The Initial-boundary Value Problem for the Ostrovsky-Vakhnenko Equation on the Half-line



We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line. This equation can be viewed as the short wave model for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can be recovered from its initial and boundary values via the solution of a vector Riemann-Hilbert problem formulated in the plane of a complex spectral parameter z.


Ostrovsky-Vakhnenko equation Initial-boundary value problem Riemann-Hilbert problem 

Mathematics Subject Classification (2010)

35Q15 30E25 31B20 


  1. 1.
    Degasperis, A., Procesi, M.: Asymptotic integrability. In: Symmetry and Perturbation Theory (Rome 1998). World Scientific Publishers, New Jersey (1999)Google Scholar
  2. 2.
    Kraenkel, R. A., Leblond, H., Manna, M. A.: An integrable evolution equation for surface waves in deep water. J. Phys. A Math. Theor. 47, 025208 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Vakhnenko, V. O.: Solitons in a nonlinear model medium. J. Phys. A Math. Gen. 25, 4181–7 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Parkes, E. J.: The stability of solutions of Vakhnenkos equation. J. Phys. A Math. Gen. 26, 6469–75 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Vakhnenko, V. O.: The existence of loop-like solutions of a model evolution equation. Ukr. J. Phys. 42, 104–10 (1997)Google Scholar
  6. 6.
    Vakhnenko, V. O.: High-frequency soliton-like waves in a relaxing medium. J. Math. Phys. 40, 2011–20 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Stepanyants, Y. A.: On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons. Chaos, Solitons Fractals 28, 193–204 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ostrovsky, L. A.: Nonlinear internal waves in a rotating ocean. Oceanology 18, 181–91 (1978)Google Scholar
  9. 9.
    Brunelli, J. C., Sakovich, S.: Hamiltonian structures for the Ostrovsky-Vakhnenko equation. Commun. Nonlinear Sci. Numer. Simul. 18, 56–62 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Davidson, M.: Continuity properties of the solution map for the generalized reduced Ostrovsky equation. J. Differ. Equ. 252, 3797–815 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khusnutdinova, K. R., Moore, K. R.: Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations. Wave Motion 48, 738–52 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Linares, F., Milan, A.: Local and global well-posedness for the Ostrovsky equation. J. Differ. Equ. 222, 325–40 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Stefanov, A., Shen, Y., Kevrekidis, P. G.: 2010 Well-posedness and small data scattering for the generalized Ostrovsky equation. J. Differ. Equ. 249, 2600–17 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Varlamov, V., Liu, Y.: Cauchy problem for the Ostrovsky equation, Discrete Contin. Dyn. Syst. 10, 731–53 (2004)MathSciNetMATHGoogle Scholar
  15. 15.
    Hone, A. N. W., Wang, J. P.: Prolongation algebras and Hamiltonian operators for peakon equations. Inverse Prob. 19, 129–45 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Boutet de Monvel, A., Shepelsky, D.: The Ostrovsky-Vakhnenko equation by a Riemann-Hilbert approach. J. Phys. A: Math. Theor. 48, 035204 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fokas, A. S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. Lond. A 453, 1411–1443 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fokas, A. S.: Integrable nonlinear evolution equations on the half-line. Commun. Math. Phys. 230, 1–39 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fokas, A. S.: A unified approach to boundary value problems. In: CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM (2008)Google Scholar
  20. 20.
    Lenells, J.: Initial-boundary value problems for integrable evolution equations with 33 Lax pairs. Phys. D 241, 857–875 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lenells, J.: The Degasperis-Procesi equation on the half-line. Nonlinear Anal. 76, 122–139 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Boutet de Monvel, A., Lenells, J., Shepelsky, D.: Long-time asymptotics for the Degasperis-Procesi equation on the half-line. arXiv:hep-th/1508:04097
  23. 23.
    Xu, J., Fan, E.: The unified transform method for the Sasa-Satsuma equation on the half-line. Proc. R. Soc. A 469, 20130068 (2013)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Xu, J., Fan, E.: The three wave equation on the half-line. Mod. Phys. Lett. A 378, 26–33 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xu, J.: Initial-boundary value problem for the two-component nonlinear schrödinger equation on the half-line. J. Non. Math. Phys. 23, 167–189 (2016)ADSCrossRefGoogle Scholar
  26. 26.
    Xu, J., Fan, E.: Initial-boundary value problem for integrable nonlinear evolution equations with 33 Lax pairs on the interval. Stud. Appl. Math. 136, 321–354 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Grimshaw, R., Pelinovsky, D.: Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete Contin. Dyn. Syst. 34, 557–66 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.College of ScienceUniversity of Shanghai for Science and TechnologyShanghaiPeople’s Republic of China
  2. 2.School of Mathematical Sciences, Key Laboratory of Mathematics for Nonlinear ScienceFudan UniversityShanghaiPeople’s Republic of China

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