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Comment on: “Double sub-equation method for complexiton solutions of nonlinear partial differential equations”
Applied Mathematics and Computation 246 (2014) 597–598
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Comment on: ‘‘Double sub-equation method for complexiton solutions of nonlinear partial differential equations’’ Hong-Zhun Liu ⇑, Sen Lin, Xiao-Quan Sun Zhijiang College, Zhejiang University of Technology, Hangzhou 310024, PR China
a r t i c l e
i n f o
a b s t r a c t In this article, we analyze the paper of Chen et al. (2013). Using the double sub-equation method, Chen et al. have constructed complexiton solutions of the KdV equation and the MKdV equation. We conclude that all Chen’s complexiton solutions can be simplified into trivial constant solutions. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Complexiton solution Double sub-equation method Common error
In [1], Chen et al. investigated the complexiton solutions of the KdV equation in the form
ut þ auux þ buxxx ¼ 0;
ð1Þ
and of the mKdV equation in the form
v t þ av 2 v x þ bv xxx ¼ 0;
ð2Þ
where a and b are nonzero constants. In Example 1 in [1], namely the KdV equation (1), Chen’s complexiton solution is expressed in the form
u ¼ a0 þ
a1 FðnÞ þ a2 GðgÞ a3 FðnÞ2 þ a4 FðnÞGðgÞ þ a5 GðgÞ2 ; þ l0 þ l1 FðnÞ þ l2 GðgÞ ½l0 þ l1 FðnÞ þ l2 GðgÞ2
ð3Þ
where n ¼ k1 ðx c1 tÞ; g ¼ k2 ðx c2 tÞ, and
F 02 ðnÞ ¼ A1 þ B1 F 2 ðnÞ þ C 1 F 4 ðnÞ; G02 ðgÞ ¼ A2 þ B2 G2 ðgÞ:
ð4Þ
Chen et al. got the following two cases: Case 1
l0 ¼ 0; a1 ¼
a4 l1 2a3 l2 þ a2 l21
l1 l2
;
a5 ¼
l2 ða4 l1 a3 l2 Þ ; l21
ð5Þ
with arbitrary nonzero constants a0 ; a2 ; a3 ; a4 ; l1 ; l2 ; A1 ; A2 ; k1 ; k2 ; c1 and c2 . Case 2
l0 ¼ A2 ¼ 0; a1 ¼
a4 l2 þ a2 l1 l2 þ 2a5 l1
l22
⇑ Corresponding author. E-mail address: [email protected] (H.-Z. Liu). http://dx.doi.org/10.1016/j.amc.2014.08.024 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
;
a3 ¼
l1 ða5 l1 a4 l2 Þ ; l22
ð6Þ
598
H.-Z. Liu et al. / Applied Mathematics and Computation 246 (2014) 597–598
with arbitrary nonzero constants a0 ; a2 ; a4 ; a5 ; l1 ; l2 ; A1 ; k1 ; k2 ; c1 and c2 . Then after substituting (5) and (6) into (3) respectively, and using some special solutions of (4), the authors presented twenty complexiton solutions and claimed these solutions were not among the elementary function solutions obtained in references. In the beginning, we point out that we replace a3 ¼ l1 ða5 ll12 a4 l2 Þ in Chen’s Case 2 by a3 ¼ l1 ða5 ll12a4 l2 Þ in solution (6). 2
2
The reasons are shown as follows. On one hand, we have verified that Chen’s Case 2 is not admitted by the original differential Eq. (1). Due to limited space, here we omit its details. On the other hand, if l0 ¼ A2 ¼ 0, we can directly obtain solution (6) by solving involved algebraic equations with the aid of Maple. We believe that the redundant negative sign in a3 in Chen’s Case 2 is just a typo, and our replacement is necessary and proper. However, if we substitute (5) into (3), we find that the right hand side of (3) can be exactly simplified to be a constant, namely
u ¼ a0 þ
a2 l21 þ a4 l1 a3 l2
l2 l21
ð7Þ
:
And if we substitute (6) into (3), we find that the right hand side of (3) can also be simplified to a constant, namely
u ¼ a0 þ
a2 l2 þ a5
l22
ð8Þ
:
