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Based on the ideas of the hyperbola function expansion method, we obtained some analytical solutions of the combined KdV-mKdV (cKdV) equations by introducing new expansion functions.One of the single soliton solutions has the kink-antikink structure, and this solution reduces to the kink-like solution and the bell-like solution under different limitations. Theoretical analysis shows that the cKdV equation has both propagated-type and non-propagated-type solitary wave solutions. We also investigated the stability of the single solitary wave solution with double kinks numerically. The results indicate that the solution may be stable or unstable, depending on different sets of parameters.
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Keywords:
- cKdV equation /
- single soliton solution with double kinks /
- stability
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[2] Wang M L, Zhou Y B, Li Z B 1996 Phys. Lett. A 216 67
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[5] Zhang G X, Li Z B, Duan Y S 2000 Sci. China (Series A) 30 1103 (in Chinese) [张桂戌、李志斌、段一士 2000 中国科学,A辑 30 1103]
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[12] Liao S J, Chen C, Xu H 2006 Beyond Perturbation:Introduction to the Homotopy Analysis Method (Beijing:Science Press) (in Chinese) [廖世俊、陈 晨、徐 航 2006 超越摄动:同伦分析方法导论(北京:科学出版社)]
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[15] Yang H J, Shi Y R, Duan W S, Lü K P 2007 Acta Phys. Sin. 56 3064 (in Chinese) [杨红娟、石玉仁、段文山、吕克璞 2007 物理学报 56 3064]
[16] Shi Y R, Wang Y H, Yang H J, Duan W S 2007 Acta Phys. Sin. 56 6791 (in Chinese) [石玉仁、汪映海、杨红娟、段文山 2007 物理学报 56 6791]
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[19] Zhou Y B, Wang M L, Miao T D 2004 Phys. Lett. A 323 77
[20] Zhang J L, Wang Y M, Wang M L 2003 Acta Phys. Sin. 52 1574 (in Chinese) [张金良、王跃明、王明亮 2003 物理学报 52 1574]
[21] Li X Z, Zhang J L, Wang Y M, Wang M L 2004 Acta Phys. Sin. 53 4045 (in Chinese) [李向正、张金良、王跃明、王明亮 2004 物理学报 53 4045]
[22] Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 13 (in Chinese) [套格图桑、斯仁道尔吉 2006 物理学报 55 13]
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[24] Li X Z, Li X Y, Zhao L Y, Zhang J L 2008 Acta Phys. Sin. 57 2203 (in Chinese) [李向正、李修勇、赵丽英、张金良 2008 物理学报 57 2203]
[25] Tang X Y, Liang Z F 2006 Phys. Lett. A 351 398
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[27] Ma H C, Ge D J, Yu Y D 2008 Chin. Phys. B 17 1448
[28] Ying J P, Lou S Y 2003 Chin. Phys. Lett. 20 1448
[29] Fang J P, Zheng C L, Zhu J M 2005 Commun. Theor. Phys. 44 203
[30] Fang J P, Zheng C L 2005 Chin. Phys. 14 670
[31] Tappert F D, Varma C M 1970 Phys. Rev. Lett. 25 1108
[32] Wadati M 1975 J. Phys. Soc. Jpn. 38 673
[33] Konno K, Ichikawa Y H 1974 J. Phys. Soc. Jpn. 37 1631
[34] Narayanamurti V, Varma C M 1970 Phys. Rev. Lett. 25 1105
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[1] Wang M L 1995 Phys. Lett. A 199 169
[2] Wang M L, Zhou Y B, Li Z B 1996 Phys. Lett. A 216 67
