Skip to main content
SpringerLink
Log in

Algebro-geometric Solutions for the Derivative Burgers Hierarchy

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

Though completely integrable Camassa–Holm (CH) equation and Degasperis–Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyperelliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we study algebro-geometric solutions for the derivative Burgers (DB) equation, which is derived by Qiao and Li (2004) as a short wave model of the DP equation with the help of functional gradient and a pair of Lenard operators. Based on the characteristic polynomial of a Lax matrix for the DB equation, we introduce a third order algebraic curve \(\mathcal {K}_{r-1}\) with genus \(r-1\), from which the associated Baker–Akhiezer functions, meromorphic function, and Dubrovin-type equations are constructed. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker–Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DB hierarchy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The computation of the genus see Dickson et al. (1999a, b).

  2. \(\sigma ^{r-1}\mathcal {K}_{r-1}\)= \(\underbrace{\mathcal {K}_{r-1}\times \cdots \times \mathcal {K}_{r-1}}_{r-1}.\)

  3. For the definition of a nonspecial divisor, see Farkas and Kra (1992).

  4. Here sums with upper limits strictly less than their lower limits are interpreted as zero.

References

  • Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974)

    MathSciNet  MATH  Google Scholar 

  • Belokolos, E.D., Bobenko, A.I., Enol’skii, V.Z., Its, A.R., Matveev, V.B.: Algebro-geometric Approach to Nonlinear Integrable Equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  • Bulla, W., Gesztesy, F., Holden, H., Teschl, G.: Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies. Am. Math. Soc. 135, 1–79 (1998)

    MathSciNet  Google Scholar 

  • Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. R. Soc. Lond. A 118, 557–583 (1928)

    Article  MATH  Google Scholar 

  • Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Date, E., Tanaka, S.: Periodic multi-soliton solutions of Korteweg–de Vries equation and Toda lattice. Progr. Theor. Phys. 59, 107–125 (1976)

    Article  Google Scholar 

  • Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, pp. 23–37. World Scientific, River Edge (1999)

    Google Scholar 

  • Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)

    Article  MathSciNet  Google Scholar 

  • Dickson, R., Gesztesy, F., Unterkofler, K.: A new approach to the Boussinesq hierarchy. Math. Nachr. 198, 51–108 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Dickson, R., Gesztesy, F., Unterkofler, K.: Algebro-geometric solutions of the Boussinesq hierarchy. Rev. Math. Phys. 11, 823–879 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Dubrovin, B.A.: Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties. Funct. Anal. Appl. 11, 265–277 (1977)

    Article  MathSciNet  Google Scholar 

  • Dubrovin, B.A.: Theta functions and nonlinear equations. Russ. Math. Surv. 36, 11–80 (1981)

    Article  MathSciNet  Google Scholar 

  • Dubrovin, B.A.: Matrix finite-gap operators. Rev. Sci. Tech. 23, 33–78 (1983)

    MathSciNet  Google Scholar 

  • Fan, E.G.: The positive and negative Camassa–Holm-\(\gamma \) hierarchies, zero curvature representations, bi-Hamiltonian structures, and algebro-geometric solutions. J. Math. Phys. 50, 013525 (2009)

    Article  MathSciNet  Google Scholar 

  • Farkas, H.M., Kra, I.: Riemann Surfaces, 2nd edn. Springer, New York (1992)

    Book  MATH  Google Scholar 

  • Geng, X.G., Wu, L.H., He, G.L.: Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions. Physica D 240, 1262–1288 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Geronimo, J.S., Gesztesy, F., Holden, H.: Algebro-geometric solutions of the Baxter–Szegö difference equation. Commun. Math. Phys. 258, 149–177 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Gesztesy, F., Ratneseelan, R.: An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev. Math. Phys. 10, 345–391 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Gesztesy, F., Holden, H.: Algebro-geometric solutions of the Camassa–Holm hierarchy. Rev. Mat. Iberoam. 19, 73–142 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-geometric Solutions, Vol. I: (1+1)-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  • Gesztesy, F., Holden, H., Michor, J., Teschl, G.: Soliton Equations and Their Algebro-geometric Solutions, Vol. II: (1+1)-Dimensional Discrete Models. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  • Hon, Y.C., Fan, E.G.: Uniformly constructing finite-band solutions for a family of derivative nonlinear Schrödinger equations. Chaos Solitons Fractals 24, 1087–1096 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Ivanov, R.I.: Water waves and integrability. Philos. Trans. R. Soc. Lond. A 365, 2267–2280 (2007)

