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On the Spectral and Orbital Stability of Spatially Periodic Stationary Solutions of Generalized Korteweg-de Vries Equations

  • Todd Kapitula
  • Bernard DeconinckEmail author
Part of the Fields Institute Communications book series (FIC, volume 75)

Abstract

In this paper we generalize previous work on the spectral and orbital stability of waves for infinite-dimensional Hamiltonian systems to include those cases for which the skew-symmetric operator \(\mathcal{J}\) is singular. We assume that \(\mathcal{J}\) restricted to the orthogonal complement of its kernel has a bounded inverse. With this assumption and some further genericity conditions we (a) derive an unstable eigenvalue count for the appropriate linearized operator, and (b) show that the spectral stability of the wave implies its orbital (nonlinear) stability, provided there are no purely imaginary eigenvalues with negative Krein signature. We use our theory to investigate the (in)stability of spatially periodic waves to the generalized KdV equation for various power nonlinearities when the perturbation has the same period as that of the wave. Solutions of the integrable modified KdV equation are studied analytically in detail, as well as solutions with small amplitudes for higher-order pure power nonlinearities. We conclude by studying the transverse stability of these solutions when they are considered as planar solutions of the generalized KP-I equation.

Notes

Acknowledgements

BD acknowledges support from the National Science Foundation through grant NSF-DMS-0604546. TK gratefully acknowledges the support of the Jack and Lois Kuipers Applied Mathematics Endowment, a Calvin Research Fellowship, and the National Science Foundation under grant DMS-0806636. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.

