Advertisement

Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1487–1530 | Cite as

Nonlinear Modulational Instability of Dispersive PDE Models

  • Jiayin Jin
  • Shasha Liao
  • Zhiwu LinEmail author
Article

Abstract

We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (for example the Whitham equation, the generalized KDV equation, the Benjamin–Ono equation), the nonlinear Schrödinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs. Second, for the KDV type equations the loss of derivative in the nonlinear terms is overcome in two complementary cases: (1) for smooth nonlinear terms and general dispersive operators, we construct higher order approximation solutions and then use energy type estimates; (2) for nonlinear terms of low regularity, with some additional assumptions on the dispersive operator, we use a bootstrap argument to overcome the loss of a derivative.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

Zhiwu Lin is supported in part by NSF Grants DMS-1411803 and DMS-1715201. Shasha Liao is partially supported by the China Scholarship Council No. 20150620040.

Compliance with Ethical Standards

Conflict of interest

There are no conflicts of interest for the research in this paper.

References

  1. 1.
    Angulo Pava J., Bona J.L., Scialom M.: Stability of cnoidal waves. Adv.Differ. Equ. 11(12), 1321–1374 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Angulo Pava J.: Nonlinear Dispersive Equations Existence and Stability of Solitary and Periodic Travelling Wave Solutions. Mathematical Surveys and Monographs, vol. 156. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  3. 3.
    Bardos C., Guo Y., Strauss W.: Stable and unstable ideal plane flows. Dedicated to the memory of Jacques-Louis Lions. Chin. Ann. Math. Ser. B 23(2), 149–164 (2002)zbMATHGoogle Scholar
  4. 4.
    Benjamin T.B., Feir J.E.: The disintegration of wave trains on deep water Part 1. Theory. J. Fluid Mech. 27(3), 417–437 (1967)ADSzbMATHGoogle Scholar
  5. 5.
    Bottman N., Deconinck B.: KdV cnoidal waves are spectrally stable. Discrete Contin. Dyn. Syst. 25(4), 1163–1180 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bronski, J.C., Hur, V.M., Johnson, M.A.: Modulational instability in equations of KdV type. New approaches to nonlinear waves, pp. 83–133. Lecture Notes in Physics, vol. 908. Springer, Cham, 2016.Google Scholar
  7. 7.
    Bronski J.C., Hur V.M.: Modulational instability and variational structure. Stud. Appl. Math. 132(4), 285–331 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bronski J.C., Johnson M.A.: The modulational instability for a generalized Korteweg–de Vries equation. Arch. Ration. Mech. Anal. 197(2), 357–400 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Deconinck B., Trichtchenko O.: High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete Contin. Dyn. Syst. A, 37(3), 1323–1358 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Deconinck B., Segal B.L.: The stability spectrumfor elliptic solutions to the focusing NLS equation. Phys. D 346, 1–19 (2017)MathSciNetGoogle Scholar
  11. 11.
    Ehrnström M., Groves M.D., Wahlén E.: On the existence and stability of solitary wave solutions to a class of evolution equations of Whitham type. Nonlinearity 25(10), 2903–2936 (2012)ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Gallay T., Haraguş M.: Stability of small periodic waves for the nonlinear Schrödinger equation. J. Differ. Equ. 234(2), 544–581 (2007)ADSzbMATHGoogle Scholar
  13. 13.
    Grenier E.: On the nonlinear instability of Euler and Prandtl equations. Commun. Pure Appl. Math. 53(9), 1067–1091 (2000)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. II. J. Funct.Anal. 94(2), 308–348 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Guo Y., Strauss W.A.: Instability of periodic BGK equilibria. Commun. Pure Appl. Math. 48(8), 861–894 (1995)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Haraguş M., Kapitula T.: On the spectra of periodic waves for infinite-dimensional Hamiltonian systems. Phys. D 237(20), 2649–2671 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    : Stability of periodic waves for the generalized BBM equation. Rev. Roum. Math. Pures Appl. 53(5-6), 445–463 (2008)MathSciNetGoogle Scholar
  18. 18.
    Hur, Vera Mikyoung., Johnson, Mathew A., (2015) Modulational instability in the Whitham equation for water waves. Stud. Appl. Math. 134(1), 120–143, 2015Google Scholar
  19. 19.
    Hur V.M., Johnson M.A.: Stability of periodic traveling waves for nonlinear dispersive equations. SIAM J. Math. Anal. 47(5), 3528–3554 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hur V.M., Pandey A.K.: Modulational instability in nonlinear nonlocal equations of regularized long wave type. Phys. D 325, 98–112 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hur, V.M., Pandey, A.K.: Modulational Instability in a Full-Dispersion Shallow Water Model. arXiv:1608.04685
  22. 22.
    Johnson M.A.: Stability of small periodic waves in fractional KdV-type equations. SIAM J. Math. Anal., 45(5), 3168–3193 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Johnson M.A.: Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg–de Vries equation. SIAM J. Math. Anal. 41(5), 1921–1947 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. Spectral theory and differential equations, pp. 25–70. Lecture Notes in Mathematics, vol. 448. Springer, Berlin, 1975Google Scholar
  25. 25.
    Kato T.: Perturbation Theory for Linear Operators Reprint of the 1980 edition. Classics in Mathematics.. Springer, Berlin (1995)Google Scholar
  26. 26.
    Kato, T.: Linear and quasi-linear equations of evolution of hyperbolic type. Hyperbolicity, pp. 125–191. C.I.M.E. Summer Sch., vol. 72. Springer, Heidelberg, 2011Google Scholar
  27. 27.
    Lighthill M.J.: Contributions to the theory of waves in non-linear dispersive systems. IMA J. Appl. Math. 1, 269–306 (1965)Google Scholar
  28. 28.
    Lin, Z., Zeng, C.: Instability, Index Theorem, and Exponential Trichotomy for Linear Hamiltonian PDEs. arXiv:1703.04016
  29. 29.
    Lin Zhiwu.: Nonlinear instability of ideal plane flows. Int. Math. Res. Not. 41, 2147–2178 (2004)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lin Z., Strauss W.: Nonlinear stability and instability of relativistic Vlasov–Maxwell systems. Commun. Pure Appl. Math. 60(6), 789–837 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Pazy, A.: Semigroups on Linear Operators and Applications to Partial Differential Equations. Springer, 1983Google Scholar
  32. 32.
    Whitham G.B.: Non-linear dispersion of water waves, J. Fluid Mech. 27, 399–412 (1967)ADSMathSciNetzbMATHGoogle Scholar
  33. 33.
    Zakharov V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9(2), 190–194 (1968)ADSGoogle Scholar
  34. 34.
    Zakharov V.E., Ostrovsky L.A.: Modulation instability: the beginning. Phys. D 238(5), 540–548 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations