Figure 1.
Left: Phase space for $ c_0 = 1 $ and $ A_0 = 1 $ . Right: Phase space for $ c_0 = 1 $ and $ A_0 = 2 $ . In both cases, the orbits in black are those for which $ \varphi_{(c_0,A_0)}>1 $
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Longtime dynamics for a type of suspension bridge equation with past history and time delay
October
2020, 19(10): 5015-5032.
doi: 10.3934/cpaa.2020225
Nonlinear stability of periodic-wave solutions for systems of dispersive equations
IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brazil |
We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.
Mathematics Subject Classification:
Primary: 35B35, 35Q51; Secondary: 35Q53.
Citation:
Fabrício Cristófani, Ademir Pastor. Nonlinear stability of periodic-wave solutions for systems of dispersive equations. Communications on Pure and Applied Analysis,
2020, 19
(10)
: 5015-5032.
doi: 10.3934/cpaa.2020225
![]()
References:
| [1] |
M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur,
Nonlinear evolution equations of physical significance, Phys. Rev. Lett., 31 (1973), 125-127.
doi: 10.1103/PhysRevLett.31.125. |
| [2] |
E. Alarcon, J. Angulo and J. F. Montenegro,
Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035.
doi: 10.1016/S0362-546X(97)00724-4. |
| [3] |
B. Alvarez-Samaniego and X. Carvajal,
On the local well-posedness for some systems of coupled KdV equations, Nonlinear Anal., 69 (2008), 692-715.
doi: 10.1016/j.na.2007.06.009. |
| [4] |
J. M. Ash, J. Cohen and G. Wang,
On strongly interacting internal solitary waves, J. Fourier Anal. Appl., 2 (1996), 507-517.
doi: 10.1007/s00041-001-4041-4. |
| [5] |
J. P. Albert, J. L. Bona and J. -C. Saut,
Model equation for waves in stratified fluids, Proc. Roy. Soc. Lodon Ser. A, 453 (1997), 1233-1260.
doi: 10.1098/rspa.1997.0068. |
| [6] |
J. P. Albert and F. Linares,
Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves, J. Math. Pure. Appl., 79 (2000), 195-226.
doi: 10.1016/S0021-7824(00)00157-4. |
| [7] |
G. Alves, F. Natali and A. Pastor,
Sufficient conditions for orbital stability of periodic traveling waves, J. Differ. Equ., 267 (2019), 879-901.
doi: 10.1016/j.jde.2019.01.029. |
| [8] |
T. P. Andrade, F. Cristófani and F. Natali, Orbital stability of periodic traveling wave solutions for the Kawahara equation, J. Math. Phys., 58 (2017), 051504.
doi: 10.1063/1.4980016. |
| [9] |
J. Angulo and F. Natali,
Positivity properties of the Fourier transform and the stability of periodic travelling-wave solutions, SIAM J. Math. Anal., 40 (2008), 1123-1151.
doi: 10.1137/080718450. |
| [10] |
J. L. Bona, H. Chen and O. Karakashian,
Stability of solitary-wave solutions of systems of dispersive equations, Appl. Math. Optim., 75 (2017), 27-53.
doi: 10.1007/s00245-015-9322-4. |
| [11] |
J. L. Bona, M. Chen and J. C. Staut,
Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Part Ⅰ. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
| [12] |
J. L. Bona, M. Chen and J. C. Staut,
Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Part Ⅱ. Nonlinear theory, Nonlinearity, 17 (2004), 925-952.
doi: 10.1088/0951-7715/17/3/010. |
| [13] |
J. L. Bona, P. E. Souganidis and W. A. Strauss,
Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A, 411 (1987), 395-412.
|
| [14] |
S. Bhattarai,
Existence of travelling-wave solutions to a coupled system of Korteweg–de Vries equations, Nonlinear Anal., 127 (2015), 182-195.
doi: 10.1016/j.na.2015.07.004. |
| [15] |
J. C. Bronski, M. A. Johnson and T. Kapitula,
An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type, P. Roy. Soc. Edinb. A, 141 (2011), 1141-1173.
doi: 10.1017/S0308210510001216. |
| [16] |
P. F. Byrd and M. D. Friedman., Handbook of elliptic integrals for engineers and scientists, Springer-Verlag, New York, 1971. |
| [17] |
A. J. Corcho and M. Panthee,
Global well-posedness for a coupled modified KdV system, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 27-57.
