Abstract
We develop a method for constructing algebro-geometric solutions of the Blaszak–Marciniak (BM) lattice hierarchy based on the theory of trigonal curves. We first derive the BM lattice hierarchy associated with a discrete (3×3)-matrix spectral problem using Lenard recurrence relations. Using the characteristic polynomial of the Lax matrix for the BM lattice hierarchy, we introduce a trigonal curve with two infinite points, which we use to establish the associated Dubrovin-type equations. We then study the asymptotic properties of the algebraic function carrying the data of the divisor and the Baker–Akhiezer function near the two infinite points on the trigonal curve. We finally obtain algebro-geometric solutions of the entire BM lattice hierarchy in terms of the Riemann theta function.
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References
I. M. Krichever, “Algebraic-geometric construction of the Zakharov–Shabat equations and their periodic solutions,” Sov. Math. Dokl., 17, 394–397 (1976).
I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funct. Anal. Appl., 11, 12–26 (1977).
B. A. Dubrovin, “Theta functions and non-linear equations,” Russian Math. Surv., 36, 11–92 (1981).
B. A. Dubrovin, “Matrix finite-zone operators,” J. Soviet Math., 28, 20–50 (1985).
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).
E. Date and S. Tanaka, “Periodic multi-soliton solutions of Korteweg–de Vries equation and Toda lattice,” Progr. Theoret. Phys. Suppl., 59, 107–125 (1976).
Y. C. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math., 65, 113–158 (1981).
E. Previato, “Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation,” Duke Math. J., 52, 329–377 (1985).
A. O. Smirnov, “Real finite-gap regular solutions of the Kaup–Boussinesq equation,” Theor. Math. Phys., 66, 19–31 (1986).
H. P. McKean, “Integrable systems and algebraic curves,” in: Global Analysis (Lect. Notes Math., Vol. 755, M. Grmela and J. E. Marsden, eds.), Springer, Berlin (1979), pp. 83–200.
S. J. Alber, “On finite-zone solutions of relativistic Toda lattices,” Lett. Math. Phys., 17, 149–155 (1989).
F. Gesztesy and R. Ratneseelan, “An alternative approach to algebro-geometric solutions of the AKNS hierarchy,” Rev. Math. Phys., 10, 345–391 (1998).
F. Gesztesy and H. Holden, “Algebro-geometric solutions of the Camassa–Holm hierarchy,” Rev. Mat. Iberoamericana, 19, 73–142 (2003).
J. S. Geronimo, F. Gesztesy, and H. Holden, “Algebro-geometric solutions of the Baxter–Szegö difference equation,” Commun. Math. Phys., 258, 149–177 (2005).
X. G. Geng and C. W Cao, “Decomposition of the (2+1)-dimensional Gardner equation and its quasi-periodic solutions,” Nonlinearity, 14, 1433–1452 (2001).
X. G. Geng, H. H. Dai, and J. Y. Zhu, “Decomposition of the discrete Ablowitz–Ladik hierarchy,” Stud. Appl. Math., 118, 281–312 (2007).
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, 25–31 (1987).
V. B. Matveev and A. O. Smirnov, “Simplest trigonal solutions of the Boussinesq and Kadomtsev–Petviashvili equations,” Sov. Phys. Dokl., 32, 202–204 (1987).
S. Baldwin, J. C. Eilbeck, J. Gibbons, and Y. Onishi, “Abelian functions for cyclic trigonal curves of genus 4,” J. Geom. Phys., 58, 450–467 (2008).
J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. Onishi, and E. Previato, “Abelian functions for trigonal curves of genus three,” Int. Math. Res. Notices, 2007, 140 (2007).
Yu. V. Brezhnev, “Finite-band potentials with trigonal curves,” Theor. Math. Phys., 133, 1657–1662 (2002).
