Algebro-geometric constructions of the Heisenberg hierarchy

References

[1] I. M. Krichever, “Algebraic-geometric construction of the Zaharov–Sabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR, vol. 227, pp. 394–397, 1976.Search in Google Scholar

[2] B. A. Dubrovin, “Theta functions and nonlinear equations,” Russ. Math. Surv., vol. 36, pp. 11–92, 1981, https://doi.org/10.1070/rm1981v036n02abeh002596.Search in Google Scholar

[3] E. Date and S. Tanaka, “Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice,” Prog. Theor. Phys. Suppl., vol. 59, pp. 107–125, 1976, https://doi.org/10.1143/ptps.59.107.Search in Google Scholar

[4] Y. C. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math., vol. 65, pp. 113–158, 1981, https://doi.org/10.1002/sapm1981652113.Search in Google Scholar

[5] M. Guo, Y. Zhang, M. Wang, Y. D. Chen, and H. W. Yang, “A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism,” Comput. Math. Appl., vol. 75, pp. 3589–3603, 2018, https://doi.org/10.1016/j.camwa.2018.02.019.Search in Google Scholar

[6] A. O. Smirnov, “Real finite-gap regular solutions of the Kaup–Boussinesq equation,” Theor. Math. Phys., vol. 66, pp. 19–31, 1986, https://doi.org/10.1007/bf01028935.Search in Google Scholar

[7] E. Previato, “Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation,” Duke Math. J., vol. 52, pp. 329–377, 1985, https://doi.org/10.1215/s0012-7094-85-05218-4.Search in Google Scholar

[8] P. D. Miller, N. M. Ercolani, I. M. Krichever, and C. D. Levermore, “Finite genus solutions to the Ablowitz–Ladik equations,” Commun. Pure Appl. Math., vol. 48, pp. 1369–1440, 1995.10.1002/cpa.3160481203Search in Google Scholar

[9] C. N. Lu, C. Fu, and H. W. Yang, “Time-fractional generalized Boussinesq Equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions,” Appl. Math. Comput., vol. 327, pp. 104–116, 2018, https://doi.org/10.1016/j.amc.2018.01.018.Search in Google Scholar

[10] M. S. Alber and Y. N. Fedorov, “Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians,” Inverse Probl., vol. 17, pp. 1017–1042, 2001, https://doi.org/10.1088/0266-5611/17/4/329.Search in Google Scholar

[11] R. G. Zhou, “The finite-band solution of Jaulent–Miodek equation,” J. Math. Phys., vol. 38, pp. 2335–2546, 1997, https://doi.org/10.1063/1.531993.Search in Google Scholar

[12] F. Gesztesy and R. Ratneseelan, “An alternative approach to algebro-geometric solutions of the AKNS hierarchy,” Rev. Math. Phys., vol. 10, pp. 345–391, 1998, https://doi.org/10.1142/s0129055x98000112.Search in Google Scholar

[13] C. W. Cao, Y. T. Wu, and X. G. Geng, “Relation between the Kadomtsev–Petviashvili equation and the Confocal involutive system,” J. Math. Phys., vol. 40, pp. 3948–3970, 1999, https://doi.org/10.1063/1.532936.Search in Google Scholar

[14] X. G. Geng and Y. T. Wu, “Finite-band solutions of the classical Boussinesq–Burgers equations,” J. Math. Phys., vol. 40, pp. 2971–2982, 1999, https://doi.org/10.1063/1.532739.Search in Google Scholar

[15] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol. I: (1+1)-Dimensional Continuous Models, Cambridge, Cambridge University Press, 2003.10.1017/CBO9780511546723Search in Google Scholar

[16] F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Vol. II: (1+1)-Dimensional Discrete Models, Cambridge, Cambridge University Press, 2008.10.1017/CBO9780511543203Search in Google Scholar

[17] X. G. Geng and C. W. Cao, “Decomposition of the (2+1)-dimensional Gardner equation and its quasi-periodic solutions,” Nonlinearity, vol. 14, pp. 1433–1452, 2001, https://doi.org/10.1088/0951-7715/14/6/302.Search in Google Scholar

[18] X. G. Geng, H. H. Dai, and J. Y. Zhu, “Decomposition of the discrete Ablowitz–Ladik hierarchy,” Stud. Appl. Math., vol. 118, pp. 281–312, 2007, https://doi.org/10.1111/j.1467-9590.2007.00374.x.Search in Google Scholar

[19] X. G. Geng and B. Xue, “Quasi-periodic solutions of mixed AKNS equations,” Nonlinear Anal., vol. 73, pp. 3662–3674, 2010, https://doi.org/10.1016/j.na.2010.07.047.Search in Google Scholar

[20] X. G. Geng, L. H. Wu, and G. L. He, “Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions,” Physica D, vol. 240, pp. 1262–1288, 2011, https://doi.org/10.1016/j.physd.2011.04.020.Search in Google Scholar

[21] Y. Y. Zhai and X. G. Geng, “Straightening out of the flows for the Hu hierarchy and its algebro-geometric solutions,” J. Math. Anal. Appl., vol. 397, pp. 561–576, 2013, https://doi.org/10.1016/j.jmaa.2012.08.023.Search in Google Scholar

