# American Institute of Mathematical Sciences

August  2011, 30(3): 851-871. doi: 10.3934/dcds.2011.30.851

## Stability for the modified and fourth-order Benjamin-Bona-Mahony equations

 1 Department of Mathematics, IME-USP, Rua do Matão 1010, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil 2 Department of Mathematics, IMECC-UNICAMP, Rua Sérgio Buarque de Holanda 651, CEP 13083-859, Campinas, SP, Brazil, Brazil

Received  November 2009 Revised  December 2010 Published  March 2011

In this work we establish new results about the existence of smooth, explicit families of periodic traveling waves for the modified and fourth-order Benjamin-Bona-Mahony equations. We also prove, under certain conditions, that these families are nonlinearly stable in the energy space. The techniques employed may be of further use in the study of the stability of periodic traveling-wave solutions of other nonlinear evolution equations.
Citation: Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851
##### References:
 [1] J. Albert, Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation,, J. Differential Equations, 63 (1986), 117.  doi: 10.1016/0022-0396(86)90057-4.  Google Scholar [2] J. Angulo, Stability of cnoidal waves to Hirota-Satsuma systems,, Mat. Contemp., 27 (2004), 189.   Google Scholar [3] J. Angulo, Non-linear stability of periodic traveling-wave solutions to the Schrödinger and the modified Korteweg-de Vries,, J. Differential Equations, 235 (2007), 1.  doi: 10.1016/j.jde.2007.01.003.  Google Scholar [4] J. Angulo, "Nonlinear Dispersive Evolution Equations: Existence and Stability of Solitary and Periodic Traveling-Waves Solutions,", Mathematical Surveys and Monographs Series (SURV), (2009).   Google Scholar [5] J. Angulo, C. Banquet and M. Scialom, Nonlinear stability of periodic traveling-wave solutions for the regularized Benjamin-Ono equation and BBM equation,, preprint, ().   Google Scholar [6] J. Angulo and F. Natali, Positivity properties of the Fourier transform and the stability of periodic traveling-wave solutions,, SIAM, 40 (2008), 1123.  doi: 10.1137/080718450.  Google Scholar [7] J. Angulo and F. Natali, Stability and instability of periodic traveling-wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations,, Phys. D, 238 (2009), 603.  doi: 10.1016/j.physd.2008.12.011.  Google Scholar [8] J. Angulo, J. Bona and M. Scialom, Stability of cnoidal waves,, Adv. Differential Equations, 11 (2006), 1321.   Google Scholar [9] T. Benjamin, Lectures on nonlinear wave motion,, Nonlinear Wave Motion, 15 (1974), 3.   Google Scholar [10] T. Benjamin, The stability of solitary waves,, Proc. R. Soc. Lond. Ser. A, 338 (1972), 153.   Google Scholar [11] T. Benjamin, J. Bona and J. Mahony, Models equations for long waves in nonlinear dispersive systems,, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar [12] J. Bona, On the stability theory of solitary waves,, Proc. R. Soc. Lond. Ser. A, 344 (1975), 363.  doi: 10.1098/rspa.1975.0106.  Google Scholar [13] J. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241.   Google Scholar [14] J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations,, Discrete Contin. Dyn. Syst., 23 (2009), 1253.   Google Scholar [15] P. Byrd and M. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," $2^{nd}$ edition,, Springer, (1971).   Google Scholar [16] K. El Dika, Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation,, Discrete Contin. Dyn. Syst., 13 (2005), 583.  doi: 10.3934/dcds.2005.13.583.  Google Scholar [17] T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation,, J. Differential Equations, 234 (2007), 544.  doi: 10.1016/j.jde.2006.12.007.  Google Scholar [18] T. 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Appl., 347 (2008), 428.  doi: 10.1016/j.jmaa.2008.06.033.  Google Scholar [24] E. Oberhettinger, "Fourier Expansions: A Collection of Formulas,", Academic Press, (1973).   Google Scholar [25] D. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321.  doi: 10.1017/S0022112066001678.  Google Scholar [26] D. Peregrine, Long waves on a beach,, J. Fluid Mech., 27 (1967), 815.  doi: 10.1017/S0022112067002605.  Google Scholar [27] P. Souganidis and W. Strauss, Instability of a class of dispersive solitary waves,, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 195.   Google Scholar [28] E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton University Press, (1970).   Google Scholar [29] M. Wadati, Wave propagation in nonlinear lattice I,, J. Phys. Soc. Japan, 38 (1975), 673.  doi: 10.1143/JPSJ.38.673.  Google Scholar [30] M. Wadati, Wave propagation in nonlinear lattice II,, J. Phys. Soc. Japan, 38 (1975), 681.  doi: 10.1143/JPSJ.38.681.  Google Scholar [31] M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Comm. PDE, 12 (1987), 1133.  doi: 10.1080/03605308708820522.  Google Scholar [32] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations,, Comm. Pure Appl. Math., 39 (1986), 51.  doi: 10.1002/cpa.3160390103.  Google Scholar [33] L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type,, J. Differential Equations, 188 (2003), 1.  doi: 10.1016/S0022-0396(02)00061-X.  Google Scholar

