The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices
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- by Jiao Wei, Xianguo Geng and Xin Zeng PDF
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Abstract:
Starting with a discrete $3\times 3$ matrix spectral problem, the hierarchy of Bogoyavlensky lattices which are pure differential-difference equations are derived with the aid of the Lenard recursion equations and the stationary discrete zero-curvature equation. By using the characteristic polynomial of Lax matrix for the hierarchy of stationary Bogoyavlensky lattices, we introduce a trigonal curve $\mathcal {K}_{m-1}$ of arithmetic genus $m-1$ and a basis of holomorphic differentials on it, from which we construct the Riemann theta function of the trigonal curve, the related Baker–Akhiezer function, and an algebraic function carrying the data of the divisor. Based on the theory of trigonal curves, the Riemann theta function representations of the Baker–Akhiezer function, the meromorphic function, and in particular, that of solutions of the hierarchy of Bogoyavlensky lattices are obtained.References
- S. V. Manakov, Complete integrability and stochastization of discrete dynamical systems, Ž. Èksper. Teoret. Fiz. 67 (1974), no. 2, 543–555 (Russian, with English summary); English transl., Soviet Physics JETP 40 (1974), no. 2, 269–274 (1975). MR 0389107
- S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. The inverse scattering method; Translated from the Russian. MR 779467
- Morikazu Toda, Theory of nonlinear lattices, Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin-New York, 1981. Translated from the Japanese by the author. MR 618652, DOI 10.1007/978-3-642-96585-2
- J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197–220. MR 375869, DOI 10.1016/0001-8708(75)90151-6
- O. I. Bogoyavlenskiĭ, Integrable dynamical systems connected with the KdV equation, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 6, 1123–1141, 1358 (Russian); English transl., Math. USSR-Izv. 31 (1988), no. 3, 435–454. MR 933958, DOI 10.1070/IM1988v031n03ABEH001084
- O. I. Bogoyavlensky, Integrable discretizations of the KdV equation, Phys. Lett. A 134 (1988), no. 1, 34–38. MR 972622, DOI 10.1016/0375-9601(88)90542-7
- Yuri B. Suris, The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics, vol. 219, Birkhäuser Verlag, Basel, 2003. MR 1993935, DOI 10.1007/978-3-0348-8016-9
- Hongwei Zhang, Gui Zhang Tu, Walter Oevel, and Benno Fuchssteiner, Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure, J. Math. Phys. 32 (1991), no. 7, 1908–1918. MR 1112724, DOI 10.1063/1.529205
- O. I. Bogoyavlensky, Algebraic constructions of integrable dynamical systems-extensions of the Volterra system, Russian Math. Surveys 46 (1991), 1–64.
- B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
- A. O. Smirnov, Real finite-gap regular solutions of the Kaup-Boussinesq equation, Teoret. Mat. Fiz. 66 (1986), no. 1, 30–46 (Russian, with English summary). MR 831416
- Igor Moiseevich Krichever, An algebraic-geometric construction of the Zaharov-Šabat equations and their periodic solutions, Dokl. Akad. Nauk SSSR 227 (1976), no. 2, 291–294 (Russian). MR 0413178
- Igor Moiseevich Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funkcional. Anal. i Priložen. 11 (1977), no. 1, 15–31, 96 (Russian). MR 0494262
- B. A. Dubrovin, Matrix finite-zone operators, J. Soviet Math. 28 (1985), 20–50.
- E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994.
- Etsuro Date and Shunichi Tanaka, Analogue of inverse scattering theory for the discrete Hill’s equation and exact solutions for the periodic Toda lattice, Progr. Theoret. Phys. 55 (1976), no. 2, 457–465. MR 403367, DOI 10.1143/PTP.55.457
- Yan Chow Ma and Mark J. Ablowitz, The periodic cubic Schrödinger equation, Stud. Appl. Math. 65 (1981), no. 2, 113–158. MR 628138, DOI 10.1002/sapm1981652113
- Emma Previato, Hyperelliptic quasiperiodic and soliton solutions of the nonlinear Schrödinger equation, Duke Math. J. 52 (1985), no. 2, 329–377. MR 792178, DOI 10.1215/S0012-7094-85-05218-4
- H. P. McKean, Integrable systems and algebraic curves, Global analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ. Calgary, Calgary, Alta., 1978) Lecture Notes in Math., vol. 755, Springer, Berlin, 1979, pp. 83–200. MR 564904
- H. Airault, H. P. McKean, and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), no. 1, 95–148. MR 649926, DOI 10.1002/cpa.3160300106
- Solomon J. Alber, On finite-zone solutions of relativistic Toda lattices, Lett. Math. Phys. 17 (1989), no. 2, 149–155. MR 993020, DOI 10.1007/BF00402329
- F. Gesztesy and R. Ratnaseelan, An alternative approach to algebro-geometric solutions of the AKNS hierarchy, Rev. Math. Phys. 10 (1998), no. 3, 345–391. MR 1626836, DOI 10.1142/S0129055X98000112
- Fritz Gesztesy, Helge Holden, Johanna Michor, and Gerald Teschl, Soliton equations and their algebro-geometric solutions. Vol. II, Cambridge Studies in Advanced Mathematics, vol. 114, Cambridge University Press, Cambridge, 2008. $(1+1)$-dimensional discrete models. MR 2446594, DOI 10.1017/CBO9780511543203
