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.
References
[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. MR2309564 10.1016/j.jde.2007.01.003[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. MR2309564 10.1016/j.jde.2007.01.003
[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. MR2567568[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. MR2567568
[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. MR2784624 10.3934/dcds.2011.30.851[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. MR2784624 10.3934/dcds.2011.30.851
[6] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. Lond. Ser. A 338 (1972), 153–183. MR338584 10.1098/rspa.1972.0074[6] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. Lond. Ser. A 338 (1972), 153–183. MR338584 10.1098/rspa.1972.0074
[7] J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. Lond. Ser. A 344 (1975), 363–374. MR386438 10.1098/rspa.1975.0106[7] J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. Lond. Ser. A 344 (1975), 363–374. MR386438 10.1098/rspa.1975.0106
[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.[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. MR277773 0213.16602[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. MR277773 0213.16602
[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. MR901236 0656.35122 10.1016/0022-1236(87)90044-9[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. MR901236 0656.35122 10.1016/0022-1236(87)90044-9
[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. MR1081647 0711.58013 10.1016/0022-1236(90)90016-E[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. MR1081647 0711.58013 10.1016/0022-1236(90)90016-E
[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. MR678151 0546.35062 10.1007/BF01208719 euclid.cmp/1103921482
[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. MR678151 0546.35062 10.1007/BF01208719 euclid.cmp/1103921482
[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. MR1361167 0855.47002[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. MR1361167 0855.47002
[14] R. J. Iorio. Jr. and V. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Stud. Adv. Math. 70, Cambridge University Press, Cambridge, 2001. MR1826392[14] R. J. Iorio. Jr. and V. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Stud. Adv. Math. 70, Cambridge University Press, Cambridge, 2001. MR1826392
[16] Z. Lin, Instability of nonlinear dispersive solitary waves, J. Funct. Anal. 255 (2008), 1191–1224. MR2455496 1157.35096 10.1016/j.jfa.2008.06.003[16] Z. Lin, Instability of nonlinear dispersive solitary waves, J. Funct. Anal. 255 (2008), 1191–1224. MR2455496 1157.35096 10.1016/j.jfa.2008.06.003
[18] E. Oberhettinger, Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973. MR352886 0265.42002[18] E. Oberhettinger, Fourier Expansions: A Collection of Formulas, Academic Press, New York, 1973. MR352886 0265.42002
[19] S. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. MR493421 0401.47001[19] S. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. MR493421 0401.47001
[21] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971. MR304972 0232.42007[21] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971. MR304972 0232.42007
[22] E. Vock and W. Hunziker, Stability of Schrödinger eigenvalue problems, Comm. Math. Phys. 83 (1982), 281–302. MR649163 10.1007/BF01976045 euclid.cmp/1103920805
[22] E. Vock and W. Hunziker, Stability of Schrödinger eigenvalue problems, Comm. Math. Phys. 83 (1982), 281–302. MR649163 10.1007/BF01976045 euclid.cmp/1103920805
[23] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), 51–67. MR820338 0594.35005 10.1002/cpa.3160390103[23] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), 51–67. MR820338 0594.35005 10.1002/cpa.3160390103