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March  2021, 41(3): 1225-1270. doi: 10.3934/dcds.2020316

On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations

Andrew Comech 1, and  Scipio Cuccagna 2,,

1. 

Texas A & M University, College Station, TX, USA, Institute for Information Transmission Problems, Moscow, Russia
2. 

University of Trieste, Trieste 34127, Italy

* Corresponding author: Scipio Cuccagna

Received  July 2019 Revised  November 2019 Published  March 2021 Early access  August 2020

We extend to a specific class of systems of nonlinear Schrödinger equations (NLS) the theory of asymptotic stability of ground states already proved for the scalar NLS. Here the key point is the choice of an adequate system of modulation coordinates and the novelty, compared to the scalar NLS, is the fact that the group of symmetries of the system is non-commutative.

Keywords: Asymptotic stability, solitary waves, nonlinear Schrödinger equation.
Mathematics Subject Classification: Primary:35B35, 35B40, 35C08, 35Q41;Secondary:37K40.
Citation: Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
References:
[1]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, vol. 250 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York-Berlin, 1983.  Google Scholar

[2]

D. Bambusi, Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry, Comm. Math. Phys., 320 (2013), 499-542.  doi: 10.1007/s00220-013-1684-3.  Google Scholar

[3]

D. Bambusi and S. Cuccagna, On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential, Amer. J. Math., 133 (2011), 1421-1468.  doi: 10.1353/ajm.2011.0034.  Google Scholar

[4]

M. Beceanu, New estimates for a time-dependent Schrödinger equation, Duke Math. J., 159 (2011), 417-477.  doi: 10.1215/00127094-1433394.  Google Scholar

[5]

S. Bhattarai, Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 36 (2016), 1789-1811.  doi: 10.3934/dcds.2016.36.1789.  Google Scholar

[6]

N. Boussaid, Stable directions for small nonlinear Dirac standing waves, Comm. Math. Phys., 268 (2006), 757-817.  doi: 10.1007/s00220-006-0112-3.  Google Scholar

[7]

N. Boussaid and S. Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37 (2012), 1001-1056.  doi: 10.1080/03605302.2012.665973.  Google Scholar

[8]

N. Boussaïd and A. Comech, Spectral stability of bi-frequency solitary waves in Soler and Dirac–Klein–Gordon models, Commun. Pure Appl. Anal., 17 (2018), 1331-1347.  doi: 10.3934/cpaa.2018065.  Google Scholar

[9]

N. Boussaïd and A. Comech, Nonlinear Dirac equation. Spectral stability of solitary waves, vol. 244 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2019. doi: 10.1090/surv/244.  Google Scholar

[10]

N. Boussaïd and A. Comech, Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity, J. Funct. Anal., 277 (2019), 108289, 68 pp. doi: 10.1016/j.jfa.2019.108289.  Google Scholar

[11]

V. S. Buslaev and G. S. Perel'man, Scattering for the nonlinear Schrödinger equation: States that are close to a soliton, Algebra i Analiz, 4 (1992), 63-102.   Google Scholar

[12]

V. S. Buslaev and G. S. Perel'man, On the stability of solitary waves for nonlinear Schrödinger equations, in Nonlinear evolution equations, vol. 164 of Amer. Math. Soc. Transl. Ser. 2, 75–98, Amer. Math. Soc., Providence, RI, 1995. doi: 10.1090/trans2/164/04.  Google Scholar

[13]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[14]

A. Comech, T. V. Phan and A. Stefanov, Asymptotic stability of solitary waves in generalized Gross–Neveu model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 157-196.  doi: 10.1016/j.anihpc.2015.11.001.  Google Scholar

[15]

S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145.  doi: 10.1002/cpa.1018.  Google Scholar

[16]

S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm. Math. Phys., 305 (2011), 279-331.  doi: 10.1007/s00220-011-1265-2.  Google Scholar

[17]

S. Cuccagna, On the Darboux and Birkhoff steps in the asymptotic stability of solitons, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 197-257.   Google Scholar

[18]

S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 366 (2014), 2827-2888.  doi: 10.1090/S0002-9947-2014-05770-X.  Google Scholar

[19]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a non-trapping potential, J. Differential Equations, 256 (2014), 1395-1466.  doi: 10.1016/j.jde.2013.11.002.  Google Scholar

[20]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a trapping potential, Discrete Contin. Dyn. Syst., 35 (2015), 3343-3376.  doi: 10.3934/dcds.2015.35.3343.  Google Scholar

[21]

S. Cuccagna and M. Maeda, On orbital instability of spectrally stable vortices of the NLS in the plane, J. Nonlinear Sci., 26 (2016), 1851-1894.  doi: 10.1007/s00332-016-9322-9.  Google Scholar

[22]

S. De Bièvre and S. Rota Nodari, Orbital stability via the energy-momentum method: The case of higher dimensional symmetry groups, Arch. Ration. Mech. Anal., 231 (2019), 233-284.  doi: 10.1007/s00205-018-1278-5.  Google Scholar

[23]

M. P. Do Carmo, Riemannian Geometry, Birkhäuser Boston, Inc., Boston, MA, 1992. Second edn.  Google Scholar

[24]

A. Galindo, A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento (2), 20 (1977), 210-212.  doi: 10.1007/BF02785129.  Google Scholar

[25]

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

[26]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.  doi: 10.1215/S0012-7094-79-04631-3.  Google Scholar

[27]

