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Algebro-geometric Quasi-periodic Solutions for a Discrete Coupled KdV Hierarchy

  • Xianguo Geng [2] ; Minxin Jia [1] ; Wei Jiao [1]
    1. [1] Zhengzhou University

      Zhengzhou University

      China

    2. [2] Henan Academy of Sciences, Zhengzhou University, North China University of Water Resources and Electric Power,
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 2, 2026
  • Idioma: inglés
  • Enlaces
  • Resumen

    • Based on the theory of tetragonal curves, we obtain algebro-geometric quasi-periodic solutions for a discrete coupled KdV (DCKdV) hierarchy, which arises from coupled five-point lattice equations. Starting from a discrete matrix spectral problem, the DCKdV hierarchy is generated via the zero-curvature equation. The associated spectral curve is a tetragonal curve and can be compactified as the four-sheeted compact Riemann surface, on which we define a Baker-Akhiezer function and study its analytical properties. Using the Abel map and Abelian differentials, all nonlinear flows of the hierarchy are linearized under Abel-Jacobi coordinates. This leads to the explicit Riemann theta function representations for the meromorphic and Baker-Akhiezer functions, thereby also obtaining the algebro-geometric quasi-periodic solutions of the DCKdV hierarchy. The work extends the finite-gap integration technique to the discrete coupled system with the Lax pair.

