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

Previous Article
Attractors for the three-dimensional incompressible Navier-Stokes equations with damping
DCDS Home
This Issue
Next Article
Existence and non-existence of global solutions for a discrete semilinear heat equation

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

Abstract
Related Papers

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 and 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.
[2]	J. Angulo Pava, J. L. Bona, and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374.
[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.
[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.
[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.
[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.
[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.
[8]	T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 338 (1972), 153-183.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	T. Kato, "Perturbation Theory for Linear Operators,'' reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[21]	H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system, Hokkaido Math. J., 37 (2008), 735-748.
[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.
[23]	W. Magnus and S. Winkler, "Hill's Equation,'' corrected reprint of the 1966 edition, Dover Publications, Inc., New York, 1979.
[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.
[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.
[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.
[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.
[28]	H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differential Integral Equations, 17 (2004), 179-214.
[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.
[30]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics," IV, Analysis of Operators, Academic Press, New York-London, 1978.
[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.
[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.
[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.
[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.
[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.

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.
[2]	J. Angulo Pava, J. L. Bona, and M. Scialom, Stability of cnoidal waves, Adv. Differential Equations, 11 (2006), 1321-1374.
[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.
[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.
[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.
[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.
[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.
[8]	T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London Ser. A, 338 (1972), 153-183.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	T. Kato, "Perturbation Theory for Linear Operators,'' reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[21]	H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system, Hokkaido Math. J., 37 (2008), 735-748.
[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.
[23]	W. Magnus and S. Winkler, "Hill's Equation,'' corrected reprint of the 1966 edition, Dover Publications, Inc., New York, 1979.
[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.
[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.
[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.
[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.
[28]	H. Pecher, Global solutions of the Klein-Gordon-Schrödinger system with rough data, Differential Integral Equations, 17 (2004), 179-214.
[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.
[30]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics," IV, Analysis of Operators, Academic Press, New York-London, 1978.
[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.
[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.
[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.
[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.
[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.

[1]	Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure and Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413
[2]	Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149
[3]	Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239
[4]	Ahmed Y. Abdallah, Taqwa M. Al-Khader, Heba N. Abu-Shaab. Attractors of the Klein-Gordon-Schrödinger lattice systems with almost periodic nonlinear part. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022006
[5]	Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations and Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041
[6]	Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic and Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
[7]	Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure and Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695
[8]	Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure and Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55
[9]	Salah Missaoui. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system. Communications on Pure and Applied Analysis, 2022, 21 (2) : 567-584. doi: 10.3934/cpaa.2021189
[10]	E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156
[11]	Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz. Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 4021-4044. doi: 10.3934/dcdsb.2018122
[12]	Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077
[13]	A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097
[14]	Zehan Lin, Chongbin Xu, Caidi Zhao, Chujin Li. Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022065
[15]	Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2187-2209. doi: 10.3934/cpaa.2021063
[16]	Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043
[17]	Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233
[18]	Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[19]	Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure and Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[20]	Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Fábio Natali Ademir Pastor
on Google Scholar
Fábio Natali Ademir Pastor

[Back to Top]