[22] L. D. Landau and E. M. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,” Phys. Z. Sowjetunion, vol. 8, pp. 153–169, 1935.10.1016/B978-0-08-036364-6.50008-9Search in Google Scholar

[23] A. R. Bishop and T. Schneider, Solitons in Condensed Matter, Berlin, Springer, 1978.10.1007/978-3-642-81291-0Search in Google Scholar

[24] P. E. Wigen, Nonlinear Phenomena and Chaos in Magnetic Materials, Singapore, World Scientific, 1994.10.1142/1686Search in Google Scholar

[25] C. C. Dantas, “An approach to loop quantum cosmology through integrable discrete Heisenberg spin chains,” Found. Phys., vol. 43, pp. 236–242, 2013, https://doi.org/10.1007/s10701-012-9692-2.Search in Google Scholar

[26] L. A. Takhtajan, “Integration of the continuous Heisenberg spin chain through the inverse scattering method,” Phys. Lett., vol. 64, pp. 235–237, 1977, https://doi.org/10.1016/0375-9601(77)90727-7.Search in Google Scholar

[27] J. Tjon and J. Wright, “Solitons in the continuous Heisenberg spin chain,” Phys. Rev. B, vol. 15, pp. 3470–3476, 1977, https://doi.org/10.1103/physrevb.15.3470.Search in Google Scholar

[28] A. Jevicki and N. Papanicolaou, “Semi-classical spectrum of the continuous Heisenberg spin chain,” Ann. Phys., vol. 120, pp. 107–128, 1979, https://doi.org/10.1016/0003-4916(79)90283-5.Search in Google Scholar

[29] G. R. W. Quispel and H. W. Capel, “The Anisotropic Heisenberg spin chain and the nonlinear Schrödinger equation,” Physica A, vol. 117, pp. 76–102, 1983, https://doi.org/10.1016/0378-4371(83)90022-5.Search in Google Scholar

[30] Y. S. Li and D. Y. Chen, “Equivalence of three kinds of nonlinear evolution equations,” Acta Math. Sin., vol. 29, pp. 264–271, 1986.Search in Google Scholar

[31] A. G. Choudhury and A. R. Chowdhury, “Nonlocal conservation laws and supersymmetric Heisenberg spin chain,” Int. J. Theor. Phys., vol. 33, pp. 2031–2036, 1994.10.1007/BF00675168Search in Google Scholar

[32] C. W. Cao, “Parametric representation of the finite-band solution of the Heisenberg equation,” Phys. Lett., vol. 184, pp. 333–338, 1994, https://doi.org/10.1016/0375-9601(94)90456-1.Search in Google Scholar

[33] N. Bhattacharya and A.R. Chowdhury, “Bethe ansatz for an open Heisenberg spin chain with impurity,” Int. J. Theor. Phys., vol. 33, pp. 679–685, 1994, https://doi.org/10.1007/bf00670523.Search in Google Scholar

[34] Z. J. Qiao, “A finite-dimensional integrable system and the involutive solutions of the higher-order Heisenberg spin chain equations,” Phys. Lett., vol. 186, pp. 97–102, 1994, https://doi.org/10.1016/0375-9601(94)90927-x.Search in Google Scholar

[35] D. L. Du, “Complex form, reduction and Lie-Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy,” Physica A, vol. 303, pp. 439–456, 2002, https://doi.org/10.1016/s0378-4371(01)00562-3.Search in Google Scholar

[36] X. G. Wang and P. Zanardi, “Quantum entanglement and Bell inequalities in Heisenberg spin chains,” Phys. Lett., vol. 301, pp. 1–6, 2002, https://doi.org/10.1016/s0375-9601(02)00885-x.Search in Google Scholar

[37] J. Wang, “Darboux transformation and soliton solutions for the Heisenberg hierarchy,” J. Phys., vol. 38, pp. 5217–5226, 2005, https://doi.org/10.1088/0305-4470/38/23/009.Search in Google Scholar

[38] B. L. Guo, M. Zeng, and F.Q. Su, “Periodic weak solutions for a classical one-dimensional isotropic biquadratic Heisenberg spin chain,” J. Math. Anal. Appl., vol. 330, pp. 729–739, 2007, https://doi.org/10.1016/j.jmaa.2006.08.010.Search in Google Scholar

[39] A. R. Its and V. E. Korepin, “Generalized entropy of the Heisenberg spin chain,” Theor. Math. Phys., vol. 164, pp. 1136–1139, 2010, https://doi.org/10.1007/s11232-010-0091-6.Search in Google Scholar

[40] H. Z. Li, B. Tian, R. Guo, Y. S. Xue, and F. H. Qi, “Gauge transformation between the first-order nonisospectral and isospectral Heisenberg hierarchies,” Appl. Math. Comput., vol. 218, pp. 7694–7699, 2012, https://doi.org/10.1016/j.amc.2012.01.018.Search in Google Scholar

[41] J. A. Miszczak, P. Gawron, and Z. Puchała, “Qubit flip game on a Heisenberg spin chain,” Quant. Inf. Process., vol. 11, pp. 1571–1583, 2012, https://doi.org/10.1007/s11128-011-0322-2.Search in Google Scholar

[42] G. Z. Tu, “The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems,” J. Math. Phys., vol. 30, pp. 330–338, 1989, https://doi.org/10.1063/1.528449.Search in Google Scholar