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Based on the idea of the hyperbola function expansion method, some analytical solutions of the modified Korteweg-de Vries (mKdV) equation are obtained by introducing new expansion functions. One of the single soliton solutions has a kink-antikink structure and it reduces to a kink-like solution and bell-like solution under different limitations. The stability of the single soliton solution with double kinks is investigated numerically. The results indicate that the soliton is stable under different disturbances.
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Keywords:
- mKdV equation /
- single soliton solution with double kinks /
- stability
[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 47 353]
[4] Fan E G 2000 Acta Phys. Sin. 49 1409 (in Chinese)[范恩贵 2000 49 1409]
[5] Zhang G X, Li Z B, Duan Y S 2000 Science In China (Series A) 30 1103 [张桂戌、 李志斌、 段一士 2000中国科学(A) 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 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 52 267]
[8] Huang D J, Zhang H Q 2004 Acta Phys. Sin. 53 2434 (in Chinese) [黄定江、 张鸿庆 2004 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 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 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 51 718]
[12] Liao S J,Chen C, Xu H 2006 Beyond Perturbation:Introduction to the Homotopy Analysis Method (BeiJing:Science Press) (in Chineses)[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 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 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 56 6791]
[17] Shi Y R, Yang H J 2010 Acta Phys. Sin. 59 67 (in Chinese) [石玉仁、 杨红娟2010 59 67]
[18] M Wadati 1973 J. Phys. Soc. Japan 34 1289
[19] R M Miura 1968 J. Math. Phys. 9 1202
[20] E F El-Shamy 2005 Chaos,Solitons and Fractals 25 665
[21] G P Agrawal 1989 Nonlinear Fibre Optics (Academic Press, Boston)
[22] Ni W S, Wei R J 1997 Solitary Wave in Water Trough,Shanghai Scientific and Technological (Shanghai:Education Publishing House) (in Chinese) [倪皖荪、 魏荣爵 1997 水槽中的孤波 (上海:上海科技教育出版社)]
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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 47 353]
[4] Fan E G 2000 Acta Phys. Sin. 49 1409 (in Chinese)[范恩贵 2000 49 1409]
[5] Zhang G X, Li Z B, Duan Y S 2000 Science In China (Series A) 30 1103 [张桂戌、 李志斌、 段一士 2000中国科学(A) 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 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 52 267]
[8] Huang D J, Zhang H Q 2004 Acta Phys. Sin. 53 2434 (in Chinese) [黄定江、 张鸿庆 2004 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 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 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 51 718]
[12] Liao S J,Chen C, Xu H 2006 Beyond Perturbation:Introduction to the Homotopy Analysis Method (BeiJing:Science Press) (in Chineses)[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 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 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 56 6791]
[17] Shi Y R, Yang H J 2010 Acta Phys. Sin. 59 67 (in Chinese) [石玉仁、 杨红娟2010 59 67]
[18] M Wadati 1973 J. Phys. Soc. Japan 34 1289
[19] R M Miura 1968 J. Math. Phys. 9 1202
[20] E F El-Shamy 2005 Chaos,Solitons and Fractals 25 665
[21] G P Agrawal 1989 Nonlinear Fibre Optics (Academic Press, Boston)
[22] Ni W S, Wei R J 1997 Solitary Wave in Water Trough,Shanghai Scientific and Technological (Shanghai:Education Publishing House) (in Chinese) [倪皖荪、 魏荣爵 1997 水槽中的孤波 (上海:上海科技教育出版社)]
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