Supercritical Zakharov–Kuznetsov equation posed on bounded rectangles

Abstract

An initial boundary value problem for the 2D generalized Zakharov–Kuznetsov equation posed on a bounded rectangle is considered. Supercritical (higher than two) integer powers in nonlinearity have been studied. Results on the existence, uniqueness and exponential decay of solutions are presented.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Bona, J.L., Smith, R.W.: The initial-value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 278, 555–601 (1975)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Castelli, M., Doronin, G.: Modified and subcritical Zakharov–Kuznetsov equations posed on rectangles. J. Differ. Equ. (In Press). https://doi.org/10.1016/j.jde.2020.11.025

  3. 3.

    Castelli, M., Doronin, G.: Modified Zakharov–Kuznetsov equation posed on a strip. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2020.1754402

    Article  Google Scholar 

  4. 4.

    Doronin, G.G., Larkin, N.A.: Stabilization of regular solutions for the Zakharov–Kuznetsov equation posed on bounded rectangles and on a strip. Proc. Edinb. Math. Soc. (2) 58, 661–682 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Evans, L. C.: Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence (2010). xxii+749 pp. ISBN: 978-0-8218-4974-3

  6. 6.

    Faminskii, A.V.: Cauchy, The, problem for the Zakharov–Kuznetsov equation (Russian), Differentsial’nye Uravneniya, 31, 1070–1081. Engl. transl. in: Differential Equations 31(1995), 1002–1012 (1995)

  7. 7.

    Faminskii, A.V.: Initial-boundary value problems in a rectangle for two-dimensional Zakharov–Kuznetsov equation. J. Math. Anal. Appl. 463(2), 760–793 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Farah, L.G., Linares, F., Pastor, A.: A note on the 2D generalized Zakharov–Kuznetsov equation: local, global, and scattering results. J. Differ. Equ. 253, 2558–2571 (2012)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Farah, L. G., Holmer, J., Roudenko, S.: Instability of solitons—revisited, II: the supercritical Zakharov–Kuznetsov equation. Nonlinear dispersive waves and fluids, pp. 89–109. Contemp. Math., 725, Amer. Math. Soc., Providence (2019)

  10. 10.

    Farah, L.G., Pastor, A.: On well-posedness and wave operator for the gKdV equation. Bull. Sci. Math. 137(3), 229–241 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Gruenrock, A.: On the generalized Zakharov–Kuznetsov equation at critical regularity. arXiv:1509.09146 [math.AP]

  12. 12.

    Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Springer, Berlin (1995). ISBN: 3-540-58661-X

  13. 13.

    Kato, Tomoya: Well-posedness for the generalized Zakharov–Kuznetsov equation on modulation spaces. J. Fourier Anal. Appl. 23(3), 612–655 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kato, Tomoya: The Cauchy problem for the generalized Zakharov–Kuznetsov equation on modulation spaces. J. Differ. Equ. 264(5), 3402–3444 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1968)

    Book  Google Scholar 

  16. 16.

    Larkin, N.A., Padilha, M.V.: Global regular solutions to one problem of Saut-Temam for the 3D Zakharov–Kuznetsov equation. Appl. Math. Optim. 77(2), 253–274 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Larkin, N., Luckesi, J.: Initial-boundary value problems for generalized dispersive equations of higher orders posed on bounded intervals, recommended to cite as : Larkin, N.A. & Luchesi. J. Appl. Math. Optim. (2019). https://doi.org/10.1007/s00245-019-09579-w

    Article  Google Scholar 

  18. 18.

    Larkin, N.: Decay of regular solutions for the critical 2D Zakharov-Kuznetsov equation posed on rectangles, Recommended to cite as: N.A. Larkin. J. Math. Phys. 61, 061509 (2020). https://doi.org/10.1063/1.5100284

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Linares, F., Pastor, A.: Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation. J. Funct. Anal. 260, 1060–1085 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Linares, F., Pastor, A.: Well-posedness for the 2D modified Zakharov–Kuznetsov equation. SIAM J. Math. Anal. 41(4), 1323–1339 (2009)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Linares, F., Pastor, A., Saut, J.-C.: Well-posedness for the ZK equation in a cylinder and on the background of a KdV Soliton. Comm. Part. Diff. Equ. 35, 1674–1689 (2010)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Linares, F., Saut, J.-C.: The Cauchy problem for the 3D Zakharov–Kuznetsov equation. Disc. Cont. Dyn. Syst. A 24, 547–565 (2009)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Sipcic, R., Benney, D.J.: Lump interactions and collapse in the modified Zakharov–Kuznetsov equation. Stud. Appl. Math. 105(4), 385–403 (2000)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Rosier, L.: Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 2, 33–55 (1997)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Chen, Mo, Rosier, L.: Exact controllability of the Zakharov–Kuznetsov equation by the flatness approach. Discrete Contin. Dyn. Syst. B. https://doi.org/10.3934/dcdsb.2020080

  26. 26.

    Saut, J.-C., Temam, R.: An initial boundary-value problem for the Zakharov–Kuznetsov equation. Adv. Differ. Equ. 15, 1001–1031 (2010)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Saut, J.-C., Temam, R., Wang, C.: An initial and boundary-value problem for the Zakharov–Kuznetsov equation in a bounded domain. J. Math. Phys. 53, 115612 (2012)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Tsutsumi, M., Mukasa, T., Iino, R.: On the generalized Korteweg–de Vries equation. Proc. Jpn. Acad. 46, 921–925 (1970)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Zakharov, V.E., Kuznetsov, E.A.: On three-dimensional solitons. Sov. Phys. JETP 39, 285–286 (1974)

    Google Scholar 

Download references

Acknowledgements

We appreciate very much fruitful and constructive comments and suggestions of the reviewer.

Author information

Affiliations

Authors

Corresponding author

Correspondence to G. Doronin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

M. Castelli: Partially supported by CAPES.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Castelli, M., Doronin, G. Supercritical Zakharov–Kuznetsov equation posed on bounded rectangles. Z. Angew. Math. Phys. 72, 37 (2021). https://doi.org/10.1007/s00033-020-01463-w

Download citation

Keywords

  • gZK equation
  • Well-posedness
  • Exponential decay

Mathematics Subject Classification

  • 35M20
  • 35Q72