-
Previous Article
Rényi entropy and recurrence
- DCDS Home
- This Issue
-
Next Article
Unique periodic orbits of a delay differential equation with piecewise linear feedback function
On the stability of standing waves of Klein-Gordon equations in a semiclassical regime
1. | Dipartimento di Matematica Applicata, Università di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy |
2. | Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France |
3. | Dipartimento di Informatica, Università di Verona, Strada Le Grazie 15, 37134 Verona, Italy |
References:
[1] |
L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Comm. Partial Differential Equations, 36 (2011), 797-818.
doi: 10.1080/03605302.2010.534684. |
[2] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[3] |
A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006. |
[4] |
A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42. |
[5] |
M. Beals and W. Strauss, $L^p$ estimates for the wave equation with a potential, Comm. Partial Differential Equations, 18 (1993), 1365-1397.
doi: 10.1080/03605309308820977. |
[6] |
V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[7] |
H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sé. I Math., 293 (1981), 489-492. |
[8] |
F. A. Berezin and M. A. Shubin, "The Schrödinger Equation," 66 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3154-4. |
[9] |
T. Cazenave, "Semilinear Schrödinger Equations," New York University - Courant Institute, New York, 2003. |
[10] |
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. |
[11] |
S. Cuccagna, On the local existence for the Maxwell-Klein-Gordon system in $\R^{3+1}$, Comm. Partial Differential Equations, 24 (1999), 851-867.
doi: 10.1080/03605309908821449. |
[12] |
T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[13] |
E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations, Phys. D, 18 (1986), 371-373. Solitons and coherent structures (Santa Barbara, Calif., 1985).
doi: 10.1016/0167-2789(86)90201-0. |
[14] |
R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, "Solitons and Nonlinear Wave Equations," Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982. |
[15] |
M. Grillakis, J. Shatah and W. A. 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. |
[16] |
______, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348. |
[17] |
I. Ianni and S. Le Coz, Orbital stability of standing waves of a semiclassical nonlinear Schrödinger-Poisson equation, Adv. Differential Equations, 14 (2009), 717-748. |
[18] |
L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. |
[19] |
______, Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, Trans. Amer. Math. Soc., 361 (2009), 5401-5416. |
[20] |
M. Keel, T. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621.
doi: 10.3934/dcds.2011.30.573. |
[21] |
S. Le Coz, Standing waves in nonlinear Schrödinger equations, in "Analytical and Numerical Aspects of Partial Differential Equations" Walter de Gruyter, Berlin, (2009), 151-192. |
[22] |
S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Phys. D, 237 (2008), 1103-1128.
doi: 10.1016/j.physd.2007.12.004. |
[23] |
T.-C. Lin and J. Wei, Orbital stability of bound states of semiclassical nonlinear Schrödinger equations with critical nonlinearity, SIAM J. Math. Anal., 40 (2008), 365-381.
doi: 10.1137/070683842. |
[24] |
Y. Liu, M. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 539-548.
doi: 10.1016/j.anihpc.2006.03.005. |
[25] |
E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law, Rev. Math. Phys., 21 (2009), 459-510.
doi: 10.1142/S0129055X09003669. |
[26] |
Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr\"odinger equations with potentials, Comm. Math. Phys., 121 (1989), 11-33. |
[27] |
M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322.
doi: 10.1137/050643015. |
[28] |
______, Instability of standing waves for nonlinear Klein-Gordon equation and related system, in "Nonlinear Dispersive Equations" 26 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkötosho, Tokyo, (2006), 189-200. |
[29] |
______, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931 (electronic). |
[30] |
J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.
doi: 10.2307/2000308. |
[31] |
J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190. |
[32] |
C. D. Sogge, Lectures on non-linear wave equations, International Press, Boston, MA, second ed., (2008). |
[33] |
C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399.
doi: 10.1007/s00032-008-0089-9. |
[34] |
D. M. A. Stuart, Modulational approach to stability of non-topological solitons in semilinear wave equations, J. Math. Pures Appl. (9), 80 (2001), 51-83.
doi: 10.1016/S0021-7824(00)01189-2. |
[35] |
T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33.
doi: 10.1090/S0273-0979-08-01228-7. |
[36] |
G. Vaira, Semiclassical states for the nonlinear Klein-Gordon-Maxwell system, J. Pure Appl. Math. Adv. Appl., 4 (2010), 59-95. |
[37] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.
|
[38] |
M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.
doi: 10.1137/0516034. |
[39] |
Y. Yu, Vortex dynamics for the nonlinear Maxwell-Klein-Gordon equation, Arch. Ration. Mech. Anal., 201 (2011), 743-776.
doi: 10.1007/s00205-011-0422-2. |
show all references
References:
[1] |
L. Aloui, S. Ibrahim and K. Nakanishi, Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Comm. Partial Differential Equations, 36 (2011), 797-818.
doi: 10.1080/03605302.2010.534684. |
[2] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[3] |
A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006. |
[4] |
A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42. |
[5] |
M. Beals and W. Strauss, $L^p$ estimates for the wave equation with a potential, Comm. Partial Differential Equations, 18 (1993), 1365-1397.