Hence we conclude that all the solution u1 ; u2 ; . . . ; u20 presented by Chen et al. [1] in their proper forms are trivial constant solutions. We also checked Example 2 in [1], namely the mKdV Eq. (2), and the same conclusion can be obtained by similar calculation. Here we present the results. The first case of Example 2 in [1] in its proper form is
l0 ¼ A2 ¼ 0; a1 ¼
a2 l1 l2 a4 l2 þ 2a5 l1
l22
;
a3 ¼
a4 l2 a5 l1
l22
l1 ;
ð9Þ
with arbitrary nonzero constants a0 ; a2 ; a4 ; a5 ; A1 ; l1 ; l2 ; k1 ; k2 ; c1 and c2 . In this case, we have
v ¼ a0 þ
a1 FðnÞ þ a2 GðgÞ a3 FðnÞ2 þ a4 FðnÞGðgÞ þ a5 GðgÞ2 a2 l2 þ a5 ¼ a0 þ : þ 2 l0 þ l1 FðnÞ þ l2 GðgÞ l22 ½l0 þ l1 FðnÞ þ l2 GðgÞ
ð10Þ
And the second case of Example 2 in [1] reads
l0 ¼ 0; a2 ¼
2a3 l2 a4 l1 þ a1 l1 l2
l21
;
a5 ¼
a4 l1 a3 l2
l21
l2 ;
ð11Þ
with arbitrary nonzero constants a0 ; a1 ; a3 ; a4 ; A1 ; A2 ; l1 ; l2 ; k1 ; k2 ; c1 and c2 . In this case, we have
v ¼ a0 þ
a1 FðnÞ þ a2 GðgÞ a3 FðnÞ2 þ a4 FðnÞGðgÞ þ a5 GðgÞ2 a3 þ a1 l1 ¼ a0 þ : þ 2 l0 þ l1 FðnÞ þ l2 GðgÞ l21 ½l0 þ l1 FðnÞ þ l2 GðgÞ
ð12Þ
Thus we conclude that all complexiton solutions in [1] in their proper forms can be simplified into certain trivial constant solutions and Chen et al. does not obtain any nontrivial complexiton solution in virtue of double sub-equation method for these two examples. In the end, we mention that all Maple calculations are available at http://pan.baidu.com/s/1dD3pe5F if necessary. Acknowledgments The work is supported by the NSF of China (11201427) and the Scientific Research Fund of Zhejiang Provincial Education Department (Y201432746). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.amc.2014.08.024. Reference [1] H.T. Chen, S.H. Yang, W.X. Ma, Double sub-equation method for complexiton solutions of nonlinear partial differential equations, Appl. Math. Comput. 219 (2013) 4775–4781.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Comment on: ‘‘Double sub-equation method for complexiton solutions of nonlinear partial differential equations’’ Hong-Zhun Liu ⇑, Sen Lin, Xiao-Quan Sun Zhijiang College, Zhejiang University of Technology, Hangzhou 310024, PR China
a r t i c l e
i n f o
a b s t r a c t In this article, we analyze the paper of Chen et al. (2013). Using the double sub-equation method, Chen et al. have constructed complexiton solutions of the KdV equation and the MKdV equation. We conclude that all Chen’s complexiton solutions can be simplified into trivial constant solutions. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Complexiton solution Double sub-equation method Common error
In [1], Chen et al. investigated the complexiton solutions of the KdV equation in the form
ut þ auux þ buxxx ¼ 0;
ð1Þ
and of the mKdV equation in the form
v t þ av 2 v x þ bv xxx ¼ 0;
ð2Þ
where a and b are nonzero constants. In Example 1 in [1], namely the KdV equation (1), Chen’s complexiton solution is expressed in the form
u ¼ a0 þ
a1 FðnÞ þ a2 GðgÞ a3 FðnÞ2 þ a4 FðnÞGðgÞ þ a5 GðgÞ2 ; þ l0 þ l1 FðnÞ þ l2 GðgÞ ½l0 þ l1 FðnÞ þ l2 GðgÞ2
ð3Þ
where n ¼ k1 ðx c1 tÞ; g ¼ k2 ðx c2 tÞ, and
F 02 ðnÞ ¼ A1 þ B1 F 2 ðnÞ þ C 1 F 4 ðnÞ; G02 ðgÞ ¼ A2 þ B2 G2 ðgÞ:
ð4Þ
Chen et al. got the following two cases: Case 1
l0 ¼ 0; a1 ¼
a4 l1 2a3 l2 þ a2 l21
l1 l2
;
a5 ¼
l2 ða4 l1 a3 l2 Þ ; l21
ð5Þ
with arbitrary nonzero constants a0 ; a2 ; a3 ; a4 ; l1 ; l2 ; A1 ; A2 ; k1 ; k2 ; c1 and c2 . Case 2
l0 ¼ A2 ¼ 0; a1 ¼
a4 l2 þ a2 l1 l2 þ 2a5 l1
l22
