Nagoya Mathematical Journal

Instability of periodic traveling waves for the symmetric regularized long wave equation

Jaime Angulo Pava and Carlos Alberto Banquet Brango

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We prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

Article information

Nagoya Math. J., Volume 219 (2015), 235-268.

Received: 28 March 2012
Revised: 30 January 2014
Accepted: 14 August 2014
First available in Project Euclid: 20 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q51: Soliton-like equations [See also 37K40]
Secondary: 35B35: Stability 35B10: Periodic solutions 35C07: Traveling wave solutions

linear instability nonlinear stability nonlinear instability symmetric regularized long wave equation


Angulo Pava, Jaime; Banquet Brango, Carlos Alberto. Instability of periodic traveling waves for the symmetric regularized long wave equation. Nagoya Math. J. 219 (2015), 235--268. doi:10.1215/00277630-2891870.

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  • [1] J. Angulo Pava, Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Differential Equations 235 (2007), 1–30.
  • [2] J. Angulo Pava, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Math. Surveys Monogr. 156, Amer. Math. Soc., Providence, 2009.
  • [3] J. Angulo Pava, C. Banquet, and M. Scialom, Stability for the modified and fourth-order Benjamin-Bona-Mahony equations, Discrete Contin. Dyn. Syst. 30 (2011), 851–871.
  • [4] J. Angulo Pava and F. Natali, Instability of periodic waves for nonlinear dispersive models, preprint, 2012.
  • [5] C. Banquet, The symmetric regularized-long-wave equation: Well-posedness and nonlinear stability, Phys. D 241 (2012), 125–133.
  • [6] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. Lond. Ser. A 338 (1972), 153–183.
  • [7] J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. Lond. Ser. A 344 (1975), 363–374.
  • [8] J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. Ser. 2 17 (1872), 55–108.
  • [9] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., revised, Grundlehren Math. Wiss. 67, Springer, New York, 1971.
  • [10] M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987), 160–197.
  • [11] M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal. 94 (1990), 308–348.
  • [12] D. B. Henry, J. F. Perez, and W. F. Wreszinski, Stability theory for solitary-wave solutions of scalar field equations, Comm. Math. Phys. 85 (1982), 351–361.
  • [13] P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory: With Application to Schrödinger Operators, Appl. Math. Sci. 113, Springer, New York, 1996.
  • [14] R. J. Iorio. Jr. and V. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Stud. Adv. Math. 70, Cambridge University Press, Cambridge, 2001.
  • [15] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1976.
  • [16] Z. Lin, Instability of nonlinear dispersive solitary waves, J. Funct. Anal. 255 (2008), 1191–1224.
  • [17] W. Magnus and S. Winkler, Hill’s Equation, Interscience Tracts in Pure Appl. Math. 20, Wiley, New York, 1966.
  • [18] E. Oberhettinger, Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973.
  • [19] S. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978.
  • [20] C. Seyler and D. Fenstermacher, A symmetric regularized-long-wave equation, Phys. Fluids 27 (1984), 4–7.
  • [21] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971.
  • [22] E. Vock and W. Hunziker, Stability of Schrödinger eigenvalue problems, Comm. Math. Phys. 83 (1982), 281–302.
  • [23] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), 51–67.