doi: 10.1080/03605309308820977. |
[6] |
V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[7] |
H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sé. I Math., 293 (1981), 489-492. |
[8] |
F. A. Berezin and M. A. Shubin, "The Schrödinger Equation," 66 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991.
doi: 10.1007/978-94-011-3154-4. |
[9] |
T. Cazenave, "Semilinear Schrödinger Equations," New York University - Courant Institute, New York, 2003. |
[10] |
T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. |
[11] |
S. Cuccagna, On the local existence for the Maxwell-Klein-Gordon system in $\R^{3+1}$, Comm. Partial Differential Equations, 24 (1999), 851-867.
doi: 10.1080/03605309908821449. |
[12] |
T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[13] |
E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations, Phys. D, 18 (1986), 371-373. Solitons and coherent structures (Santa Barbara, Calif., 1985).
doi: 10.1016/0167-2789(86)90201-0. |
[14] |
R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, "Solitons and Nonlinear Wave Equations," Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982. |
[15] |
M. Grillakis, J. Shatah and W. A. 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. |
[16] |
______, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348. |
[17] |
I. Ianni and S. Le Coz, Orbital stability of standing waves of a semiclassical nonlinear Schrödinger-Poisson equation, Adv. Differential Equations, 14 (2009), 717-748. |
[18] |
L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. |
[19] |
______, Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, Trans. Amer. Math. Soc., 361 (2009), 5401-5416. |
[20] |
M. Keel, T. Roy and T. Tao, Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm, Discrete Contin. Dyn. Syst., 30 (2011), 573-621.
doi: 10.3934/dcds.2011.30.573. |
[21] |
S. Le Coz, Standing waves in nonlinear Schrödinger equations, in "Analytical and Numerical Aspects of Partial Differential Equations" Walter de Gruyter, Berlin, (2009), 151-192. |
[22] |
S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Phys. D, 237 (2008), 1103-1128.
doi: 10.1016/j.physd.2007.12.004. |
[23] |
T.-C. Lin and J. Wei, Orbital stability of bound states of semiclassical nonlinear Schrödinger equations with critical nonlinearity, SIAM J. Math. Anal., 40 (2008), 365-381.
doi: 10.1137/070683842. |
[24] |
Y. Liu, M. Ohta and G. Todorova, Instabilité forte d'ondes solitaires pour des équations de Klein-Gordon non linéaires et des équations généralisées de Boussinesq, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 539-548.
doi: 10.1016/j.anihpc.2006.03.005. |
[25] |
E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law, Rev. Math. Phys., 21 (2009), 459-510.
doi: 10.1142/S0129055X09003669. |
[26] |
Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr\"odinger equations with potentials, Comm. Math. Phys., 121 (1989), 11-33. |
[27] |
M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 12 (2005), 315-322.
doi: 10.1137/050643015. |
[28] |
______, Instability of standing waves for nonlinear Klein-Gordon equation and related system, in "Nonlinear Dispersive Equations" 26 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkötosho, Tokyo, (2006), 189-200. |
[29] |
______, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal., 38 (2007), 1912-1931 (electronic). |
[30] |
J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710.
doi: 10.2307/2000308. |
[31] |
J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190. |
[32] |
C. D. Sogge, Lectures on non-linear wave equations, International Press, Boston, MA, second ed., (2008). |
[33] |
C. A. Stuart, Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation, Milan J. Math., 76 (2008), 329-399.
doi: 10.1007/s00032-008-0089-9. |
[34] |
D. M. A. Stuart, Modulational approach to stability of non-topological solitons in semilinear wave equations, J. Math. Pures Appl. (9), 80 (2001), 51-83.
doi: 10.1016/S0021-7824(00)01189-2. |
[35] |
T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33.
doi: 10.1090/S0273-0979-08-01228-7. |
[36] |
G. Vaira, Semiclassical states for the nonlinear Klein-Gordon-Maxwell system, J. Pure Appl. Math. Adv. Appl., 4 (2010), 59-95. |
[37] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.
|
[38] |
M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.
doi: 10.1137/0516034. |
[39] |
Y. Yu, Vortex dynamics for the nonlinear Maxwell-Klein-Gordon equation, Arch. Ration. Mech. Anal., 201 (2011), 743-776.
doi: 10.1007/s00205-011-0422-2. |
[1] |
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 |
[2] |
Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359 |
[3] |
Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315 |
[4] |
Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889 |
[5] |
Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525 |
[6] |
Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043 |
[7] |
Yanheng Ding, Xiaojing Dong, Qi Guo. On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030 |
[8] |
Xiaoming An, Xian Yang. Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1649-1672. doi: 10.3934/cpaa.2022038 |
[9] |
Marcelo M. Cavalcanti, Leonel G. Delatorre, Daiane C. Soares, Victor Hugo Gonzalez Martinez, Janaina P. Zanchetta. Uniform stabilization of the Klein-Gordon system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5131-5156. doi: 10.3934/cpaa.2020230 |
[10] |
Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071 |
[11] |
Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753 |
[12] |
Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 |
[13] |
Necdet Bildik, Sinan Deniz. New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 503-518. doi: 10.3934/dcdss.2020028 |
[14] |
Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541 |
[15] |
Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure and Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030 |
[16] |
Oana Ivanovici. Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5707-5742. doi: 10.3934/dcds.2021093 |
[17] |
Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903 |
[18] |
Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076 |
[19] |
Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679 |
[20] |
Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]