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October  2020, 19(10): 5015-5032. doi: 10.3934/cpaa.2020225

Nonlinear stability of periodic-wave solutions for systems of dispersive equations

Fabrício Cristófani , and  Ademir Pastor

IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brazil

* Corresponding author

Received  February 2020 Revised  May 2020 Published  August 2020

Fund Project: F. C. is supported by FAPESP/Brazil grant 2017/20760-0. A. P. is partially supported by CNPq/Brazil grants 402849/2016-7 and 303098/2016-3

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.

Keywords: Orbital stability, dispersive system, periodic waves.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. F. Byrd and M. D. Friedman., Handbook of elliptic integrals for engineers and scientists, Springer-Verlag, New York, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22] M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, London, 1973.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

T. Kato, Perturbation theory for linear Operators, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

V. F. Nesterenko, Dynamics of heterogeneous materials, Springer-Verlag, New York, 2001. Google Scholar

[32]

T. Oh, Diophantine conditions in global well-posedness for coupled KdV systems, Electron. J. Differ. Equ., 2009 (2009), 1-48.   Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. F. Byrd and M. D. Friedman., Handbook of elliptic integrals for engineers and scientists, Springer-Verlag, New York, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22] M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, London, 1973.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

T. Kato, Perturbation theory for linear Operators, Springer-Verlag, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

V. F. Nesterenko, Dynamics of heterogeneous materials, Springer-Verlag, New York, 2001. Google Scholar

[32]

T. Oh, Diophantine conditions in global well-posedness for coupled KdV systems, Electron. J. Differ. Equ., 2009 (2009), 1-48.   Google Scholar

[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.   Google Scholar

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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