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March  2011, 31(1): 221-238. doi: 10.3934/dcds.2011.31.221

Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system

Fábio Natali 1, and  Ademir Pastor 2,

1. 

Universidade Estadual de Maringá - UEM, Avenida Colombo, 5790, CEP 87020-900, Maringá
2. 

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

Received  December 2009 Revised  April 2011 Published  June 2011

This article deals with the existence and orbital stability of a two--parameter family of periodic traveling-wave solutions for the Klein-Gordon-Schrödinger system with Yukawa interaction. The existence of such a family of periodic waves is deduced from the Implicit Function Theorem, and the orbital stability is obtained from arguments due to Benjamin, Bona, and Weinstein.
Keywords: Periodic waves, orbital stability, Klein-Gordon-Schrödinger system..
Mathematics Subject Classification: Primary: 35B10, 35B35; Secondary: 35Q9.
Citation: Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221
References:
[1]

J. Angulo Pava, Nonlinear stability of periodic traveling wave solutions to the Schrödinger and modified Korteweg-de Vries equations, J. Differential Equations, 235 (2007), 1-30.  Google Scholar

[2]

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

[3]

J. Angulo Pava, C. Matheus and D. Pilod, Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system, Commun. Pure Appl. Anal., 8 (2009), 815-844.  Google Scholar

[4]

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

[5]

J. Angulo Pava and F. Natali, Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and non-linear Schrödinger equations, Physica D, 238 (2009), 603-621. doi: 10.1016/j.physd.2008.12.011.  Google Scholar

[6]

J. Angulo Pava and A. Pastor Ferreira, Stability of periodic optical solitons for a nonlinear Schrödinger system, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 927-959.  Google Scholar

[7]

J.-B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations, in "Contemporary Developments in Continuum Mechanics and Partial Differential Equations,'' Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, (1978), 37-44. doi: 10.1016/S0304-0208(08)70857-0.  Google Scholar

[8]

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 338 (1972), 153-183.  Google Scholar

[9]

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

[10]

P. F. Byrd and M. D. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists,'' $2^{nd}$ edition, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar

[11]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.  Google Scholar

[12]

J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.  Google Scholar

[13]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations. II, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[18]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math., 41 (1988), 747-774. doi: 10.1002/cpa.3160410602.  Google Scholar

[19]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.  Google Scholar

[20]

T. Kato, "Perturbation Theory for Linear Operators,'' reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[21]

H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system, Hokkaido Math. J., 37 (2008), 735-748.  Google Scholar

[22]

H. Kikuchi and M. Ohta, Stability of standing waves for the Klein-Gordon-Schrödinger system, J. Math. Anal. Appl., 365 (2010), 109-114. doi: 10.1016/j.jmaa.2009.10.024.  Google Scholar

[23]

W. Magnus and S. Winkler, "Hill's Equation,'' corrected reprint of the 1966 edition, Dover Publications, Inc., New York, 1979.  Google Scholar

[24]

F. Natali and A. Pastor Ferreira, Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system, Commun. Pure Appl. Anal., 9 (2010), 413-430. doi: 10.3934/cpaa.2010.9.413.  Google Scholar

[25]

F. Natali and A. Pastor Ferreira, 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.  Google Scholar

[26]

M. Ohta, Stability of solitary waves for coupled Klein-Gordon-Schrödinger equations in one space dimension, Variational Problems and Related Topics, (Japanese) (Kyoto, 1998), S/=urikaisekikenkyūsho Kōkyūroku, 1076 (1999), 83-92.  Google Scholar

[27]

M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schrdinger equations, Nonlinear Anal., 27 (1996), 455-461. doi: 10.1016/0362-546X(95)00017-P.  Google Scholar

[28]

H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differential Integral Equations, 17 (2004), 179-214.  Google Scholar

[29]

S. Rabsztyn, On the Cauchy problem for the coupled Schrödinger-Klein-Gordon equations in one space dimension, J. Math. Phys., 25 (1984), 1262-1265. doi: 10.1063/1.526281.  Google Scholar

[30]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics," IV, Analysis of Operators, Academic Press, New York-London, 1978.  Google Scholar

[31]

X.-Y. Tang and W. Ding, The general Klein-Gordon-Schrödinger system: Modulational instability and exact solutions, Phys. Scr., 77 (2008), 1-8. doi: 10.1088/0031-8949/77/01/015004.  Google Scholar

[32]