References

  1. 1.
    Angulo, J.: Non-linear stability of periodic travelling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations. J. Differ. Equ. 235, 1–30 (2007)zbMATHCrossRefGoogle Scholar
  2. 2.
    Angulo, J., Quintero, J.: Existence and orbital stability of cnoidal waves for a 1D Boussinesq equation. Int. J. Math. Math. Sci. 2007, 52020 (2007)Google Scholar
  3. 3.
    Benzoni-Gavage, S.: Planar traveling waves in capillary fluids. Differ. Integr. Equ. 26(3/4), 439–485 (2013)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bona, J., Souganidis, P., Strauss, W.: Stability and instability of solitary waves of Korteweg-de Vries type. Proc. R. Soc. Lond. A 411, 395–412 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bottman, N., Deconinck, B.: KdV cnoidal waves are linearly stable. Disc. Cont. Dyn. Sys. A 25, 1163–1180 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bottman, N., Nivala, M., Deconinck, B.: Elliptic solutions of the defocusing NLS equation are stable. J. Phys. A 44, 285201 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bronski, J., Johnson, M.: The modulational instability for a generalized KdV equation. Arch. Ration. Mech. Anal. 197 (2), 357–400 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bronski, J., Johnson, M., Kapitula, T.: An index theorem for the stability of periodic traveling waves of KdV type. Proc. Roy. Soc. Edinburgh Sect. A 141(6), 1141–1173 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bronski, J., Johnson, M., Kapitula, T.: An instability index theory for quadratic pencils and applications. Comm. Math. Phys. 327(2), 521–550 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Carter, J., Deconinck, B., Kiyak, F., Kutz, J.N.: SpectrUW: a laboratory for the numerical exploration of spectra of linear operators. Math. Comp. Sim. 74, 370–379 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, M., Curtis, C., Deconinck, B., Lee, C., Nguyen, N.: Spectral stability of stationary solutions of a boussinesq system describing long waves in dispersive media. SIAM J. Appl. Dyn. Sys.9, 999–1018 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp global well-posedness for KdV and modified KdV on \(\mathbb{R}\) and \(\mathbb{T}\). J. Am. Math. Soc. 16(3), 705–749 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Deconinck, B., Kapitula, T.: The orbital stability of the cnoidal waves of the Korteweg-de Vries equation. Phys. Lett. A 374, 4018–4022 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Deconinck, B., Kutz, J.N.: Computing spectra of linear operators using Hill’s method. J. Comput. Phys. 219, 296–321 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Deconinck, B., Nivala, B.: The stability analysis of the periodic traveling wave solutions of the mkdv equation. Stud. Appl. Math. 126, 17–48 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Farah, L., Scialom, M.: On the periodic “good” Boussinesq equation. Proc. Am. Math. Soc. 138(3), 953–964 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Finkel, F., González-López, A., Rodríguez, M.: A new algebraization of the Lamé equation. J. Phys. A Math. Gen. 33, 1519–1542 (2000)zbMATHCrossRefGoogle Scholar
  18. 18.
    Gallay, T., Hǎrǎguş, M.: Orbital stability of periodic waves for the nonlinear Schrödinger equation. J. Dyn. Differ. Equ. 19, 825–865 (2007)zbMATHCrossRefGoogle Scholar
  19. 19.
    Gallay, T., Hǎrǎguş, M.: Stability of small periodic waves for the nonlinear Schrödinger equation. J. Differ. Equ. 234, 544–581 (2007)zbMATHCrossRefGoogle Scholar
  20. 20.
    Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry, II. J. Funct. Anal. 94, 308–348 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Hakkaev, S., Stanislavova, M., Stefanov, A.: Transverse instability for periodic waves of KP-I and Schrödinger equations. Indiana Univ. Math. J. 61(2), 461–492 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Hakkaev, S., Stanislavova, M., Stefanov, A.: Linear stability analysis for periodic traveling waves of the Boussinesq equation and the Klein-Gordon-Zakharov system. Proc. Roy. Soc. Edinburgh A 144(3), 455–489 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Hǎrǎguş, M.: Transverse spectral stability of small periodic traveling waves for the KP equation. Stud. Appl. Math. 126, 157–185 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hǎrǎguş, M., Kapitula, T.: On the spectra of periodic waves for infinite-dimensional Hamiltonian systems. Phys. D 237(20), 2649–2671 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jian-Ping, Y., Yong-Li, S.: Weierstrass elliptic function solutions to nonlinear evolution equations. Commun. Theor. Phys. 50(2), 295–298 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Johnson, M.: Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation. SIAM J. Math. Anal. 41(5), 1921–1947 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Johnson, M.: The transverse instability of periodic waves in Zakharov–Kuznetsov type equations. Stud. Appl. Math. 124(4), 323–345 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Johnson, M., Zumbrun, K.: Rigorous justification of the Whitham Modulation Equations for the generalized Korteweg-de Vries equation. Stud. Appl. Math. 125(1), 69–89 (2010)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Johnson, M., Zumbrun, K.: Transverse instability of periodic traveling waves in the generalized Kadomtsev-Petviashvili equation. SIAM J. Math. Anal. 42(6), 2681–2702 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Johnson, M., Zumbrun, K., Bronski, J.: On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions. Phys. D 239(23–24), 2057–2065 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Kapitula, T., Promislow, K.: Stability indices for constrained self-adjoint operators. Proc. Am. Math. Soc. 140(3), 865–880 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Kapitula, T., Promislow, K.: Spectral and Dynamical Stability of Nonlinear Waves. Springer, New York (2013)zbMATHCrossRefGoogle Scholar
  33. 33.
    Kapitula, T., Stefanov, A.: A Hamiltonian-Krein (instability) index theory for KdV-like eigenvalue problems. Stud. Appl. Math. 132(3), 183–211 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Kapitula, T., Kevrekidis, P., Sandstede, B.: Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems. Phys. D 195(3&4), 263–282 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Kapitula, T., Kevrekidis, P., Sandstede, B.: Addendum: Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems. Phys. D 201(1&2), 199–201 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Kapitula, T., Law, K., Kevrekidis, P.: Interaction of excited states in two-species Bose-Einstein condensates: a case study. SIAM J. Appl. Dyn. Sys. 9(1), 34–61 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Magnus, W., Winkler, S.: Hill’s Equation, volume 20 of Interscience Tracts in Pure and Applied Mathematics. Interscience Publishers, New York (1966)Google Scholar
  38. 38.
    Nivala, M., Deconinck, B.: Periodic finite-genus solutions of the KdV equation are orbitally stable. Phys. D 239(13), 1147–1158 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Olver, F., Lozier, D., Boisvert, R., Clark, C. (eds.): NIST handbook of mathematical functions. US Department of Commerce National Institute of Standards, Washington, DC (2010)zbMATHGoogle Scholar
  40. 40.
    Pava, J., Natali, F.: (non)linear instability of periodic traveling waves: Klein–Gordon and KdV type equations. Adv. Nonlinear Anal. 3(2), 95–123 (2014)Google Scholar
  41. 41.
    Pava, J., Bona, J., Scialom, M.: Stability of cnoidal waves. Adv. Differ. Equ. 11(12), 1321–1374 (2006)zbMATHGoogle Scholar
  42. 42.
    Pelinovsky, D.: Spectral stability of nonlinear waves in KdV-type evolution equations. In: Kirillov, O., Pelinovsky, D. (eds.) Nonlinear Physical Systems: Spectral Analysis, Stability, and Bifurcations, pp. 377–400. Wiley-ISTE, Hoboken (2014)CrossRefGoogle Scholar
  43. 43.
    Rademacher, J., Sandstede, B., Scheel, A.: Computing absolute and essential spectra using continuation. Phys. D 229(1&2), 166–183 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Rousset, F., Tzvetkov, N.: A simple criterion of transverse linear instability for solitary waves. Math. Res. Lett. 17(1) 157–169 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Stanislavova, M., Stefanov, A.: Stability analysis for traveling waves of second order in time PDE’s. Nonlinearity 25, 2625–2654 (2012)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsGrand RapidsUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

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