American Institute of Mathematical Sciences

Advanced
June  2013, 33(6): 2389-2401. doi: 10.3934/dcds.2013.33.2389

On the stability of standing waves of Klein-Gordon equations in a semiclassical regime

Marco Ghimenti 1, , Stefan Le Coz 2, and  Marco Squassina 3,

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

Received  January 2012 Revised  September 2012 Published  December 2012

We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
Keywords: stability, semi-classical limit., Klein-Gordon equations.
Mathematics Subject Classification: 35D99, 35J62, 58E05, 35J7.
Citation: Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389
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.  Google Scholar

[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.  Google Scholar

[3]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[4]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Cazenave, "Semilinear Schrödinger Equations," New York University - Courant Institute, New York, 2003. Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

______, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

______, Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, Trans. Amer. Math. Soc., 361 (2009), 5401-5416. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr\"odinger equations with potentials, Comm. Math. Phys., 121 (1989), 11-33.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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). Google Scholar

[30]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710. doi: 10.2307/2000308.  Google Scholar

[31]

J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190.  Google Scholar

[32]

C. D. Sogge, Lectures on non-linear wave equations, International Press, Boston, MA, second ed., (2008).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33. doi: 10.1090/S0273-0979-08-01228-7.  Google Scholar

[36]

G. Vaira, Semiclassical states for the nonlinear Klein-Gordon-Maxwell system, J. Pure Appl. Math. Adv. Appl., 4 (2010), 59-95.  Google Scholar

[37]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. Ambrosetti and A. Malchiodi, "Perturbation Methods and Semilinear Elliptic Problems on $R^n$," 240 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2006.  Google Scholar

[4]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Cazenave, "Semilinear Schrödinger Equations," New York University - Courant Institute, New York, 2003. Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

______, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal., 94 (1990), 308-348. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

______, Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, Trans. Amer. Math. Soc., 361 (2009), 5401-5416. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr\"odinger equations with potentials, Comm. Math. Phys., 121 (1989), 11-33.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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). Google Scholar

[30]

J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc., 290 (1985), 701-710. doi: 10.2307/2000308.  Google Scholar

[31]

J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys., 100 (1985), 173-190.  Google Scholar

[32]

C. D. Sogge, Lectures on non-linear wave equations, International Press, Boston, MA, second ed., (2008).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

T. Tao, Why are solitons stable?, Bull. Amer. Math. Soc., 46 (2009), 1-33. doi: 10.1090/S0273-0979-08-01228-7.  Google Scholar

[36]

G. Vaira, Semiclassical states for the nonlinear Klein-Gordon-Maxwell system, J. Pure Appl. Math. Adv. Appl., 4 (2010), 59-95.  Google Scholar

[37]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2021, 41 (9) : 4105-4123. doi: 10.3934/dcds.2021030
[8]

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 & Applied Analysis, 2020, 19 (11) : 5131-5156. doi: 10.3934/cpaa.2020230
[9]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071
[10]

Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753
[11]

Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370
[12]

Necdet Bildik, Sinan Deniz. New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 503-518. doi: 10.3934/dcdss.2020028
[13]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[14]

Radhia Ghanmi, Tarek Saanouni. Well-posedness issues for some critical coupled non-linear Klein-Gordon equations. Communications on Pure & Applied Analysis, 2019, 18 (2) : 603-623. doi: 10.3934/cpaa.2019030
[15]

Oana Ivanovici. Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021093
[16]

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
[17]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076
[18]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679
[19]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[20]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences