Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 15, 2017

Algebro-Geometric Solutions of the Harry Dym Hierarchy

  • Zhu Li EMAIL logo


The Harry Dym hierarchy is derived with the help of Lenard recursion equations and zero curvature equation. Based on the Lax matrix, an algebraic curve Kn of arithmetic genus n is introduced, from which the corresponding meromorphic function ϕ and Dubrovin-type equations are given. Further, the divisor and asymptotic properties of ϕ are studied. Finally, algebro-geometric solutions for the entire hierarchy are obtained according to above results and the theory of algebraic curve.

MSC 2010: 37K10; 14K70

Funding statement: This work was supported by the National Natural Science Foundation of China (Grant No. 11301487), a Foundation for the Author of National Excellent Doctoral Dissertation of P.R. China (No. 201313), the Key Scientific Research Projects of Henan Institution of Higher Education (No. 17A110029) and Nanhu Scholars Program for Young Scholars of XYNU.


[1] Kruskal M., Nonlinear wave equations, in: Dynamical Systems, Theory and Applications, (Lecture Notes in Phys., 38.), pp. 310–354, Springer-Verlag, Berlin, 1975.10.1007/3-540-07171-7_9Search in Google Scholar

[2] Sabatier P. G., On some spectral problems and isospectral evolutions connected with the classical string problem, Lett. Nuovo Cimento 26 (1979), 477–486.10.1007/BF02750260Search in Google Scholar

[3] Vasconcelos G. L. and Kadanoff L. P., Stationary solution for the Saffman-Taylor problem with surface tension, Phys. Rev. A 44, (1991), 6490–6495.10.1103/PhysRevA.44.6490Search in Google Scholar PubMed

[4] Goldstein R. E. and Petrich D. M., The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67 (1991), 3203–3206.10.1103/PhysRevLett.67.3203Search in Google Scholar PubMed

[5] Wadati M., Ichikawa Y. H. and Shimizu T., Cusp soliton of a new integrable nonlinear evolution equation, Prog. Theory Phys. 64 (1980), 1959-1966.10.1143/PTP.64.1959Search in Google Scholar

[6] Dmitrieva L. A., N-loop solitons and their link with the complex Harry Dym equation, J. Phys. A 27 (1994), 8197–8205.10.1088/0305-4470/27/24/026Search in Google Scholar

[7] Wadati M., Konno K. and Ichikawa Y. H., New integrable nonlinear evolution equations, J. Phys. Soc. Japan 41 (1979), 1698–1700.10.1143/JPSJ.47.1698Search in Google Scholar

[8] Pedroni M., V. Shiacha and Zubelli J. P., The bi-Hamiltonian theory of the Harry Dym equation, Teoret. Mat. Fiz. 133 (2002), 311–326.10.4213/tmf400Search in Google Scholar

[9] Rogers C. and Nucci M. C., On reciprocal Bäcklund transformations and the Korteweg-de Vries hierarchy, Phys. Scr. 33 (1986), 289–292.10.1088/0031-8949/33/4/001Search in Google Scholar

[10] Kawamoto S., An exact transformation from the Harry Dym equation to the modified KdV equation, J. Phys. Sac. Japan 54 (1985), 2055–2056.10.1143/JPSJ.54.2055Search in Google Scholar

[11] Leo M., Leo R. A., Soliani G., Solombrino L. and Martina L., Lie-Bäcklund symmetries for the Harry-Dym equation, Phys. Rev. D 27 (1983), 1406–1408.10.1103/PhysRevD.27.1406Search in Google Scholar

[12] Chowdhury A. R. and Mukherjee R., Elliptic solutions, recursion operators and complete Lie-Bäcklund symmetry for the Harry Dym equation, Phys. Scripta 29 (1984), 293–295.10.1088/0031-8949/29/4/002Search in Google Scholar

[13] Dmitrieva L. A., The higher-times approach to multisoliton solutions of the Harry Dym equation, J. Phys. A 26 (1993), 6005–6020.10.1088/0305-4470/26/21/037Search in Google Scholar

[14] Fuchssteiner B., Schulze T. and Carillo S., Explicit solutions for the Harry Dym equation, J. Phys. A 25 (1992), 223–230.10.1088/0305-4470/25/1/025Search in Google Scholar

[15] Konopelchenko B. G. and Lee J. H., The Harry Dym equation on the complex plane: inverse spectral transform and exact solutions, Teoret. Mat. Fiz. 99 (1994), 337–344.10.1007/BF01016150Search in Google Scholar

[16] Cao C. W., Stationary Harry-Dym’s equation and its relation with geodesics on ellipsoid, Acta Math. Sinica 6 (1990), 35–41.10.1007/BF02108861Search in Google Scholar

[17] Zakharov D. V., Isoperiodic deformations of an acoustic operator, and periodic solutions of the Harry Dym equation, Teoret. Mat. Fiz. 153 (2007), 46–57.10.1007/s11232-007-0122-0Search in Google Scholar

[18] Drnitriew L. A., Finite-gap solutions of the Harry Dym equation, Phys. Lett. A 182 (1993), 65–70.10.1016/0375-9601(93)90054-4Search in Google Scholar

