American Institute of Mathematical Sciences

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

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

Jaime Angulo Pava 1, , Carlos Banquet 2, and  Márcia Scialom 2,

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.
Keywords: Periodic traveling waves, Fourth BBM equation., Nonlinear stability, Modified BBM equation.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B35, 35B1.
Citation: Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete and Continuous Dynamical Systems, 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), 117-134. doi: 10.1016/0022-0396(86)90057-4.

[2]

J. Angulo, Stability of cnoidal waves to Hirota-Satsuma systems, Mat. Contemp., 27 (2004), 189-223.

[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), 1-30. doi: 10.1016/j.jde.2007.01.003.

[4]

J. Angulo, "Nonlinear Dispersive Evolution Equations: Existence and Stability of Solitary and Periodic Traveling-Waves Solutions," Mathematical Surveys and Monographs Series (SURV), 156, American Mathematical Society, Providence, RI, 2009.

[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, (). 

[6]

J. Angulo and F. Natali, Positivity properties of the Fourier transform and the stability of periodic traveling-wave solutions, SIAM, J. Math. Anal., 40 (2008), 1123-1151. doi: 10.1137/080718450.

[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), 603-621. doi: 10.1016/j.physd.2008.12.011.

[8]

J. Angulo, J. Bona and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374.

[9]

T. Benjamin, Lectures on nonlinear wave motion, Nonlinear Wave Motion, AMS, Providence, R. I., 15 (1974), 3-47.

[10]

T. Benjamin, The stability of solitary waves, Proc. R. Soc. Lond. Ser. A, 338 (1972), 153-183.

[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), 47-78. doi: 10.1098/rsta.1972.0032.

[12]

J. Bona, On the stability theory of solitary waves, Proc. R. Soc. Lond. Ser. A, 344 (1975), 363-374. doi: 10.1098/rspa.1975.0106.

[13]

J. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252.

[14]

J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations, Discrete Contin. Dyn. Syst., 23 (2009), 1253-1275.

[15]

P. Byrd and M. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," $2^{nd}$ edition, Springer, NY, 1971.

[16]

K. El Dika, Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst., 13 (2005), 583-622. doi: 10.3934/dcds.2005.13.583.

[17]

T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Differential Equations, 234 (2007), 544-581. doi: 10.1016/j.jde.2006.12.007.

[18]

T. Gallay and M. Hărăguş, Orbital stability of periodic waves for the nonlinear Schrödinger equation, J. Dynam. Differential Equations, 19 (2007), 825-865. doi: 10.1007/s10884-007-9071-4.

[19]

M. Hărăguş, Stability of periodic waves for the generalized BBM equation, Rev. Roumaine Math. Pures Appl., 53 (2008), 445-463.

[20]

R. Iorio Jr. and V. Iorio, "Fourier Analysis and Partial Differential Equations," Cambridge Stud. Adv. Math. 70, Cambridge University Press, Cambridge, UK, 2001.

[21]

S. Karlin, "Total Positivity," Stanford University Press, 1968.

[22]

J. Miller and M. Weinstein, Asymptotic stability of solitary waves for the regularized long-wave equation, Comm. Pure Appl. Math., 495 (1996), 399-441. doi: 10.1002/(SICI)1097-0312(199604)49:4<399::AID-CPA4>3.0.CO;2-7.

[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), 428-441. doi: 10.1016/j.jmaa.2008.06.033.

[24]

E. Oberhettinger, "Fourier Expansions: A Collection of Formulas," Academic Press, New York, London, 1973.

[25]

D. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330. doi: 10.1017/S0022112066001678.

[26]

D. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. doi: 10.1017/S0022112067002605.

[27]

P. Souganidis and W. Strauss, Instability of a class of dispersive solitary waves, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 195-212.

[28]

E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton University Press, Princeton, N. J., 1970.

[29]

M. Wadati, Wave propagation in nonlinear lattice I, J. Phys. Soc. Japan, 38 (1975), 673-680. doi: 10.1143/JPSJ.38.673.

[30]

M. Wadati, Wave propagation in nonlinear lattice II, J. Phys. Soc. Japan, 38 (1975), 681-686. doi: 10.1143/JPSJ.38.681.

[31]

M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. PDE, 12 (1987), 1133-1173. doi: 10.1080/03605308708820522.

[32]

M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations, Comm. Pure Appl. Math., 39 (1986), 51-68. doi: 10.1002/cpa.3160390103.