⇑ Corresponding author. E-mail address: [email protected] (H.-Z. Liu). http://dx.doi.org/10.1016/j.amc.2014.08.024 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
;
a3 ¼
l1 ða5 l1 a4 l2 Þ ; l22
ð6Þ
598
H.-Z. Liu et al. / Applied Mathematics and Computation 246 (2014) 597–598
with arbitrary nonzero constants a0 ; a2 ; a4 ; a5 ; l1 ; l2 ; A1 ; k1 ; k2 ; c1 and c2 . Then after substituting (5) and (6) into (3) respectively, and using some special solutions of (4), the authors presented twenty complexiton solutions and claimed these solutions were not among the elementary function solutions obtained in references. In the beginning, we point out that we replace a3 ¼ l1 ða5 ll12 a4 l2 Þ in Chen’s Case 2 by a3 ¼ l1 ða5 ll12a4 l2 Þ in solution (6). 2
2
The reasons are shown as follows. On one hand, we have verified that Chen’s Case 2 is not admitted by the original differential Eq. (1). Due to limited space, here we omit its details. On the other hand, if l0 ¼ A2 ¼ 0, we can directly obtain solution (6) by solving involved algebraic equations with the aid of Maple. We believe that the redundant negative sign in a3 in Chen’s Case 2 is just a typo, and our replacement is necessary and proper. However, if we substitute (5) into (3), we find that the right hand side of (3) can be exactly simplified to be a constant, namely
u ¼ a0 þ
a2 l21 þ a4 l1 a3 l2
l2 l21
ð7Þ
:
And if we substitute (6) into (3), we find that the right hand side of (3) can also be simplified to a constant, namely
u ¼ a0 þ
a2 l2 þ a5
l22
ð8Þ
:
Hence we conclude that all the solution u1 ; u2 ; . . . ; u20 presented by Chen et al. [1] in their proper forms are trivial constant solutions. We also checked Example 2 in [1], namely the mKdV Eq. (2), and the same conclusion can be obtained by similar calculation. Here we present the results. The first case of Example 2 in [1] in its proper form is
l0 ¼ A2 ¼ 0; a1 ¼
a2 l1 l2 a4 l2 þ 2a5 l1
l22
;
a3 ¼
a4 l2 a5 l1
l22
l1 ;
ð9Þ
with arbitrary nonzero constants a0 ; a2 ; a4 ; a5 ; A1 ; l1 ; l2 ; k1 ; k2 ; c1 and c2 . In this case, we have
v ¼ a0 þ
a1 FðnÞ þ a2 GðgÞ a3 FðnÞ2 þ a4 FðnÞGðgÞ þ a5 GðgÞ2 a2 l2 þ a5 ¼ a0 þ : þ 2 l0 þ l1 FðnÞ þ l2 GðgÞ l22 ½l0 þ l1 FðnÞ þ l2 GðgÞ
ð10Þ
And the second case of Example 2 in [1] reads
l0 ¼ 0; a2 ¼
2a3 l2 a4 l1 þ a1 l1 l2
l21
;
a5 ¼
a4 l1 a3 l2
l21
l2 ;
ð11Þ
with arbitrary nonzero constants a0 ; a1 ; a3 ; a4 ; A1 ; A2 ; l1 ; l2 ; k1 ; k2 ; c1 and c2 . In this case, we have
v ¼ a0 þ
a1 FðnÞ þ a2 GðgÞ a3 FðnÞ2 þ a4 FðnÞGðgÞ þ a5 GðgÞ2 a3 þ a1 l1 ¼ a0 þ : þ 2 l0 þ l1 FðnÞ þ l2 GðgÞ l21 ½l0 þ l1 FðnÞ þ l2 GðgÞ
ð12Þ
Thus we conclude that all complexiton solutions in [1] in their proper forms can be simplified into certain trivial constant solutions and Chen et al. does not obtain any nontrivial complexiton solution in virtue of double sub-equation method for these two examples. In the end, we mention that all Maple calculations are available at http://pan.baidu.com/s/1dD3pe5F if necessary. Acknowledgments The work is supported by the NSF of China (11201427) and the Scientific Research Fund of Zhejiang Provincial Education Department (Y201432746). Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.amc.2014.08.024. Reference [1] H.T. Chen, S.H. Yang, W.X. Ma, Double sub-equation method for complexiton solutions of nonlinear partial differential equations, Appl. Math. Comput. 219 (2013) 4775–4781.