Nonlocal Symmetries and Geometric Integrability of Multi-Component Camassa–Holm and Hunter–Saxton Systems
YAN Lu1, SONG Jun-Feng1,2, QU Chang-Zheng1**
1Department of Mathematics, Northwest University, Xi'an 710069 2College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062
Nonlocal Symmetries and Geometric Integrability of Multi-Component Camassa–Holm and Hunter–Saxton Systems
YAN Lu1, SONG Jun-Feng1,2, QU Chang-Zheng1**
1Department of Mathematics, Northwest University, Xi'an 710069 2College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062
摘要We present the multi-component Hunter–Saxton and μ−Camassa–Holm systems. It is shown that the multi-component Camassa–Holm, Hunter–Saxton and μ-Camassa–Holm systems are geometrically integrable, namely they describe pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws can be directly constructed. For the three-component Camassa–Holm and Hunter–Saxton systems, their nonlocal symmetries depending on the pseudo-potentials are obtained.
Abstract：We present the multi-component Hunter–Saxton and μ−Camassa–Holm systems. It is shown that the multi-component Camassa–Holm, Hunter–Saxton and μ-Camassa–Holm systems are geometrically integrable, namely they describe pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws can be directly constructed. For the three-component Camassa–Holm and Hunter–Saxton systems, their nonlocal symmetries depending on the pseudo-potentials are obtained.
YAN Lu;SONG Jun-Feng;QU Chang-Zheng**
. Nonlocal Symmetries and Geometric Integrability of Multi-Component Camassa–Holm and Hunter–Saxton Systems[J]. 中国物理快报, 2011, 28(5): 50204-050204.
YAN Lu, SONG Jun-Feng, QU Chang-Zheng**
. Nonlocal Symmetries and Geometric Integrability of Multi-Component Camassa–Holm and Hunter–Saxton Systems. Chin. Phys. Lett., 2011, 28(5): 50204-050204.
[1] Camassa R and Holm D 1993 Phys. Rev. Lett. 71 1661
[2] Fokas A and Fuchssteiner B 1981 Physica D 4 47
[3] Hunter J K and Saxton R 1991 SIAM J. Appl. Math. 51 1498
[4] Reyes E G 2002 Lett. Math. Phys. 59 17
[5] Chou K S and Qu C Z 2002 Physica D 162 9
[6] Kouranbaeva S 1999 J. Math. Phys. 40 857
[7] Lenells J 2007 J. Geom. Phys. 57 2049
[8] Khesin B, Lenells J and Misiolek G 2008 Math. Ann. 342 617
[9] Fu Y, Liu Y and Qu C Z 2010 arXiv:1009.2466v2
[10] Olver P J and Rosenau P 1996 Phys. Rev. E 53 1900
[11] Chen M, Liu S and Zhang Y 2006 Lett. Math. Phys. 75 1
[12] Constantin A and Ivanov R I 2008 Phys. Lett. A 372 7129
[13] Guha P and Olver P J 2006 SIGMA Integrability Geom. Methods Appl. 2 paper 054
[14] Kohlmann M 2010 arXiv:1010.2363v1 [math-ph], preprint.
[15] Zuo D F 2010 Inverse Problems 26 085003
[16] Ivanov R I 2006 Z. Naturforsch. 61a 133
[17] Holm D D and Ivanov R I 2010 arXiv:1009.5368v1[nlin.SI], 27 Sep.
[18] Chern S S and Tenenblat K 1986 Stud. Appl. Math. 74 55
[19] Cavalcante J A and Tenenblat K 1988 J. Math. Phys. 29 1044
[20] Sasaki R 1979 Nucl. Phys. 13 343
[21] Song J F et al 2010 Comm. Theor. Phys. (accepted)
[22] Reyes E G 2005 J. Math. Phys. 46 073507.
[23] Galas F 1999 J. Phys. A: Math. Gen. 25 L981
2011, Vol. 28 