  • Referencias bibliográficas
    • 1. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary...
    • 2. Lou, S.Y., Huang, F.: Alice-Bob physics: Coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)
    • 3. Jagtap, A.D., Kharazmi, E., Karniadakis, G.E.: Conservative physics-informed neural networks on discrete domains for conservation laws:...
    • 4. Volterra, V., Brelot, M.: Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, Paris (1931)
    • 5. Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons: The Inverse Scattering Method. Consultants Bureau,...
    • 6. Hirota, R., Satsuma, J.: Soliton solutions of a coupled Korteweg-de Vries equation. Phys. Lett. A 85, 407–408 (1981)
    • 7. Satsuma, J., Hirota, R.: A coupled KdV equation is one case of the four-reduction of the KP hierarchy. J. Phys. Soc. Jpn. 51, 3390–3397...
    • 8. Foursov, M.: On integrable coupled KdV-type systems. Inverse Probl. 16, 259–274 (2000)
    • 9. Lou, S.Y., Tong, B., Hu, H.C., Tang, X.Y.: Coupled KdV equations derived from two-layer fluids. J. Phys. A: Math. Gen. 39, 513 (2006)
    • 10. Liu, P., Fu, P.K.: A coupled hybrid lattice: Its related continuous equation and symmetries. Commun. Theor. Phys. 56, 5–10 (2011)
    • 11. Liu, N.,Wen, X.Y.: Modulational instability and dynamics of rational soliton solutions for the coupled Volterra lattice equation associated...
    • 12. Zhang, Y.N., Hu, X.B., Sun, J.Q.: Numerical calculation of N-periodic wave solutions to coupled KdV-Toda-type equations. Proc. R. Soc....
    • 13. Liu, X.K., Wen, X.Y.: A discrete KdV equation hierarchy: Continuous limit, diverse exact solutions and their asymptotic state analysis....
    • 14. Jia, M.X., Geng, X.G.: Coupled five-point lattices: Lax pairs and Hamiltonian structures. Appl. Math. Lett. 164, 109484 (2025)
    • 15. Its, A.R., Kotlyarov, V.P.: Explicit formulas for solutions of the nonlinear Schrödinger equation. Dokl. Akad. Nauk Ukr. SSR Ser. A 11,...
    • 16. Krichever, I.M.: Algebraic curves and commuting matrix differential operators. Funct. Anal. Appl. 10, 144–146 (1976)
    • 17. Ma, Y.C., Ablowitz, M.J.: The periodic cubic Schrödinger equation. Stud. Appl. Math. 65, 113–158 (1981)
    • 18. Previato, E.: Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation. Duke Math. J. 52, 329–377 (1985)
    • 19. Rand, D., Steiglitz, K., Prucnal, P.R.: Multicomponent gap solitons in superposed grating structures. Opt. Lett. 30, 1695–1697 (2005)
    • 20. Flaschka, H., McLaughlin, D.W.: Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic...
    • 21. Kac, M., van Moerbeke, P.: On some periodic Toda lattices. Proc. Natl. Acad. Sci. U.S.A. 72, 1627– 1629 (1975)
    • 22. Date, E., Tanaka, S.: Analogue of inverse scattering theory for the discrete Hill’s equation and exact solutions for the periodic Toda...
    • 23. McKean, H.P.: The sine-Gordon and sinh-Gordon equations on the circle. Commun. Pure Appl.Math. 34, 197–257 (1981)
    • 24. Harnad, J., Wisse, M.A.: Isospectral flow in loop algebras and quasiperiodic solutions of the sine- Gordon equation. J. Math. Phys. 34,...
    • 25. Gesztesy, F., Holden, H., Michor, J., Teschl, G.: Algebro-geometric finite-band solutions of the Ablowitz–Ladik hierarchy. Int. Math....
    • 26. Miller, P.D., Ercolani, N.M., Krichever, I.M., Levermore, C.D.: Finite genus solutions to the Ablowitz- Ladik equations. Commun. Pure...
    • 27. Baldwin, S., Eilbeck, J.C., Gibbons, J., Ônishi, Y.: Abelian functions for cyclic trigonal curves of genus 4. J. Geom. Phys. 58, 450–467...
    • 28. Wei, J., Geng, X.G., Zeng, X.: Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices. Trans. Amer. Math. Soc. 371,...
    • 29. Xue, B., Du, H.L., Geng, X.G.: A Degasperis-Procesi equation II with multi-peakon solutions. Appl. Math. Lett. 121, 107463 (2021)
    • 30. Geng, X.G.,Wei, J., Zeng, X.: Algebro-geometric integration of the modified Belov-Chaltikian lattice hierarchy. Theor. Math. Phys. 199,...
    • 31. Wei, J., Geng, X.G., Wang, X.: Finite genus solutions of the generalized Merola-Ragnisco-Tu lattice hierarchy. J. Math. Phys. 63, 083503...
    • 32. Geng, X.G., Liu, W., Xue, B.: Explicit solutions of the coupled Bogoyavlensky lattice 1(2) hierarchy. Results Math. 76, 67 (2021)
    • 33. Zhai, Y.Y., Geng, X.G., Xue, B.: Riemann theta function solutions to the coupled long wave-short wave resonance equations. Anal. Math....
    • 34. Ônishi, Y.: Abelian functions for trigonal curves of degree four and determinantal formulae in the purely trigonal case. Int. J. Math....
    • 35. Geng, X.G., Zeng, X.: Quasi-periodic solutions of the Belov-Chaltikian lattice hierarchy. Rev. Math. Phys. 29, 1750025 (2017a)
    • 36. Geng, X.G., Zeng, X.: Application of the trigonal curve to the Blaszak-Marciniak lattice hierarchy. Theor. Math. Phys. 190, 18–42 (2017b)
    • 37. Zeng, X., Geng, X.G.: Finite-band solutions for the hierarchy of coupled Toda lattices. Acta Appl. Math. 154, 59–81 (2018)
    • 38. Wu, L.H., He, G.L., Geng, X.G.: A note on the quasi-periodic solutions of the modified Boussinesq hierarchy. J. Geom. Phys. 96, 133–145...
    • 39. Jia, M.X., Geng, X.G., Wei, J.: Algebro-geometric quasi-periodic solutions to the Bogoyavlensky lattice 2(3) equations. J. Nonlinear Sci....
    • 40. Jia, M.X., Geng, X.G., Liu, H.,Wei, J.: Application of the theory of tetragonal curves to the hierarchy of extended Volterra lattices....
    • 41. Jia, M.X., Geng, X.G., Li, R.M.: Algebro-geometric initial value problems for integrable nonlinear lattices: Tetragonal curves and Riemann...
    • 42. Zhao, Q.L., Li, C.X., Li, X.Y.: Application of tetragonal curve theory to the four-field Blaszak- Marciniak lattice hierarchy. Physica...
    • 43. Zeng, X., Jia, M.X., Geng, X.G.: Algebro-geometric quasi-periodic solutions to the generalized Drinfel’d-Sokolov hierarchy. J. Geom. Anal....
    • 44. Geng, X.G., Jia, M.X., Zeng, X.,Wei, J.: Riemann theta function solutions to a hierarchy of integrable semi-discrete equations. Russ....
    • 45. Geng, X.G., Zeng, X., Wei, J.: Applications of the theory of trigonal curves to the discrete coupled nonlinear Schrödinger hierarchy....
    • 46. Garifullin, R.N., Yamilov, R.I., Levi, D.: Classification of five-point differential-difference equations II. J. Phys. A: Math. Theor....
    • 47. Yamilov, R.I.: Symmetries as integrability criteria for differential-difference equations. J. Phys. D Appl. Phys. 39, R541–R623 (2006)
    • 48. Farkas, H.M., Kra, I.: Riemann Surfaces, 2nd edn. Springer, New York (1992)
    • 49. Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: Integrable Systems I. Springer, Berlin (1990)
    • 50. Miranda, R.: Algebraic Curves and Riemann Surfaces. American Mathematical Society, Providence, RI (1995)
    • 51. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1994)
    • 52. Deconinck, B., van Hoeij, M.: Computing Riemannmatrices of algebraic curves. PhysicaD152, 28–46 (2001)
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