[3] Fan E G, Zhang H Q 1998 Acta Phys. Sin. 47 353 (in Chinese) [范恩贵、张鸿庆 1998 物理学报 47 353]
[4] Fan E G 2000 Acta Phys. Sin. 49 1409 (in Chinese) [范恩贵 2000 物理学报 49 1409]
[5] Zhang G X, Li Z B, Duan Y S 2000 Sci. China (Series A) 30 1103 (in Chinese) [张桂戌、李志斌、段一士 2000 中国科学,A辑 30 1103]
[6] Lü K P, Shi Y R, Duan W S, Zhao J B 2001 Acta Phys. Sin. 50 2074 (in Chinese) [吕克璞、石玉仁、段文山、赵金保 2001 物理学报 50 2074]
[7] Shi Y R, Lü K P, Duan W S, Hong X R, Zhao J B, Yang H J 2003 Acta Phys. Sin. 52 267 (in Chinese) [石玉仁、吕克璞、段文山、洪学仁、赵金保、杨红娟 2003 物理学报 52 267]
[8] Huang D J, Zhang H Q 2004 Acta Phys. Sin. 53 2434 (in Chinese) [黄定江、张鸿庆 2004 物理学报 53 2434]
[9] Liu S K, Fu Z T, Liu S D, Zhao Q 2001 Acta Phys. Sin. 50 2068 (in Chinese) [刘式适、傅遵涛、刘式达、赵 强 2001 物理学报 50 2068]
[10] Liu S K, Fu Z T, Liu S D, Zhao Q 2002 Acta Phys. Sin. 51 10 (in Chinese) [刘式适、傅遵涛、刘式达、赵 强 2002 物理学报 51 10]
[11] Liu S D, Fu Z T, Liu S K, Zhao Q 2002 Acta Phys. Sin. 51 718 (in Chinese) [刘式达、傅遵涛、刘式适、赵 强 2002 物理学报 51 718]
[12] Liao S J, Chen C, Xu H 2006 Beyond Perturbation:Introduction to the Homotopy Analysis Method (Beijing:Science Press) (in Chinese) [廖世俊、陈 晨、徐 航 2006 超越摄动:同伦分析方法导论(北京:科学出版社)]
[13] Liao S J, Pop I 2004 Int. J. Non-Linear Mech. 39 271
[14] Shi Y R, Xu X J, Wu Z X, Wang Y H, Yang H J, Duan W S, Lü K P 2006 Acta Phys. Sin. 55 1555 (in Chinese) [石玉仁、许新建、吴枝喜、汪映海、杨红娟、段文山、吕克璞 2006 物理学报 55 1555]
[15] Yang H J, Shi Y R, Duan W S, Lü K P 2007 Acta Phys. Sin. 56 3064 (in Chinese) [杨红娟、石玉仁、段文山、吕克璞 2007 物理学报 56 3064]
[16] Shi Y R, Wang Y H, Yang H J, Duan W S 2007 Acta Phys. Sin. 56 6791 (in Chinese) [石玉仁、汪映海、杨红娟、段文山 2007 物理学报 56 6791]
[17] Shi Y R, Yang H J 2010 Acta Phys. Sin. 59 67 (in Chinese) [石玉仁、杨红娟 2009 物理学报 59 67]
[18] Wang M L, Zhou Y B, Li Z B 2003 Phys. Lett. A 318 84
[19] Zhou Y B, Wang M L, Miao T D 2004 Phys. Lett. A 323 77
[20] Zhang J L, Wang Y M, Wang M L 2003 Acta Phys. Sin. 52 1574 (in Chinese) [张金良、王跃明、王明亮 2003 物理学报 52 1574]
[21] Li X Z, Zhang J L, Wang Y M, Wang M L 2004 Acta Phys. Sin. 53 4045 (in Chinese) [李向正、张金良、王跃明、王明亮 2004 物理学报 53 4045]
[22] Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 13 (in Chinese) [套格图桑、斯仁道尔吉 2006 物理学报 55 13]
[23] Taogetusang, Sirendaoerji 2006 Acta Phys. Sin. 55 3246 (in Chinese) [套格图桑、斯仁道尔吉 2006 物理学报 55 3246]
[24] Li X Z, Li X Y, Zhao L Y, Zhang J L 2008 Acta Phys. Sin. 57 2203 (in Chinese) [李向正、李修勇、赵丽英、张金良 2008 物理学报 57 2203]
[25] Tang X Y, Liang Z F 2006 Phys. Lett. A 351 398
[26] Zhang S L, Lou S Y, Qu C Z 2006 Chin. Phys. 15 2765
[27] Ma H C, Ge D J, Yu Y D 2008 Chin. Phys. B 17 1448
[28] Ying J P, Lou S Y 2003 Chin. Phys. Lett. 20 1448
[29] Fang J P, Zheng C L, Zhu J M 2005 Commun. Theor. Phys. 44 203
[30] Fang J P, Zheng C L 2005 Chin. Phys. 14 670
[31] Tappert F D, Varma C M 1970 Phys. Rev. Lett. 25 1108
[32] Wadati M 1975 J. Phys. Soc. Jpn. 38 673
[33] Konno K, Ichikawa Y H 1974 J. Phys. Soc. Jpn. 37 1631
[34] Narayanamurti V, Varma C M 1970 Phys. Rev. Lett. 25 1105
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