    Article  MATH  Google Scholar 

  • Johnson, R.S.: Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid. Mech. 455, 63–82 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Kohlenberg, J., Lundmark, H., Szmigielski, J.: The inverse spectral problem for the discrete cubic string. Inverse Probl. 23, 99–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Krichever, I.M.: Algebraic-geometric construction of the Zaharov–Sabat equations and their periodic solutions. Dokl. Akad. Nauk SSSR 227, 394–397 (1976)

    Google Scholar 

  • Krichever, I.M.: Integration of nonlinear equations by the methods of algebraic geometry. Funct. Anal. Appl. 11, 12–26 (1977)

    Article  MATH  Google Scholar 

  • Lundmark, H.: Formation and dynamics of shock waves in the Degasperis–Procesi equation H Lundmark. J. Nonlinear Sci. 17, 169–198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, Y.C., Ablowitz, M.J.: The periodic cubic Schrödinger equation. Stud. Appl. Math. 65, 11–80 (1981)

    MathSciNet  Google Scholar 

  • Matsuno, Y.: Cusp and loop soliton solutions of short-wave models for the Camassa–Holm and Degasperis–Procesi equations. Phys. Lett. A 359, 451–457 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Matveev, V.B., Yavor, M.I.: Solutions presque périodiques et à \(N\)-solitons de l’équation hydrodynamique non linéaire de Kaup. Ann. Inst. H. Poincaré Sect. A 31, 25–41 (1979)

    MathSciNet  MATH  Google Scholar 

  • Mumford, D.: Tata Lectures on Theta II. Birkhäuser, Boston (1984)

    MATH  Google Scholar 

  • Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons: The Inverse Scattering Methods. Consultants Bureau, New York (1984)

    Google Scholar 

  • Qiao, Z.J.: r-matrix and algebraic-geometric solution for an integrable symplectic map, preprint 1996. Chin. Sci. Bull. 43, 1149–1153 (1998)

    Google Scholar 

  • Qiao, Z.J.: Generalized r-matrix structure and algebro-geometric solution for integrable system. Rev. Math. Phys. 13, 545–586 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao, Z.J.: r-matrix and algebro-geometric solution for the AKNS system. Theor. Math. Phys. 127, 827–834 (2001)

    Article  MATH  Google Scholar 

  • Qiao, Z.J.: The Camassa–Holm hierarchy, \(N\)-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold. Commun. Math. Phys. 239, 309–341 (2003)

    Article  MATH  Google Scholar 

  • Qiao, Z.J.: Integrable hierarchy, \(3 \times 3\) constrained systems and parametric solutions. Acta Appl. Math. 83, 199–220 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Qiao, Z.J., Li, S.T.: A new integrable hierarchy, parametric solutions and traveling wave solutions. Math. Phys. Anal. Geom. 7, 289–308 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou, R.G.: The finite-band solution of the Jaulent–Miodek equation. J. Math. Phys. 38, 2535–2546 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their sincerest thanks to referees and editors for constructive suggestions and helpful comments. This work was partially supported by grants from the National Science Foundation of China (Project Nos. 10971031; 11271079; 11075055; 11171295), Doctoral Programs Foundation of the Ministry of Education of China, and the Shanghai Shuguang Tracking Project (Project 08GG01). Hou and Qiao was partially supported by the Norman Hackerman Advanced Research Program under Grant Number 003599-0001-2009.

Author information

Authors and Affiliations

  1. School of Mathematical Science, Institute of Mathematics and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai, 200433, People’s Republic of China

    Yu Hou & Engui Fan

  2. Department of Mathematics, University of Texas-Pan American, Edinburg, TX, 78539, USA

    Yu Hou & Zhijun Qiao

  3. School of Mathematical Sciences, Zhaoqing University, Zhaoqing, 526061, Guangdong, People’s Republic of China

    Zhong Wang

Authors
  1. Yu Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Engui Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhijun Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhijun Qiao.

Additional information

Communicated by Paul Newton.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hou, Y., Fan, E., Qiao, Z. et al. Algebro-geometric Solutions for the Derivative Burgers Hierarchy. J Nonlinear Sci 25, 1–35 (2015). https://doi.org/10.1007/s00332-014-9219-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-014-9219-4

Keywords

Mathematics Subject Classification

Search

Navigation