Results in Mathematics

, 74:11 | Cite as

Finite Genus Solutions to the Coupled Burgers Hierarchy

  • Xianguo Geng
  • Wei Liu
  • Bo XueEmail author


The coupled Burgers hierarchy is derived with the aid of Lenard recursion sequences. Based on the characteristic polynomial of Lax matrix, a trigonal curve of arithmetic genus \(m-2\) is introduced, from which the meromorphic functions \(\phi _2,\phi _3\) and the Baker–Akhiezer \(\psi \) function are defined. The finite genus solutions for the coupled Burgers hierarchy are achieved by using asymptotic expansion of \(\phi _2,\phi _3\) and their Riemann theta function representation.


Finite genus solutions coupled Burgers hierarchy trigonal curves 

Mathematics Subject Classification

35Q51 35Q55 37K10 37K20 



  1. 1.
    Burgers, J.M.: Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verh. Nederl. Akad. Wetensch. Afd. Natuurk. Sect. 1(17), 281–334 (1939)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Burgers, J.M.: Application of a model system to illustrate some points of the statistical theory of free turbulence. Nederl. Akad. Wetensch. Proc. 43, 2–12 (1940)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Burgers, J.M.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1948)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Rosenblatt, M.: Remarks on the Burgers equation. J. Math. Phys. 9, 1129–1136 (1968)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ma, W.X., Zhou, Z.X.: Coupled integrable systems associated with a polynomial spectral problem and their Virasoro symmetry algebras. Progr. Theor. Phys. 96, 449–457 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Öziş, T., Aksan, E.N., Özdeş, A.: A finite element approach for solution of Burgers equation. Appl. Math. Comput. 139, 417–428 (2003)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Mukundan, V., Awasthi, A.: Efficient numerical techniques for Burgers equation. Appl. Math. Comput. 262, 282–297 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dubrovin, B.A.: Theta functions and non-linear equations. Rus. Math. Surv. 36, 11–92 (1981)CrossRefGoogle Scholar
  9. 9.
    Its, A.R., Matveev, V.B.: Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg–de Vries equation. Theor. Math. Phys. 23, 343–355 (1975)CrossRefGoogle Scholar
  10. 10.
    McKean, H.P.: Integrable systems and algebraic curves. In: Grmela, M., Marsden, J.E. (eds.) Global Analysis. Springer, Berlin (1979)Google Scholar
  11. 11.
    Matveev, V.B., Smirnov, A.O.: On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations. Lett. Math. Phys. 14, 25–31 (1987)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Date, E., Tanaka, S.: Periodic multi-soliton solutions of Korteweg–de Vries equation and Toda lattice. Progr. Theor. Phys. Suppl. 59, 107–125 (1976)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ma, Y.C., Ablowitz, M.J.: The periodic cubic Schrödinger equation. Stud. Appl. Math. 65, 113–158 (1981)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cao, C.W., Wu, Y.T., Geng, X.G.: Relation between the Kadometsev–Petviashvili equation and the confocal involutive system. J. Math. Phys. 40, 3948–3970 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Geng, X.G., Cao, C.W.: Decomposition of the (2+1)-dimensional Gardner equation and its quasi-periodic solutions. Nonlinearity 14, 1433–1452 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gesztesy, F., Holden, H.: Algebro-geometric solutions of the Camassa–Holm hierarchy. Rev. Mat. Iberoam. 19, 73–142 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Previato, E.: Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation. Duke Math. J. 52, 329–377 (1985)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions. Volume I: (1+1)-Dimensional Continuous Models. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  19. 19.
    Gesztesy, F., Ratnaseelan, R.: An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev. Math. Phys. 10, 345–391 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons, The Inverse Scattering Method. Consultants Bureau, New York (1984)zbMATHGoogle Scholar
  21. 21.
    Lundmark, H., Szmigielski, J.: An inverse spectral problem related to the Geng–Xue two-component peakon equation. Mem. Am. Math. Soc. 244, X+87 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ma, W.X.: Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs. J. Geom. Phys. 133, 10–16 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lundmark, H., Szmigielski, J.: Dynamics of interlacing peakons (and shockpeakons) in the Geng–Xue equation. J. Integr. Syst. 2, xyw014 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Geng, X.G., Liu, H.: The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation. J. Nonlinear Sci. 28, 739–763 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264, 2633–2659 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Dickson, R., Gesztesy, F., Unterkofler, K.: A new approach to the Boussinesq hierarchy. Math. Nachr. 198, 51–108 (1999)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Dickson, R., Gesztesy, F., Unterkofler, K.: Algebro-geometric solutions of the Boussinesq hierarchy. Rev. Math. Phys. 11, 823–879 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Geng, X.G., Wu, L.H., He, G.L.: Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions. Phys. D 240, 1262–1288 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    He, G.L., Wu, L.H., Geng, X.G.: Finite genus solutions to the mixed Boussinesq equation. Sci. Sin. Math. 42, 711–734 (2012)CrossRefGoogle Scholar
  30. 30.
    Geng, X.G., Zhai, Y.Y., Dai, H.H.: Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy. Adv. Math. 263, 123–153 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    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 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wu, L.H., Geng, X.G., He, G.L.: Algebro-geometric solutions to the Manakov hierarchy. Appl. Anal. 95, 769–800 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wei, J., Geng, X.G., Zeng, X.: Quasi-periodic solutions to the hierarchy of four-component Toda lattices. J. Geom. Phys. 106, 26–41 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Geng, X.G., Zeng, X.: Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy. Theor. Math. Phys. 190, 18–42 (2017)CrossRefGoogle Scholar
  35. 35.
    Ma, W.X.: Trigonal curves and algebro-geometric solutions to soliton hierarchies I. Proc. R. Soc. A 473, 20170232 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ma, W.X.: Trigonal curves and algebro-geometric solutions to soliton hierarchies II. Proc. R. Soc. A 473, 20170233 (2017)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1994)CrossRefGoogle Scholar
  38. 38.
    Mumford, D.: Tata Lectures on Theta II. Birkhäuser, Boston (1984)zbMATHGoogle Scholar
  39. 39.
    Farkas, H.M., Kra, I.: Riemann Surfaces. Springer, New York (1992)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China
  2. 2.Department of Mathematics and PhysicsShijiazhuang Tiedao UniversityShijiazhuangPeople’s Republic of China

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