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To research the bounded damped oscillatory solutions of nonlinear evolution equation, we choose the Fisher equation as an example. The solutions with negative amplitudes of Fisher equation may become meaningful in the context of nonscalar models describing excitable media (e.g. the Belousov-Zhabotinsky (BZ) reaction). The theory of planar dynamical systems is used to study the existence conditions of bounded traveling wave solutions of Fisher equation. The bounded approximate damped oscillatory analytic solution is given by using LS method and linearization method. And its error is also estimated.
[1] Ablowitz M J, Clarkson P A 1991 Solitons, Nonlinear evolution equations and inverse scattering transform (Cambridge: Cambridge University Press)
[2] Hirota R A 1974 Progr. Theor. Phys. 52 1498
[3] Wang M L 1995 Phys. Lett. A 199 169
[4] Li Z B 2002 Acta Mathematica Scientia 22B 138
[5] Li X Z, Zhang X Y, Zhao L Y 2007 J. Henan Univ. Sci. Techn. 28(2) 70 (in Chinese) [李向正, 张小勇, 赵丽英 2007 河南科技大学学报 28(2) 70]
[6] Lan H B, Wang K L 1990 J. Phys. A: Math. Gen. 23 4097
[7] Fan E G, Zhang H Q 1998 Phys. Lett. A 246 403
[8] Liu S K, Fu Z T, Liu S D, Zhao Q 2001 Phys. Lett. A 289 69
[9] Wang M L, Zhou Y B 2003 Phys. Lett. A 318 84
[10] Zhou Y B, Wang M L, Wang Y M 2003 Phys. Lett. A 308 31
[11] 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 53 4045]
[12] Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372(4) 417
[13] Wang M L, Zhang J L, Li X Z 2008 Appl. Math. Comp. 206 321
[14] Zhao Y L, Liu Y P, Li Z B 2010 Chin. Phys. B 19 030306
[15] Li X Z, Zhang W G, Yuan S L 2010 Acta Phys. Sin. 59 744 (in Chinese) [李向正, 张卫国, 原三领 2010 59 744]
[16] Zhang W G, Bian L Y, Zhao Y 2010 Proceed. Royal Soc. Edinburgh 140A 241
[17] Zhao Y, Zhang W G 2010 Acta Math. Appl. Sinica 26(3) 415
[18] Fisher R A 1937 Ann. Eugen. 7 353
[19] Bi H, Lu T C 2004 J. Shandong Univ. (Sci. Edit.) 39(2) 41 (in Chinese) [闭海, 鲁统超 2004 山东大学学报(理学版) 39(2) 41]
[20] Tao T, Zhang W G, Feng L P 2004 J. Univ. Shanghai Sci. Techn. 26(2) 111 (in Chinese) [陶涛, 张卫国, 冯丽萍 2004 上海理工大学学报 26(2) 111]
[21] Pavel K B, John J T 1999 SIAM J. Appl. Math. 60(2) 371
[22] Tan Y, Xu H, Liao S J 2007 Chaos, Solitons and Fractals 31 462
[23] Ablowitz M J, Zeppetella A 1979 Bull. Math. Biol. 41 835
[24] Ma Z E, Zhou Y C 2001 Qualitative and Stability Methods of Ordinary Differential Equations (Beijing: Science Press) (in Chinese) [马知恩, 周义仓编著 2001 常微分方程定性与稳定性方法(北京:科学出版社)]
[25] Liang K M 2006 Methods of Mathematical Physics (Beijing: Higher Education Press) p206 (in Chinese) [梁昆淼 2006 数学物理方法(北京:高等教育出版社) 第206页]
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[1] Ablowitz M J, Clarkson P A 1991 Solitons, Nonlinear evolution equations and inverse scattering transform (Cambridge: Cambridge University Press)
[2] Hirota R A 1974 Progr. Theor. Phys. 52 1498
[3] Wang M L 1995 Phys. Lett. A 199 169
[4] Li Z B 2002 Acta Mathematica Scientia 22B 138
[5] Li X Z, Zhang X Y, Zhao L Y 2007 J. Henan Univ. Sci. Techn. 28(2) 70 (in Chinese) [李向正, 张小勇, 赵丽英 2007 河南科技大学学报 28(2) 70]
[6] Lan H B, Wang K L 1990 J. Phys. A: Math. Gen. 23 4097
[7] Fan E G, Zhang H Q 1998 Phys. Lett. A 246 403
[8] Liu S K, Fu Z T, Liu S D, Zhao Q 2001 Phys. Lett. A 289 69
[9] Wang M L, Zhou Y B 2003 Phys. Lett. A 318 84
[10] Zhou Y B, Wang M L, Wang Y M 2003 Phys. Lett. A 308 31
[11] 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 53 4045]
[12] Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372(4) 417
[13] Wang M L, Zhang J L, Li X Z 2008 Appl. Math. Comp. 206 321
[14] Zhao Y L, Liu Y P, Li Z B 2010 Chin. Phys. B 19 030306
[15] Li X Z, Zhang W G, Yuan S L 2010 Acta Phys. Sin. 59 744 (in Chinese) [李向正, 张卫国, 原三领 2010 59 744]
[16] Zhang W G, Bian L Y, Zhao Y 2010 Proceed. Royal Soc. Edinburgh 140A 241
[17] Zhao Y, Zhang W G 2010 Acta Math. Appl. Sinica 26(3) 415
[18] Fisher R A 1937 Ann. Eugen. 7 353
[19] Bi H, Lu T C 2004 J. Shandong Univ. (Sci. Edit.) 39(2) 41 (in Chinese) [闭海, 鲁统超 2004 山东大学学报(理学版) 39(2) 41]
[20] Tao T, Zhang W G, Feng L P 2004 J. Univ. Shanghai Sci. Techn. 26(2) 111 (in Chinese) [陶涛, 张卫国, 冯丽萍 2004 上海理工大学学报 26(2) 111]
[21] Pavel K B, John J T 1999 SIAM J. Appl. Math. 60(2) 371
[22] Tan Y, Xu H, Liao S J 2007 Chaos, Solitons and Fractals 31 462
[23] Ablowitz M J, Zeppetella A 1979 Bull. Math. Biol. 41 835
[24] Ma Z E, Zhou Y C 2001 Qualitative and Stability Methods of Ordinary Differential Equations (Beijing: Science Press) (in Chinese) [马知恩, 周义仓编著 2001 常微分方程定性与稳定性方法(北京:科学出版社)]
[25] Liang K M 2006 Methods of Mathematical Physics (Beijing: Higher Education Press) p206 (in Chinese) [梁昆淼 2006 数学物理方法(北京:高等教育出版社) 第206页]
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