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Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system
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 |
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.
|
[2] |
J. Angulo Pava, J. L. Bona, and M. Scialom, Stability of cnoidal waves,, Adv. Differential Equations, 11 (2006), 1321.
|
[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.
|
[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.
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.
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.
|
[7] |
J.-B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations,, in, (1978), 37.
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.
|
[9] |
J. L. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. London Ser. A, 344 (1975), 363.
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, (1971).
|
[11] |
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549.
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.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1002/cpa.3160430302. |
[20] |
T. Kato, "Perturbation Theory for Linear Operators,'', reprint of the 1980 edition, (1980).
|
[21] |
H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system,, Hokkaido Math. J., 37 (2008), 735.
|
[22] |
H. Kikuchi and M. Ohta, Stability of standing waves for the Klein-Gordon-Schrödinger system,, J. Math. Anal. Appl., 365 (2010), 109.
doi: 10.1016/j.jmaa.2009.10.024. |
[23] |
W. Magnus and S. Winkler, "Hill's Equation,'', corrected reprint of the 1966 edition, (1966).
|
[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.
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.
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, 1076 (1999), 83.
|
[27] |
M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schrdinger equations,, Nonlinear Anal., 27 (1996), 455.
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.
|
[29] |
S. Rabsztyn, On the Cauchy problem for the coupled Schrödinger-Klein-Gordon equations in one space dimension,, J. Math. Phys., 25 (1984), 1262.
doi: 10.1063/1.526281. |
[30] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics,", \textbf{IV}, IV (1978).
|
[31] |
X.-Y. Tang and W. Ding, The general Klein-Gordon-Schrödinger system: Modulational instability and exact solutions,, Phys. Scr., 77 (2008), 1.
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.
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.
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.
doi: 10.1137/0516034. |
[35] |
M. Y. Yu and P. K. Shukla, On the formation of upper-hybrid solitons,, Plasma Phys., 19 (1977), 889.
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.
|
[2] |
J. Angulo Pava, J. L. Bona, and M. Scialom, Stability of cnoidal waves,, Adv. Differential Equations, 11 (2006), 1321.
|
[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.
|
[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.
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.
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.
|
[7] |
J.-B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations,, in, (1978), 37.
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.
|
[9] |
J. L. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. London Ser. A, 344 (1975), 363.
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, (1971).
|
[11] |
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549.
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.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1002/cpa.3160430302. |
[20] |
T. Kato, "Perturbation Theory for Linear Operators,'', reprint of the 1980 edition, (1980).
|
[21] |
H. Kikuchi and M. Ohta, Instability of standing waves for the Klein-Gordon-Schrödinger system,, Hokkaido Math. J., 37 (2008), 735.
|
[22] |
H. Kikuchi and M. Ohta, Stability of standing waves for the Klein-Gordon-Schrödinger system,, J. Math. Anal. Appl., 365 (2010), 109.
doi: 10.1016/j.jmaa.2009.10.024. |
[23] |
W. Magnus and S. Winkler, "Hill's Equation,'', corrected reprint of the 1966 edition, (1966).
|
[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.
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.
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, 1076 (1999), 83.
|
[27] |
M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schrdinger equations,, Nonlinear Anal., 27 (1996), 455.
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.
|
[29] |
S. Rabsztyn, On the Cauchy problem for the coupled Schrödinger-Klein-Gordon equations in one space dimension,, J. Math. Phys., 25 (1984), 1262.
doi: 10.1063/1.526281. |
[30] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics,", \textbf{IV}, IV (1978).
|
[31] |
X.-Y. Tang and W. Ding, The general Klein-Gordon-Schrödinger system: Modulational instability and exact solutions,, Phys. Scr., 77 (2008), 1.
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.
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.
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.
doi: 10.1137/0516034. |
[35] |
M. Y. Yu and P. K. Shukla, On the formation of upper-hybrid solitons,, Plasma Phys., 19 (1977), 889.
doi: 10.1088/0032-1028/19/9/008. |
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