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Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities
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 |
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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