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##### References:
 [1] J. Albert, Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation,, J. Differential Equations, 63 (1986), 117.  doi: 10.1016/0022-0396(86)90057-4.  Google Scholar [2] J. Angulo, Stability of cnoidal waves to Hirota-Satsuma systems,, Mat. Contemp., 27 (2004), 189.   Google Scholar [3] J. Angulo, Non-linear stability of periodic traveling-wave solutions to the Schrödinger and the modified Korteweg-de Vries,, J. Differential Equations, 235 (2007), 1.  doi: 10.1016/j.jde.2007.01.003.  Google Scholar [4] J. Angulo, "Nonlinear Dispersive Evolution Equations: Existence and Stability of Solitary and Periodic Traveling-Waves Solutions,", Mathematical Surveys and Monographs Series (SURV), (2009).   Google Scholar [5] J. Angulo, C. Banquet and M. Scialom, Nonlinear stability of periodic traveling-wave solutions for the regularized Benjamin-Ono equation and BBM equation,, preprint, ().   Google Scholar [6] J. Angulo and F. Natali, Positivity properties of the Fourier transform and the stability of periodic traveling-wave solutions,, SIAM, 40 (2008), 1123.  doi: 10.1137/080718450.  Google Scholar [7] J. Angulo and F. Natali, Stability and instability of periodic traveling-wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations,, Phys. D, 238 (2009), 603.  doi: 10.1016/j.physd.2008.12.011.  Google Scholar [8] J. Angulo, J. Bona and M. Scialom, Stability of cnoidal waves,, Adv. Differential Equations, 11 (2006), 1321.   Google Scholar [9] T. Benjamin, Lectures on nonlinear wave motion,, Nonlinear Wave Motion, 15 (1974), 3.   Google Scholar [10] T. Benjamin, The stability of solitary waves,, Proc. R. Soc. Lond. Ser. A, 338 (1972), 153.   Google Scholar [11] T. Benjamin, J. Bona and J. Mahony, Models equations for long waves in nonlinear dispersive systems,, Philos. Trans. R. Soc. Lond. Ser. A, 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar [12] J. Bona, On the stability theory of solitary waves,, Proc. R. Soc. Lond. Ser. A, 344 (1975), 363.  doi: 10.1098/rspa.1975.0106.  Google Scholar [13] J. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241.   Google Scholar [14] J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations,, Discrete Contin. Dyn. Syst., 23 (2009), 1253.   Google Scholar [15] P. Byrd and M. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," $2^{nd}$ edition,, Springer, (1971).   Google Scholar [16] K. El Dika, Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation,, Discrete Contin. Dyn. Syst., 13 (2005), 583.  doi: 10.3934/dcds.2005.13.583.  Google Scholar [17] T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation,, J. Differential Equations, 234 (2007), 544.  doi: 10.1016/j.jde.2006.12.007.  Google Scholar [18] T. Gallay and M. Hărăguş, Orbital stability of periodic waves for the nonlinear Schrödinger equation,, J. Dynam. Differential Equations, 19 (2007), 825.  doi: 10.1007/s10884-007-9071-4.  Google Scholar [19] M. Hărăguş, Stability of periodic waves for the generalized BBM equation,, Rev. Roumaine Math. Pures Appl., 53 (2008), 445.   Google Scholar [20] R. Iorio Jr. and V. Iorio, "Fourier Analysis and Partial Differential Equations,", Cambridge Stud. Adv. Math. \textbf{70}, 70 (2001).   Google Scholar [21] S. Karlin, "Total Positivity,", Stanford University Press, (1968).   Google Scholar [22] J. Miller and M. Weinstein, Asymptotic stability of solitary waves for the regularized long-wave equation,, Comm. Pure Appl. Math., 495 (1996), 399.  doi: 10.1002/(SICI)1097-0312(199604)49:4<399::AID-CPA4>3.0.CO;2-7.  Google Scholar [23] F. Natali and A. Pastor, Stability and instability of periodic standing wave solutions for some Klein-Gordon equations,, J. Math. Anal. Appl., 347 (2008), 428.  doi: 10.1016/j.jmaa.2008.06.033.  Google Scholar [24] E. Oberhettinger, "Fourier Expansions: A Collection of Formulas,", Academic Press, (1973).   Google Scholar [25] D. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321.  doi: 10.1017/S0022112066001678.  Google Scholar [26] D. Peregrine, Long waves on a beach,, J. Fluid Mech., 27 (1967), 815.  doi: 10.1017/S0022112067002605.  Google Scholar [27] P. Souganidis and W. Strauss, Instability of a class of dispersive solitary waves,, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 195.   Google Scholar [28] E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton University Press, (1970).   Google Scholar [29] M. Wadati, Wave propagation in nonlinear lattice I,, J. Phys. Soc. Japan, 38 (1975), 673.  doi: 10.1143/JPSJ.38.673.  Google Scholar [30] M. Wadati, Wave propagation in nonlinear lattice II,, J. Phys. Soc. Japan, 38 (1975), 681.  doi: 10.1143/JPSJ.38.681.  Google Scholar [31] M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Comm. PDE, 12 (1987), 1133.  doi: 10.1080/03605308708820522.  Google Scholar [32] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations,, Comm. Pure Appl. Math., 39 (1986), 51.  doi: 10.1002/cpa.3160390103.  Google Scholar [33] L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type,, J. Differential Equations, 188 (2003), 1.  doi: 10.1016/S0022-0396(02)00061-X.  Google Scholar
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