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

Part of: Representations of solutions Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  11 January 2016

Jaime Angulo Pava  and
Carlos Alberto Banquet Brango
Jaime Angulo Pava
Affiliation:
Department of Mathematics, Instituto de Matemática e Estatística, Universidade de São Paulo, CEP 05508-090, São Paulo, SP, Brazil, angulo@ime.usp.br
Carlos Alberto Banquet Brango
Affiliation:
Departamento de Matemáticas y Estadística, Universidad de Córdoba Código, Postal 230002, Colombia, cbanquet@correo.unicordoba.edu.co
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Abstract

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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.

MSC classification

Secondary: 35Q51: Soliton-like equations 35B35: Stability 35B10: Periodic solutions 35C07: Traveling wave solutions

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 235 - 268
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Angulo Pava, J., Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations, J. Differential Equations 235 (2007), 1–30. MR 2309564. DOI 10.1016/j.jde.2007.01.003.CrossRefGoogle Scholar
[2] Angulo Pava, J., Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Math. Surveys Monogr. 156, Amer. Math. Soc., Providence, 2009. MR 2567568. DOI 10.1090/surv/156.Google Scholar
[3] Angulo Pava, J., Banquet, C., and Scialom, M., Stability for the modified and fourth-order Benjamin-Bona-Mahony equations, Discrete Contin. Dyn. Syst. 30 (2011), 851–871. MR 2784624. DOI 10.3934/dcds.2011.30.851.CrossRefGoogle Scholar
[4] Angulo Pava, J. and Natali, F., Instability of periodic waves for nonlinear dispersive models, preprint, 2012.Google Scholar
[5] Banquet, C., The symmetric regularized-long-wave equation: Well-posedness and nonlinear stability, Phys. D 241 (2012), 125–133.Google Scholar
[6] Benjamin, T. B., The stability of solitary waves, Proc. Roy. Soc. Lond. Ser. A 338 (1972), 153–183. MR 0338584.Google Scholar
[7] Bona, J., On the stability theory of solitary waves, Proc. Roy. Soc. Lond. Ser. A 344 (1975), 363–374. MR 0386438.Google Scholar
[8] Boussinesq, J., 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. 217 (1872), 55–108.Google Scholar
[9] Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., revised, Grundlehren Math. Wiss. 67, Springer, New York, 1971. MR 0277773.Google Scholar
[10] Grillakis, M., Shatah, J., and Strauss, W., Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987), 160–197. MR 0901236. DOI 10.1016/0022-1236(87)90044-9.CrossRefGoogle Scholar
[11] Grillakis, M., Shatah, J., and Strauss, W., Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal. 94 (1990), 308–348. MR 1081647. DOI 10.1016/0022-1236(90)90016-E.Google Scholar
[12] Henry, D. B., Perez, J. F., and Wreszinski, W. F., Stability theory for solitary-wave solutions of scalar field equations, Comm. Math. Phys. 85 (1982), 351–361. MR 0678151.Google Scholar
[13] Hislop, P. D. and Sigal, I. M., Introduction to Spectral Theory: With Application to Schrödinger Operators, Appl. Math. Sci. 113, Springer, New York, 1996. MR 1361167. DOI 10.1007/978-1-4612-0741-2.Google Scholar
[14] Iorio, R. J. Jr. and Iorio, V., Fourier Analysis and Partial Differential Equations, Cambridge Stud. Adv. Math. 70, Cambridge University Press, Cambridge, 2001. MR 1826392. DOI 10.1017/CBO9780511623745.Google Scholar
[15] Kato, T., Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1976. MR 0407617.Google Scholar
[16] Lin, Z., Instability of nonlinear dispersive solitary waves, J. Funct. Anal. 255 (2008), 1191–1224. MR 2455496. DOI 10.1016/j.jfa.2008.06.003.CrossRefGoogle Scholar
[17] Magnus, W. and Winkler, S., Hill's Equation, Interscience Tracts in Pure Appl. Math. 20, Wiley, New York, 1966. MR 0197830.Google Scholar
[18] Oberhettinger, E., Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973. MR 0352886.Google Scholar
[19] Reed, S. and Simon, B., Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. MR 0493421.Google Scholar
[20] Seyler, C. and Fenstermacher, D., A symmetric regularized-long-wave equation, Phys. Fluids 27 (1984), 4–7.Google Scholar
[21] Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971. MR 0304972.Google Scholar
[22] Vock, E. and Hunziker, W., Stability of Schrödinger eigenvalue problems, Comm. Math. Phys. 83 (1982), 281–302. MR 0649163.CrossRefGoogle Scholar
[23] Weinstein, M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), 51–67. MR 0820338. DOI 10.1002/cpa.3160390103.Google Scholar