[19] Novikov D. P., Algebro-geometric solutions of the Harry Dym equation, Sibirsk. Mat. Zh. 40 (1999), 159–163.Search in Google Scholar

[20] Qiao Z. J., Qiao X. B., Cusp solitons and cusp-like singular solutions for nonlinear equations, Chaos, Solitons and Fractals 25 (2005), 153–163.10.1016/j.chaos.2004.09.074Search in Google Scholar

[21] Ma W. X., An extended Harry Dym hierarchy, J. Phys. A 43, 165202 (2010).10.1088/1751-8113/43/16/165202Search in Google Scholar

[22] Ma W. X. and Zeng Y. B., Binary constrained flows and separation of variables for soliton equations, ANZIAM 44 (2002), 129–139.10.1017/S1446181100007987Search in Google Scholar

[23] Marciniak K. and Błaszak M., Construction of coupled Harry Dym hierarchy and its solutions from Stäckel systems, Nonlinear Anal. 73 (2010), 3004–3017.10.1016/ in Google Scholar

[24] Gesztesy F. and Ratnaseelan R., An alternative approach to algebro-geometric solutions of the AKNS hierarchy, Rev. Math. Phys. 10 (1998), 345–391.10.1142/S0129055X98000112Search in Google Scholar

[25] Dickson R., Gesztesy F. and Unterkofler K., Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys. 11 (1999), 823–879.10.1142/S0129055X9900026XSearch in Google Scholar

[26] Dickson R., Gesztesy F. and Unterkofler K., A new approach to the Boussinesq hierarchy, Math. Nachr. 198 (1999), 51–108.10.1002/mana.19991980105Search in Google Scholar

[27] Gesztesy F. and Holden H., Soliton equations and their algebro-geometric solutions, Cambridge University Press, Cambridge, 2003.10.1017/CBO9780511546723Search in Google Scholar

[28] Geng X. G. and Xue B., Quasi-periodic solutions of mixed AKNS equations, Nonlinear Anal. 73 (2010), 3662–3674.10.1016/ in Google Scholar

[29] Gesztesy F. and Holden H., Algebro-geometric solutions of the Camassa-Holm hierarchy, Rev. Mat. Iberoam. 19 (2003), 73–142.10.4171/RMI/339Search in Google Scholar

[30] Geng X. G. and Xue B., Soliton solutions and quasiperiodic solutions of modified Korteweg-de Vries type equations, J. Math. Phys. 51, 063516 (2010).10.1063/1.3409345Search in Google Scholar

[31] Geng X. G., Wu L. H. and He G. L., Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions, Phys. D 240 (2011), 1262–1288.10.1016/j.physd.2011.04.020Search in Google Scholar

[32] Geng X. G., Wu L. H. and He G. L., Quasi-periodic solutions of nonlinear evolution equa- tions associated with a 3×3 matrix spectral problem, Stud. Appl. Math. 127 (2011), 107–140.10.1111/j.1467-9590.2010.00513.xSearch in Google Scholar

[33] Zhai Y. Y. and Geng X. G., Straightening out of the flows for the Hu hierarchy and its algebro-geometric solutions, J. Math. Anal. Appl. 397 (2013), 561–576.10.1016/j.jmaa.2012.08.023Search in Google Scholar

[34] Geng X. G., Zeng X. and B. Xue, Algebro-geometric solutions of the TD Hierarchy, Math. Phys. Anal. Geom. 16 (2013), 229–251.10.1007/s11040-013-9129-ySearch in Google Scholar

[35] He G. L., Geng X. G. and Wu L. H., Algebro-geometric quasi-periodic solutions to the three-wave resonant interaction hierarchy, SIAM J. Math. Anal. 46 (2014), 1348–1384.10.1137/130918794Search in Google Scholar

[36] Geng X. G., Zhai Y. Y. and Dai H. H., Algebro-geometric solutions of the coupled modified Korteweg-de Vries hierarchy, Adv. Math. 263 (2014), 123–153.10.1016/j.aim.2014.06.013Search in Google Scholar

[37] Dubrovin B. A., Theta functions and nonlinear equations, Russian Math. Surveys 36 (1981), 11–92.10.1070/RM1981v036n02ABEH002596Search in Google Scholar

[38] Zhou R. G. and Ma W. X., Algebro-geometric solutions of the (2+1)-dimensional Gardner equation, Nuovo Cimento B 115 (2000), 1419–1431.Search in Google Scholar

[39] Ma W. X., Trigonal curves and algebro-geometric solutions to soliton hierarchies (2016) arXiv: 1607.07532.Search in Google Scholar

[40] Ma W. X., Some Hamiltonian operators in the infinite-dimensional Hamiltonian systems, Acta Math. Appl. Sinica 13 (1990), 484–496.Search in Google Scholar

[41] Antonowicz M. and Fordy A. P., Coupled Harry Dym equations with multi-Hamiltonian structures, J. Phys. A 21 (1988), L269–L275.10.1088/0305-4470/21/5/001Search in Google Scholar

Received: 2016-4-13
Accepted: 2016-12-29
Published Online: 2017-2-15
Published in Print: 2017-4-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 11.12.2023 from
Scroll to top button