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There are various nonlinear solutions in the anisotropic Heisenberg spin chain model (AHSCM), such as soliton solutions. In consideration of high-order nonlinear terms, a good modified nonlinear analytical solution can be obtained under reasonable simplification conditions. The purpose of this paper is to find the nonlinear solutions other than soliton of AHSCM. We use Holstein-Primakoff representation to study the AHSCM. Under the semi-classical approximation, considering the high order nonlinear term and the periodic boundary condition, an improved nonlinear Schrodinger equation and its wave solutions of the hyper-elliptic function expressed by the combination of the inverse function of Jacobi elliptic function are obtained through using the coherent state. These solutions can be expressed by the combination of the inverse functions of the first kind of elliptic functions. In the limit case, these solutions are reduced to wave solutions of sinusoidal (or cosine) functions, or wave solutions that can be represented by hyperbolic tangent functions. The energy levels of these nonlinear solutions can be obtained theoretically by the normalized conditions, but even by using hyper-elliptic functions, it is difficult to express them as analytic expressions.
[1] Nakamura K, Sasada T 1974 Phys. Lett. A 48 321
[2] Lakshmanan M 1977 Phys. Lett. A 61 53
[3] Pushkarov D I, Pushkarov K I 1977 Phys. Lett. A 61 339
[4] Jauslin H R, Schneider T 1982 Phys. Rev. B 26 5153
[5] Mead L R, Papanicolaoy N 1983 Phys. Rev. B 28 1633
[6] Borsa F, Pini M G, Rettori A, Tognetti V 1983 Phys. Rev. B 28 5173
[7] Kopinqa K, Tinus A M C, de Jonge W J M 1984 Phys. Rev. B 29 2868
[8] Skrinjar M J, Kapor D V, Stojanovic S D 1987 Sol. State Phys. 12 2243
[9] Mikeska H J, Steiner M 1991 Adv. Phys. 40 191
[10] Daniel M, Kavitha L 2002 Phys. Rev. B 66 184433
[11] Xie Y D 2016 Acta Phys. Sin. 65 207501 (in Chinese) [谢元栋 2016 物理学报 65 207501]
[12] Kazumi M, Pradeep K 1976 Phys. Rev. B 9 3920
[13] Holstein T, Primakoff H 1940 Phys. Rev. 58 1098
[14] Glauber R J 1963 Phys. Rev. 131 2766
[15] Tsoy E N 2010 Phys. Rev. A 82 063829
[16] Ablowitz M J, Clarkson P A 1991 Soliton, Nonlinear Evolution Equations Scattering (New York:Cambridge University Press) pp98-102
[17] Xie Y D 2012 Acta Phys. Sin. 61 210305 (in Chinese) [谢元栋 2012 物理学报 61 210305]
[18] Daniel M, Kavitha L 1999 Phys. Rev. B 59 13774
[19] Daniel M, Beula J 2008 Phys. Rev. B 77 144416
[20] Gao B Q 1991 Elliptic Functions and Their Applications (Beijing:National Defense Industry Press) pp142-146 (in Chinese) [高本庆 1991 椭圆函数及其应用(北京:国防工业出版社) 第142146页]
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[1] Nakamura K, Sasada T 1974 Phys. Lett. A 48 321
[2] Lakshmanan M 1977 Phys. Lett. A 61 53
[3] Pushkarov D I, Pushkarov K I 1977 Phys. Lett. A 61 339
[4] Jauslin H R, Schneider T 1982 Phys. Rev. B 26 5153
[5] Mead L R, Papanicolaoy N 1983 Phys. Rev. B 28 1633
[6] Borsa F, Pini M G, Rettori A, Tognetti V 1983 Phys. Rev. B 28 5173
[7] Kopinqa K, Tinus A M C, de Jonge W J M 1984 Phys. Rev. B 29 2868
[8] Skrinjar M J, Kapor D V, Stojanovic S D 1987 Sol. State Phys. 12 2243
[9] Mikeska H J, Steiner M 1991 Adv. Phys. 40 191
[10] Daniel M, Kavitha L 2002 Phys. Rev. B 66 184433
[11] Xie Y D 2016 Acta Phys. Sin. 65 207501 (in Chinese) [谢元栋 2016 物理学报 65 207501]
[12] Kazumi M, Pradeep K 1976 Phys. Rev. B 9 3920
[13] Holstein T, Primakoff H 1940 Phys. Rev. 58 1098
[14] Glauber R J 1963 Phys. Rev. 131 2766
[15] Tsoy E N 2010 Phys. Rev. A 82 063829
[16] Ablowitz M J, Clarkson P A 1991 Soliton, Nonlinear Evolution Equations Scattering (New York:Cambridge University Press) pp98-102
[17] Xie Y D 2012 Acta Phys. Sin. 61 210305 (in Chinese) [谢元栋 2012 物理学报 61 210305]
[18] Daniel M, Kavitha L 1999 Phys. Rev. B 59 13774
[19] Daniel M, Beula J 2008 Phys. Rev. B 77 144416
[20] Gao B Q 1991 Elliptic Functions and Their Applications (Beijing:National Defense Industry Press) pp142-146 (in Chinese) [高本庆 1991 椭圆函数及其应用(北京:国防工业出版社) 第142146页]
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