doi: 10.1007/s00574-012-0004-4. |
| [18] |
F. Cristófani, F. Natali and A. Pastor,
Orbital stability of periodic traveling-wave solutions for the Log-KdV equation, J. Differ. Equ., 263 (2017), 2630-2660.
doi: 10.1016/j.jde.2017.04.004. |
| [19] |
F. Cristófani, F. Natali and A. Pastor,
Periodic traveling-wave solutions for regularized dispersive equations: sufficient conditions for orbital stability with applications, Commun. Math. Sci., 18 (2020), 613-634.
doi: 10.4310/CMS.2020.v18.n3.a2. |
| [20] |
T. P. de Andrade and A. Pastor,
Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications, J. Math. Anal. Appl., 475 (2019), 1242-1275.
doi: 10.1016/j.jmaa.2019.03.011. |
| [21] |
E. Dumas and D. Pelinovsky,
Justification of the log-KdV equation in granular chains: the case of precompression, SIAM J. Math. Anal., 46 (2014), 4075-4103.
doi: 10.1137/140969270. |
| [22] |
M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, London, 1973.
|
| [23] |
J. Gear and R. Grinshaw,
Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258.
doi: 10.1002/sapm1984703235. |
| [24] |
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.
doi: 10.1016/0022-1236(87)90044-9. |
| [25] |
M. Johnson,
Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation, SIAM J. Math. Anal., 41 (2009), 1921-1947.
doi: 10.1137/090752249. |
| [26] |
G. James and D. Pelinovsky, Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20pp.
doi: 10.1098/rspa.2013.0462. |
| [27] |
T. Kato, Perturbation theory for linear Operators, Springer-Verlag, Berlin, 1976. |
| [28] |
A. Liu, T. Kubota and D. Ko,
Resonant transfer of energy between nonlinear waves in neighboring pycnoclines, Stud. Appl. Math., 63 (1980), 25-45.
doi: 10.1002/sapm198063125. |
| [29] |
A. Majda and J. Biello,
The nonlinear interaction of barotropic and equatorial baroclinic Rossby Waves, J. Atmos. Sci., 60 (2003), 1809-1821.
doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2. |
| [30] |
F. Natali and A. Pastor,
The fourth-order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1347.
doi: 10.1137/151004884. |
| [31] |
V. F. Nesterenko, Dynamics of heterogeneous materials, Springer-Verlag, New York, 2001. |
| [32] |
T. Oh,
Diophantine conditions in global well-posedness for coupled KdV systems, Electron. J. Differ. Equ., 2009 (2009), 1-48.
|
| [33] |
C. A. Stuart,
Lectures on the orbital stability of standing waves and applications to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399.
|
show all references
References:
| [1] |
M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur,
Nonlinear evolution equations of physical significance, Phys. Rev. Lett., 31 (1973), 125-127.
doi: 10.1103/PhysRevLett.31.125. |
| [2] |
E. Alarcon, J. Angulo and J. F. Montenegro,
Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035.
doi: 10.1016/S0362-546X(97)00724-4. |
| [3] |
B. Alvarez-Samaniego and X. Carvajal,
On the local well-posedness for some systems of coupled KdV equations, Nonlinear Anal., 69 (2008), 692-715.
doi: 10.1016/j.na.2007.06.009. |
| [4] |
J. M. Ash, J. Cohen and G. Wang,
On strongly interacting internal solitary waves, J. Fourier Anal. Appl., 2 (1996), 507-517.
doi: 10.1007/s00041-001-4041-4. |
| [5] |
J. P. Albert, J. L. Bona and J. -C. Saut,
Model equation for waves in stratified fluids, Proc. Roy. Soc. Lodon Ser. A, 453 (1997), 1233-1260.
doi: 10.1098/rspa.1997.0068. |
| [6] |
J. P. Albert and F. Linares,
Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves, J. Math. Pure. Appl., 79 (2000), 195-226.
doi: 10.1016/S0021-7824(00)00157-4. |
| [7] |
G. Alves, F. Natali and A. Pastor,
Sufficient conditions for orbital stability of periodic traveling waves, J. Differ. Equ., 267 (2019), 879-901.
doi: 10.1016/j.jde.2019.01.029. |
| [8] |
T. P. Andrade, F. Cristófani and F. Natali, Orbital stability of periodic traveling wave solutions for the Kawahara equation, J. Math. Phys., 58 (2017), 051504.