[33]

L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Differential Equations, 188 (2003), 1-32. doi: 10.1016/S0022-0396(02)00061-X.

show all references

References:
[1]

J. Albert, Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation, J. Differential Equations, 63 (1986), 117-134. doi: 10.1016/0022-0396(86)90057-4.

[2]

J. Angulo, Stability of cnoidal waves to Hirota-Satsuma systems, Mat. Contemp., 27 (2004), 189-223.

[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), 1-30. doi: 10.1016/j.jde.2007.01.003.

[4]

J. Angulo, "Nonlinear Dispersive Evolution Equations: Existence and Stability of Solitary and Periodic Traveling-Waves Solutions," Mathematical Surveys and Monographs Series (SURV), 156, American Mathematical Society, Providence, RI, 2009.

[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, (). 

[6]

J. Angulo and F. Natali, Positivity properties of the Fourier transform and the stability of periodic traveling-wave solutions, SIAM, J. Math. Anal., 40 (2008), 1123-1151. doi: 10.1137/080718450.

[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), 603-621. doi: 10.1016/j.physd.2008.12.011.

[8]

J. Angulo, J. Bona and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374.

[9]

T. Benjamin, Lectures on nonlinear wave motion, Nonlinear Wave Motion, AMS, Providence, R. I., 15 (1974), 3-47.

[10]

T. Benjamin, The stability of solitary waves, Proc. R. Soc. Lond. Ser. A, 338 (1972), 153-183.

[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), 47-78. doi: 10.1098/rsta.1972.0032.

[12]

J. Bona, On the stability theory of solitary waves, Proc. R. Soc. Lond. Ser. A, 344 (1975), 363-374. doi: 10.1098/rspa.1975.0106.

[13]

J. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252.

[14]

J. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations, Discrete Contin. Dyn. Syst., 23 (2009), 1253-1275.

[15]

P. Byrd and M. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists," $2^{nd}$ edition, Springer, NY, 1971.

[16]

K. El Dika, Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst., 13 (2005), 583-622. doi: 10.3934/dcds.2005.13.583.

[17]

T. Gallay and M. Hărăguş, Stability of small periodic waves for the nonlinear Schrödinger equation, J. Differential Equations, 234 (2007), 544-581. doi: 10.1016/j.jde.2006.12.007.

[18]

T. Gallay and M. Hărăguş, Orbital stability of periodic waves for the nonlinear Schrödinger equation, J. Dynam. Differential Equations, 19 (2007), 825-865. doi: 10.1007/s10884-007-9071-4.

[19]

M. Hărăguş, Stability of periodic waves for the generalized BBM equation, Rev. Roumaine Math. Pures Appl., 53 (2008), 445-463.

[20]

R. Iorio Jr. and V. Iorio, "Fourier Analysis and Partial Differential Equations," Cambridge Stud. Adv. Math. 70, Cambridge University Press, Cambridge, UK, 2001.

[21]

S. Karlin, "Total Positivity," Stanford University Press, 1968.

[22]

J. Miller and M. Weinstein, Asymptotic stability of solitary waves for the regularized long-wave equation, Comm. Pure Appl. Math., 495 (1996), 399-441. doi: 10.1002/(SICI)1097-0312(199604)49:4<399::AID-CPA4>3.0.CO;2-7.

[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), 428-441. doi: 10.1016/j.jmaa.2008.06.033.

[24]

E. Oberhettinger, "Fourier Expansions: A Collection of Formulas," Academic Press, New York, London, 1973.

[25]

D. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330. doi: 10.1017/S0022112066001678.

[26]

D. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. doi: 10.1017/S0022112067002605.

[27]

P. Souganidis and W. Strauss, Instability of a class of dispersive solitary waves, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 195-212.

[28]

E. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton University Press, Princeton, N. J., 1970.

[29]

M. Wadati, Wave propagation in nonlinear lattice I, J. Phys. Soc. Japan, 38 (1975), 673-680. doi: 10.1143/JPSJ.38.673.

[30]

M. Wadati, Wave propagation in nonlinear lattice II, J. Phys. Soc. Japan, 38 (1975), 681-686. doi: 10.1143/JPSJ.38.681.

[31]

M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. PDE, 12 (1987), 1133-1173. doi: 10.1080/03605308708820522.

[32]

M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations, Comm. Pure Appl. Math., 39 (1986), 51-68. doi: 10.1002/cpa.3160390103.

[33]

L. Zeng, Existence and stability of solitary-wave solutions of equations of Benjamin-Bona-Mahony type, J. Differential Equations, 188 (2003), 1-32. doi: 10.1016/S0022-0396(02)00061-X.

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