- Jeffrey S. Geronimo, Fritz Gesztesy, and Helge Holden, Algebro-geometric solutions of the Baxter-Szegő difference equation, Comm. Math. Phys. 258 (2005), no. 1, 149–177. MR 2166844, DOI 10.1007/s00220-005-1305-x
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
- David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776, DOI 10.1007/978-0-8176-4578-6
- Xianguo Geng, H. H. Dai, and Junyi Zhu, Decomposition of the discrete Ablowitz-Ladik hierarchy, Stud. Appl. Math. 118 (2007), no. 3, 281–312. MR 2305780, DOI 10.1111/j.1467-9590.2007.00374.x
- A. R. Its and V. B. Matveev, Schrödinger operators with the finite-band spectrum and the $N$-soliton solutions of the Korteweg-de Vries equation, Teoret. Mat. Fiz. 23 (1975), no. 1, 51–68 (Russian, with English summary). MR 479120
- S. Baldwin, J. C. Eilbeck, J. Gibbons, and Y. Ônishi, Abelian functions for cyclic trigonal curves of genus 4, J. Geom. Phys. 58 (2008), no. 4, 450–467. MR 2406408, DOI 10.1016/j.geomphys.2007.12.001
- J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. Ônishi, and E. Previato, Abelian functions for trigonal curves of genus three, Int. Math. Res. Not. IMRN 1 (2008), Art. ID rnm 140, 38. MR 2417791, DOI 10.1093/imrn/rnm140
- Yu. V. Brezhnev, Finite-gap potentials with trigonal curves, Teoret. Mat. Fiz. 133 (2002), no. 3, 398–404 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 133 (2002), no. 3, 1657–1662. MR 2001550, DOI 10.1023/A:1021310208404
- Emma Previato, The Calogero-Moser-Krichever system and elliptic Boussinesq solitons, Hamiltonian systems, transformation groups and spectral transform methods (Montreal, PQ, 1989) Univ. Montréal, Montreal, QC, 1990, pp. 57–67. MR 1110372
- V. B. Matveev and A. O. Smirnov, On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations, Lett. Math. Phys. 14 (1987), no. 1, 25–31. MR 901696, DOI 10.1007/BF00403466
- V. B. Matveev and A. O. Smirnov, On the simplest trigonal solutions of the Boussinesq equation and the Kadomtsev-Petviashvili equation, Dokl. Akad. Nauk SSSR 293 (1987), no. 1, 78–82 (Russian). MR 882082
- Emma Previato, Monodromy of Boussinesq elliptic operators, Acta Appl. Math. 36 (1994), no. 1-2, 49–55. MR 1303855, DOI 10.1007/BF01001542
- Emma Previato and Jean-Louis Verdier, Boussinesq elliptic solitons: the cyclic case, Proceedings of the Indo-French Conference on Geometry (Bombay, 1989) Hindustan Book Agency, Delhi, 1993, pp. 173–185. MR 1274502
- A. O. Smirnov, A matrix analogue of a theorem of Appell and reductions of multidimensional Riemann theta-functions, Mat. Sb. (N.S.) 133(175) (1987), no. 3, 382–391, 416 (Russian); English transl., Math. USSR-Sb. 61 (1988), no. 2, 379–388. MR 909858, DOI 10.1070/SM1988v061n02ABEH003213
- Yoshihiro Ônishi, Abelian functions for trigonal curves of degree four and determinantal formulae in purely trigonal case, Internat. J. Math. 20 (2009), no. 4, 427–441. MR 2515048, DOI 10.1142/S0129167X09005340
- Ronnie Dickson, Fritz Gesztesy, and Karl Unterkofler, A new approach to the Boussinesq hierarchy, Math. Nachr. 198 (1999), 51–108. MR 1670365, DOI 10.1002/mana.19991980105
- R. Dickson, F. Gesztesy, and K. Unterkofler, Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys. 11 (1999), no. 7, 823–879. MR 1702719, DOI 10.1142/S0129055X9900026X
- Yongtang Wu and Xianguo Geng, A finite-dimensional integrable system associated with the three-wave interaction equations, J. Math. Phys. 40 (1999), no. 7, 3409–3430. MR 1696992, DOI 10.1063/1.532896
- Xianguo Geng, Lihua Wu, and Guoliang He, Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions, Phys. D 240 (2011), no. 16, 1262–1288. MR 2813828, DOI 10.1016/j.physd.2011.04.020
- Xianguo Geng, Lihua Wu, and Guoliang He, Quasi-periodic solutions of the Kaup-Kupershmidt hierarchy, J. Nonlinear Sci. 23 (2013), no. 4, 527–555. MR 3079668, DOI 10.1007/s00332-012-9160-3
- Xianguo Geng, Yunyun Zhai, and H. H. Dai, Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy, Adv. Math. 263 (2014), 123–153. MR 3239136, DOI 10.1016/j.aim.2014.06.013
- Guoliang He, Xianguo Geng, and Lihua Wu, Algebro-geometric quasi-periodic solutions to the three-wave resonant interaction hierarchy, SIAM J. Math. Anal. 46 (2014), no. 2, 1348–1384. MR 3180851, DOI 10.1137/130918794
Additional Information
- Jiao Wei
- Affiliation: School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China
- MR Author ID: 1128395
- Email: weijiaozzu@sohu.com
- Xianguo Geng
- Affiliation: School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China
- MR Author ID: 257090
- Email: xggeng@zzu.edu.cn
- Xin Zeng
- Affiliation: School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China
- MR Author ID: 765917
- Email: xzeng@zzu.edu.cn
- Received by editor(s): May 4, 2016
- Received by editor(s) in revised form: November 24, 2016, and June 5, 2017
- Published electronically: September 10, 2018
- Additional Notes: This work was supported by National Natural Science Foundation of China (Grant nos. 11331008 and 11871440
The second author (Xianguo Geng) is the corresponding author. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1483-1507
- MSC (2010): Primary 37K10, 37K20, 14H42, 37K40
- DOI: https://doi.org/10.1090/tran/7349
- MathSciNet review: 3885186