doi: 10.1063/1.4980016. |
| [9] |
J. Angulo and F. Natali,
Positivity properties of the Fourier transform and the stability of periodic travelling-wave solutions, SIAM J. Math. Anal., 40 (2008), 1123-1151.
doi: 10.1137/080718450. |
| [10] |
J. L. Bona, H. Chen and O. Karakashian,
Stability of solitary-wave solutions of systems of dispersive equations, Appl. Math. Optim., 75 (2017), 27-53.
doi: 10.1007/s00245-015-9322-4. |
| [11] |
J. L. Bona, M. Chen and J. C. Staut,
Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Part Ⅰ. Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
| [12] |
J. L. Bona, M. Chen and J. C. Staut,
Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Part Ⅱ. Nonlinear theory, Nonlinearity, 17 (2004), 925-952.
doi: 10.1088/0951-7715/17/3/010. |
| [13] |
J. L. Bona, P. E. Souganidis and W. A. Strauss,
Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A, 411 (1987), 395-412.
|
| [14] |
S. Bhattarai,
Existence of travelling-wave solutions to a coupled system of Korteweg–de Vries equations, Nonlinear Anal., 127 (2015), 182-195.
doi: 10.1016/j.na.2015.07.004. |
| [15] |
J. C. Bronski, M. A. Johnson and T. Kapitula,
An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type, P. Roy. Soc. Edinb. A, 141 (2011), 1141-1173.
doi: 10.1017/S0308210510001216. |
| [16] |
P. F. Byrd and M. D. Friedman., Handbook of elliptic integrals for engineers and scientists, Springer-Verlag, New York, 1971. |
| [17] |
A. J. Corcho and M. Panthee,
Global well-posedness for a coupled modified KdV system, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 27-57.
doi: 10.1007/s00574-012-0004-4. |
| [18] |
F. Cristófani, F. Natali and A. Pastor,
Orbital stability of periodic traveling-wave solutions for the Log-KdV equation, J. Differ. Equ., 263 (2017), 2630-2660.
doi: 10.1016/j.jde.2017.04.004. |
| [19] |
F. Cristófani, F. Natali and A. Pastor,
Periodic traveling-wave solutions for regularized dispersive equations: sufficient conditions for orbital stability with applications, Commun. Math. Sci., 18 (2020), 613-634.
doi: 10.4310/CMS.2020.v18.n3.a2. |
| [20] |
T. P. de Andrade and A. Pastor,
Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications, J. Math. Anal. Appl., 475 (2019), 1242-1275.
doi: 10.1016/j.jmaa.2019.03.011. |
| [21] |
E. Dumas and D. Pelinovsky,
Justification of the log-KdV equation in granular chains: the case of precompression, SIAM J. Math. Anal., 46 (2014), 4075-4103.
doi: 10.1137/140969270. |
| [22] |
M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, London, 1973.
|
| [23] |
J. Gear and R. Grinshaw,
Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258.
doi: 10.1002/sapm1984703235. |
| [24] |
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.
doi: 10.1016/0022-1236(87)90044-9. |
| [25] |
M. Johnson,
Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation, SIAM J. Math. Anal., 41 (2009), 1921-1947.
doi: 10.1137/090752249. |
| [26] |
G. James and D. Pelinovsky, Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20pp.
doi: 10.1098/rspa.2013.0462. |
| [27] |
T. Kato, Perturbation theory for linear Operators, Springer-Verlag, Berlin, 1976. |
| [28] |
A. Liu, T. Kubota and D. Ko,
Resonant transfer of energy between nonlinear waves in neighboring pycnoclines, Stud. Appl. Math., 63 (1980), 25-45.
doi: 10.1002/sapm198063125. |
| [29] |
A. Majda and J. Biello,
The nonlinear interaction of barotropic and equatorial baroclinic Rossby Waves, J. Atmos. Sci., 60 (2003), 1809-1821.
doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2. |
| [30] |
F. Natali and A. Pastor,
The fourth-order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1347.
doi: 10.1137/151004884. |
| [31] |
V. F. Nesterenko, Dynamics of heterogeneous materials, Springer-Verlag, New York, 2001. |
| [32] |
T. Oh,
Diophantine conditions in global well-posedness for coupled KdV systems, Electron. J. Differ. Equ., 2009 (2009), 1-48.
|
| [33] |
C. A. Stuart,
Lectures on the orbital stability of standing waves and applications to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399.
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