V. M. Buchstaber, D. V. Leikin, and V. Z. Ènol’skii, “Uniformization of jacobi varieties of trigonal curves and nonlinear differential equations,” Funct. Anal. Appl., 34, 159–171 (2000).
R. Inoue, “The extended Lotka–Volterra lattice and affine Jacobi varieties of spectral curves,” J. Math. Phys., 44, 338–351 (2003).
E. Previato, “The Calogero–Moser–Krichever system and elliptic Boussinesq solitons,” in: Hamiltonian Systems, Transformation Groups, and Spectral Transform Methods (Proc. CRM Workshop, Montréal, Canada, 20–26 October 1989, J. Harnad and J. E. Marsden, eds.), CRM, Montréal (1990), pp. 57–67.
E. Previato, “Monodromy of Boussinesq elliptic operators,” Acta Appl. Math., 36, 49–55 (1994).
E. Previato and J.-L. Verdier, “Boussinesq elliptic solitons: The cyclic case,” in: Proc. Indo-French Conference on Geometry (Bombay, India, 1989, S. Ramanan and A. Beauville, eds.), Hindustan Book Agency, Delhi (1993), pp. 173–185.
A. O. Smirnov, “A matrix analogue of Appell’s theorem and reductions of multidimensional Riemann thetafunctions,” Math. USSR-Sb., 61, 379–388 (1988).
H. Airault, H. P. Mckean, and J. Moser, “Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem,” Commun. Pure Appl. Math., 30, 95–148 (1977).
R. Dickson, F. Gesztesy, and K. Unterkofler, “A new approach to the Boussinesq hierarchy,” Math. Nachr., 198, 51–108 (1999).
R. Dickson, F. Gesztesy, and K. Unterkofler, “Algebro-geometric solutions of the Boussinesq hierarchy,” Rev. Math. Phys., 11, 823–879 (1999).
X. G. Geng, L. H. Wu, and G. L. He, “Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions,” Phys. D, 240, 1262–1288 (2011).
X. G. Geng, L. H. Wu, and G. L. He, “Quasi-periodic solutions of the Kaup–Kupershmidt hierarchy,” J. Nonlinear Sci., 23, 527–555 (2013).
X. G. Geng, Y. Y. Zhai, and H. H. Dai, “Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy,” Adv. Math., 263, 123–153 (2014).
M. Blaszak and K. Marciniak, “R-matrix approach to lattice integrable systems,” J. Math. Phys., 35, 4661–4682 (1994).
X.-B. Hu and Z.-N. Zhu, “Some new results on the Blaszak–Marciniak lattice: Bäcklund transformation and nonlinear superposition formula,” J. Math. Phys., 39, 4766–4772 (1998).
W.-X. Ma and B. Fuchssteiner, “Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations,” J. Math. Phys., 40, 2400–2418 (1999).
X. B. Hu, D. L. Wang, and H. Tam, “Integrable extended Blaszak–Marciniak lattice and another extended lattice with their Lax pairs,” Theor. Math. Phys., 127, 738–743 (2001).
Z.-N. Zhu, X.Wu,_W. Xue, and Q. Ding, “Infinitely many conservation laws of two Blaszak–Marciniak three-field lattice hierarchies,” Phys. Lett. A, 297, 387–395 (2002).
R. Sahadevan and S. Khousalya, “Master symmetries for Volterra equation, Belov–Chaltikian and Blaszak–Marciniak lattice equations,” J. Math. Anal. Appl., 280, 241–251 (2003).
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York (1994).
D. Mumford, Tata Lectures on Theta II, Birkhäuser, Boston, Mass. (1984).
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This research was supported by the National Natural Science Foundation of China (Grant No. 11331008).
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 190, No. 1, pp. 21–47, January, 2017.
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Geng, X., Zeng, X. Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy. Theor Math Phys 190, 18–42 (2017). https://doi.org/10.1134/S0040577917010020
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DOI: https://doi.org/10.1134/S0040577917010020