called the first step Darboux transformation for the BK equation. Using the iteration of the Darboux transformation, one can easily obtain the n step Darboux transformation expressed by the Wronskian determinant. That is to say, if φ1 , φ2 , . . . , φn are solutions of the Lax pair (2.2) and (2.3) at λ = λ1 , . . . , λn , respectively, then
u [ N ] = u + 2 ln W (φ1 , φ2 , . . . , φn ) xx , v [ N ] = v + 2 ln W (φ1 , φ2 , . . . , φn ) xy ,
where W (φ1 , φ2 , . . . , φn ) is the usual Wronskian determinant, are also the solution of the BK equation. 3. Positon, negaton and complexiton solutions for the BK equation In this section, we study the positon, negaton and complexiton solutions of the BK equation in detail by means of the Darboux transformation given in the last section.
Fig. 1. Singular positon solution for the BK equation (1.1) with (3.2) and (3.3) at t = 0.
Substituting the seed solution {u = 0, v = 0} into the Lax pair (2.2) and (2.3), we can easily obtain the wave function in the form of
φ = F1
4βλt − y
e
4βλ
√ λ(α y −β x) β
+ F2
4βλt − y 4βλ
e
√ λ(β x−α y ) β
,
(3.1)
where F 1 and F 2 are two arbitrary functions of the indicated variable. Because of the two arbitrary functions F 1 and F 2 in (3.1), one can select the arbitrary functions properly to construct different singular or analytical positon solutions for the BK equation. Here we give out two kinds of the positon solutions below. Selecting the two arbitrary functions F 1 and F 2 as
F 1 = cn
4βλt − y 4βλ
, 0.99 ,
F 2 = sn
4βλt − y 4βλ
, 0.88
(3.2)
and the parameters
α = 2,
β = 4,
λ = 25,
(3.3)
where “cn” and “sn” are the usual Jacobi elliptic functions. Then we obtain a type of singular positon solution for the BK equation by the first step Darboux transformation and Fig. 1 shows the detail structure. When the two arbitrary functions F 1 and F 2 are chosen to be
F 1 = cn
4βλt − y
4βλ
,
F2 = e
2 sinh(
4βλt − y ) 4βλ
,
(3.4)
and
α = 2,
β = 4,
λ = 9,
(3.5)
an analytical positon solution is obtained for the BK equation (1.1) with the constant selection (3.5) and the detail structure is shown in Fig. 2. 3.2. Negaton solution, λ < 0 In order to obtain the negaton solution for the BK equation, one only need to substitute λ = −k2 into (3.1) and select the arbitrary functions F 1 and F 2 properly. Selecting the two arbitrary functions as
F 1 = F 2 = arctanh
4βλt − y 4βλ
,
(3.6)
and the constants
3.1. Positon solution, λ > 0
α = 4,
Based on the first step Darboux transformation, the positon solution for the BK equation can be constructed directly by taking the seed solution {u = 0, v = 0}.
By substituting (3.6) and (3.7) into (3.1) and (2.4), we have the negaton solution for the BK equation, which was given out graphically in Fig. 3.
β = 4,
λ = −1.
(3.7)
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H.C. Hu / Physics Letters A 373 (2009) 1750–1753
Fig. 5. Singular complexiton solution for the BK equation (1.1) with (3.9), (3.10) and (3.11) at t = 0.
3.3. Complexiton solution
Fig. 2. Analytical positon solution for the BK equation (1.1) with (3.4) and (3.5) at t = 0.
To obtain the complexiton solutions for the BK equation under the seed solution {u = 0, v = 0}, one can select the spectral parameter λ as complex number. When we fix the spectral parameter and constants as
λ = 1 + 2I ,
α = 3,
β = 1,
(3.9)
where I 2 = −1, the corresponding wave function is given by
φ=
C2 6
+
C4 60
x−
C2 2
y−
C4 2
t−
C1 + C3 + C5
2
C2 C4 + − + x + C 2 y + C 4t + C 1 + C 3 + C 5 I , 3
30
(3.10)
and {C 1 , C 2 , C 3 , C 4 , C 5 } are arbitrary constants. Then from the first Darboux transformation (2.4), one can construct the complexiton solution for the BK equation immediately by choosing the five constants. If the five constants in (3.10) are chosen as C 1 = 1, Fig. 3. Negaton solution for the BK equation (1.1) with (3.1), (3.6) and (3.7) at t = 0.
C 2 = 1,
C 3 = 4,
C 4 = 2,
C 5 = 1,
(3.11)
the singular complexiton solution for the BK equation is arrived and given out in Fig. 5. In this section, with the help of Darboux transformation for the BK equation, we have obtained three types of exact solutions, positon, negaton and complexiton solutions, by selecting the real or complex spectral parameters in the Lax pair from the constant seed solution. Due to the two arbitrary functions F 1 and F 2 in the wave function, we can construct the analytical or singular positon, negaton and complexiton solutions by choosing the two arbitrary functions properly. 4. Summary and discussion
Fig. 4. Second type of the analytical negaton solution for the BK equation (1.1) with (3.1), (3.7) and (3.8) at t = 0.
When we choose the arbitrary functions F 1 and F 2 as
F 1 = arctanh
4βλt − y 4βλ
,
F 2 = arcsinh
4βλt − y 4βλ
,
(3.8)
then another type of the negaton solution is obtained with the constant selection (3.7) and given out directly in Fig. 4.
In this Letter, starting from the trivial constant seed solution, the positon, negaton and complexiton solutions for the BK equation are considered from the Darboux transformation by selecting the real or complex spectral parameters in its Lax pair. It should be pointed out that the obtained positon, negaton and complexiton solutions are analytical or singular due to the proper selection of the two arbitrary functions in the wave function. The detail structures of the positon, negaton and complexiton solutions are given out analytically and graphically. As we know, a family of solutions of the BK equation have been constructed in Ref. [21] which was expressed by exponential function. But the analytical or singular positon, negaton and complexiton solutions for the BK equation are firstly given out to our knowledge. We believe that these new positon, negaton and complexiton solutions for the BK equation will help us to learn the internal property of the nonlinear evolution equation deeply.
H.C. Hu / Physics Letters A 373 (2009) 1750–1753
Acknowledgements The author would like to thank Prof. S.Y. Lou for helpful discussions and the anonymous referees very much for their valuable comments and suggestions. The work was supported by the National Natural Science Foundation of China (No. 10601033). References [1] [2] [3] [4]
V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer, 1991. M. Wadati, H. Sanuki, K. Konno, Prog. Theor. Phys. 53 (1975) 419. K. Konno, M. Wadati, Prog. Theor. Phys. 52 (1975) 1652. C.H. Gu, H.S. Hu, Z.X. Zhou, Darboux Transformation in Soliton Theory and Its Geometric Applications, Shanghai Science and Technology Publishing House, Shanghai, 2005. [5] H.C. Hu, Q.P. Liu, Chaos Solitons Fractals 17 (2003) 921.
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
1753
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