Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations

doi:10.1088/1674-1056/23/3/030506

中国物理B ›› 2014, Vol. 23 ›› Issue (3): 30506-030506.doi: 10.1088/1674-1056/23/3/030506

Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations

杨云青^a, 王云虎^b, 李昕^c, 程雪苹^a

a School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316004, China;
b Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China;
c School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China

收稿日期:2013-07-09 修回日期:2013-08-08 出版日期:2014-03-15 发布日期:2014-03-15
基金资助:
Project supported by the Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LQ12A01008 and LY12A01010).

Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations

Yang Yun-Qing (杨云青)^a, Wang Yun-Hu (王云虎)^b, Li Xin (李昕)^c, Cheng Xue-Ping (程雪苹)^a

a School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316004, China;
b Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China;
c School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China

Received:2013-07-09 Revised:2013-08-08 Online:2014-03-15 Published:2014-03-15
Contact: Yang Yun-Qing E-mail:yqyang@amss.ac.cn
Supported by:
Project supported by the Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LQ12A01008 and LY12A01010).

摘要/Abstract

摘要： We extend the method of constructing Bäcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg–de Vries (KdV) equations as examples, their Bäcklund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Especially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.

关键词: Bä, cklund transformation, Lax pair, conservation law, Cole–, Hopf transformation

Abstract: We extend the method of constructing Bäcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg–de Vries (KdV) equations as examples, their Bäcklund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Especially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.

Key words: Bäcklund transformation, Lax pair, conservation law, Cole–Hopf transformation

中图分类号: (Solitons)

05.45.Yv

02.30.Ik (Integrable systems)

杨云青, 王云虎, 李昕, 程雪苹. Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations[J]. 中国物理B, 2014, 23(3): 30506-030506.

Yang Yun-Qing (杨云青), Wang Yun-Hu (王云虎), Li Xin (李昕), Cheng Xue-Ping (程雪苹). Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations[J]. Chin. Phys. B, 2014, 23(3): 30506-030506.

参考文献 32

[1]	Rogers C and Shadwick W F 1982 Bäcklund Transformations and Their Applications (New York: Academic Press) pp. 12–22
[2]	Miura R M 1976 Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications (Berlin: Springer-Verlag) pp. 162–183
[3]	Wahlquist H D and Estabrook F B 1975 J. Math. Phys. 16 1
[4]	Fan E G, Zhang H Q and Lin G 1998 Acta Phys. Sin. (Overseas Edition) 7 649
[5]	Yan Z Y and Zhang H Q 1999 Acta Phys. Sin. (Overseas Edition) 8 889
[6]	Chen H H 1974 Phys. Rev. Lett. 33 925
[7]	Weiss J 1983 J. Math. Phys. 24 1405
[8]	Wadati M, Sanuki H and Konno K 1975 Prog. Theor. Phys. 53 419
[9]	Clainrin J 1902 Ann. Sci. Ecole. Norm. Sup. 3 1
[10]	Reid W T 1972 Riccati Differential Equations (New York: Academic Press) pp. 9–28
[11]	Grundland A M and Levi D 1999 J. Phys. A: Math. Gen. 32 3931
[12]	Chan W L and Li K S 1989 J. Math. Phys. 30 2521
[13]	Zhang C Y, Gao Y T, Meng X H, Li J, Xu T, Wei G M and Zhu H W 2006 J. Phy. A: Math. Gen. 39 14353
[14]	Nirmala N, Vedan M J and Baby B V 1986 J. Math. Phys. 27 2640
[15]	Hirota R 1979 J. Phys. Soc. Jpn. 46 1681
[16]	Lou S Y and Ruan H Y 1992 Acta Phys. Sin. 41 182 (in Chinese)
[17]	Zhang D J 2007 Chin. Phys. Lett. 24 3021
[18]	Zhang Y, Cheng Z L and Hao X H 2012 Chin. Phys. B 21 120203
[19]	Wahlquist H D and Estabrook F B 1973 Phys. Rev. Lett. 31 1386
[20]	Frantzeskakis D J, Proukakis N P and Kevrekidis P G 2004 Phys. Rev. A 70 015601
[21]	Polyanin A D and Zaitsev V E 2003 Handcook of Exact Solutions for Ordinary Differential Equations (Boca Raton: CRC Press) pp. 392–398
[22]	Hirota R 2004 The Direct Methods in Soliton Theory (Cambridge: Cambridge University Press) pp. 18–40
[23]	Fan E G 2011 Phys. Lett. A 375 493
[24]	Joshi N 1987 Phys. Lett. A 125 456
[25]	Maxon S and Viecelli J 1974 Phys. Fluids 17 1614
[26]	Hershkowitz N and Romesser T 1974 Phys. Rev. Lett. 32 581
[27]	Calogero F and Degasperis A 1978 Lett. Nuovo Cimento 23 155
[28]	Nakamura A 1980 J. Phys. Soc. Jpn. 49 2380
[29]	Steeb W H, Klole M, Spieker B M and Oevel W 1983 J. Phy. A: Math. Gen. 16 L447
[30]	Yu F J 2012 Chin. Phys. B 21 010201
[31]	Ames W 1968 Nonlinear Ordinary Differential Equations in Transport Processes (New York: Academic press) pp. 199–125
[32]	Lou S Y, Hu X R and Chen Y 2012 J. phys. A: Math. Gen. 45 155209

Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations

Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations

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