N. Tzirakis, The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space, Comm. Partial Differential Equations, 30 (2005), 605-641. doi: 10.1081/PDE-200059260.  Google Scholar

[33]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

[34]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.  Google Scholar

[35]

M. Y. Yu and P. K. Shukla, On the formation of upper-hybrid solitons, Plasma Phys., 19 (1977), 889-893. doi: 10.1088/0032-1028/19/9/008.  Google Scholar

show all references

References:
[1]

J. Angulo Pava, Nonlinear stability of periodic traveling wave solutions to the Schrödinger and modified Korteweg-de Vries equations, J. Differential Equations, 235 (2007), 1-30.  Google Scholar

[2]

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

[3]

J. Angulo Pava, C. Matheus and D. Pilod, Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system, Commun. Pure Appl. Anal., 8 (2009), 815-844.  Google Scholar

[4]

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

[5]

J. Angulo Pava and F. Natali, Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and non-linear Schrödinger equations, Physica D, 238 (2009), 603-621. doi: 10.1016/j.physd.2008.12.011.  Google Scholar

[6]

J. Angulo Pava and A. Pastor Ferreira, Stability of periodic optical solitons for a nonlinear Schrödinger system, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 927-959.  Google Scholar

[7]

J.-B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations, in "Contemporary Developments in Continuum Mechanics and Partial Differential Equations,'' Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, (1978), 37-44. doi: 10.1016/S0304-0208(08)70857-0.  Google Scholar

[8]

T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 338 (1972), 153-183.  Google Scholar

[9]

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

[10]

P. F. Byrd and M. D. Friedman, "Handbook of Elliptic Integrals for Engineers and Scientists,'' $2^{nd}$ edition, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar

[11]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.  Google Scholar

[12]

J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.  Google Scholar

[13]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations. II, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[18]

M. Grillakis, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math., 41 (1988), 747-774. doi: 10.1002/cpa.3160410602.  Google Scholar

[19]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.  Google Scholar

[20]

T. Kato, "Perturbation Theory for Linear Operators,'' reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[21]

H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system, Hokkaido Math. J., 37 (2008), 735-748.  Google Scholar

[22]

H. Kikuchi and M. Ohta, Stability of standing waves for the Klein-Gordon-Schrödinger system, J. Math. Anal. Appl., 365 (2010), 109-114. doi: 10.1016/j.jmaa.2009.10.024.  Google Scholar

[23]

W. Magnus and S. Winkler, "Hill's Equation,'' corrected reprint of the 1966 edition, Dover Publications, Inc., New York, 1979.  Google Scholar

[24]

F. Natali and A. Pastor Ferreira, Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system, Commun. Pure Appl. Anal., 9 (2010), 413-430. doi: 10.3934/cpaa.2010.9.413.  Google Scholar

[25]

F. Natali and A. Pastor Ferreira, 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.  Google Scholar

[26]

M. Ohta, Stability of solitary waves for coupled Klein-Gordon-Schrödinger equations in one space dimension, Variational Problems and Related Topics, (Japanese) (Kyoto, 1998), S/=urikaisekikenkyūsho Kōkyūroku, 1076 (1999), 83-92.  Google Scholar

[27]

M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schrdinger equations, Nonlinear Anal., 27 (1996), 455-461. doi: 10.1016/0362-546X(95)00017-P.  Google Scholar

[28]

H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differential Integral Equations, 17 (2004), 179-214.  Google Scholar

[29]

S. Rabsztyn, On the Cauchy problem for the coupled Schrödinger-Klein-Gordon equations in one space dimension, J. Math. Phys., 25 (1984), 1262-1265. doi: 10.1063/1.526281.  Google Scholar

[30]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics," IV, Analysis of Operators, Academic Press, New York-London, 1978.  Google Scholar

[31]

X.-Y. Tang and W. Ding, The general Klein-Gordon-Schrödinger system: Modulational instability and exact solutions, Phys. Scr., 77 (2008), 1-8. doi: 10.1088/0031-8949/77/01/015004.  Google Scholar

[32]

N. Tzirakis, The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space, Comm. Partial Differential Equations, 30 (2005), 605-641. doi: 10.1081/PDE-200059260.  Google Scholar

[33]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

[34]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.  Google Scholar

[35]

M. Y. Yu and P. K. Shukla, On the formation of upper-hybrid solitons, Plasma Phys., 19 (1977), 889-893. doi: 10.1088/0032-1028/19/9/008.  Google Scholar

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