K. Nakanishi and W. Schlag, Global dynamics above the ground state for the nonlinear Klein–Gordon equation without a radial assumption, Arch. Ration. Mech. Anal., 203 (2012), 809-851.  doi: 10.1007/s00205-011-0462-7.  Google Scholar

[28]

P. J. Olver, Applications of Lie groups to differential equations, vol. 107 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[29]

D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys., 53 (2012), 073705, 27 pp. doi: 10.1063/1.4731477.  Google Scholar

[30]

G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 1051-1095.  doi: 10.1081/PDE-200033754.  Google Scholar

[31]

C. Radford, Dirac equation revisited: Charge quantization as a consequence of the Dirac equation, Phys. Rev. D, 27 (1983), 1970-1971.  doi: 10.1103/PhysRevD.27.1970.  Google Scholar

[32] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.   Google Scholar
[33] W. Rossmann, Lie Groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, 5. Oxford University Press, Oxford, 2002.   Google Scholar
[34]

I. M. Sigal, Nonlinear wave and Schrödinger equations. Ⅰ. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys., 153 (1993), 297-320.  doi: 10.1007/BF02096645.  Google Scholar

[35]

A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136 (1999), 9-74.  doi: 10.1007/s002220050303.  Google Scholar

[36]

W. A. Strauss, Nonlinear Wave Equations, vol. 73 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989. doi: 10.1090/cbms/073.  Google Scholar

[37]

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

[38]

M. I. Weinstein, Localized states and dynamics in the nonlinear Schrödinger / Gross–Pitaevskii equation, Dynamics of Partial Differential Equations. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 3. Springer, Cham. 41–79. doi: 10.1007/978-3-319-19935-1_2.  Google Scholar

[39]

J. A. Wolf, Harmonic Analysis on Commutative Spaces, vol. 142 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/142.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, vol. 250 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York-Berlin, 1983.  Google Scholar

[2]

D. Bambusi, Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry, Comm. Math. Phys., 320 (2013), 499-542.  doi: 10.1007/s00220-013-1684-3.  Google Scholar

[3]

D. Bambusi and S. Cuccagna, On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential, Amer. J. Math., 133 (2011), 1421-1468.  doi: 10.1353/ajm.2011.0034.  Google Scholar

[4]

M. Beceanu, New estimates for a time-dependent Schrödinger equation, Duke Math. J., 159 (2011), 417-477.  doi: 10.1215/00127094-1433394.  Google Scholar

[5]

S. Bhattarai, Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 36 (2016), 1789-1811.  doi: 10.3934/dcds.2016.36.1789.  Google Scholar

[6]

N. Boussaid, Stable directions for small nonlinear Dirac standing waves, Comm. Math. Phys., 268 (2006), 757-817.  doi: 10.1007/s00220-006-0112-3.  Google Scholar

[7]

N. Boussaid and S. Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37 (2012), 1001-1056.  doi: 10.1080/03605302.2012.665973.  Google Scholar

[8]

N. Boussaïd and A. Comech, Spectral stability of bi-frequency solitary waves in Soler and Dirac–Klein–Gordon models, Commun. Pure Appl. Anal., 17 (2018), 1331-1347.  doi: 10.3934/cpaa.2018065.  Google Scholar

[9]

N. Boussaïd and A. Comech, Nonlinear Dirac equation. Spectral stability of solitary waves, vol. 244 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2019. doi: 10.1090/surv/244.  Google Scholar

[10]

N. Boussaïd and A. Comech, Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity, J. Funct. Anal., 277 (2019), 108289, 68 pp. doi: 10.1016/j.jfa.2019.108289.  Google Scholar

[11]

V. S. Buslaev and G. S. Perel'man, Scattering for the nonlinear Schrödinger equation: States that are close to a soliton, Algebra i Analiz, 4 (1992), 63-102.   Google Scholar

[12]

V. S. Buslaev and G. S. Perel'man, On the stability of solitary waves for nonlinear Schrödinger equations, in Nonlinear evolution equations, vol. 164 of Amer. Math. Soc. Transl. Ser. 2, 75–98, Amer. Math. Soc., Providence, RI, 1995. doi: 10.1090/trans2/164/04.  Google Scholar

[13]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[14]

A. Comech, T. V. Phan and A. Stefanov, Asymptotic stability of solitary waves in generalized Gross–Neveu model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 157-196.  doi: 10.1016/j.anihpc.2015.11.001.  Google Scholar

[15]

S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 54 (2001), 1110-1145.  doi: 10.1002/cpa.1018.  Google Scholar

[16]

S. Cuccagna, The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states, Comm. Math. Phys., 305 (2011), 279-331.  doi: 10.1007/s00220-011-1265-2.  Google Scholar

[17]

S. Cuccagna, On the Darboux and Birkhoff steps in the asymptotic stability of solitons, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 197-257.   Google Scholar

[18]

S. Cuccagna, On asymptotic stability of moving ground states of the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 366 (2014), 2827-2888.  doi: 10.1090/S0002-9947-2014-05770-X.  Google Scholar

[19]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a non-trapping potential, J. Differential Equations, 256 (2014), 1395-1466.  doi: 10.1016/j.jde.2013.11.002.  Google Scholar

[20]

S. Cuccagna and M. Maeda, On weak interaction between a ground state and a trapping potential, Discrete Contin. Dyn. Syst., 35 (2015), 3343-3376.  doi: 10.3934/dcds.2015.35.3343.  Google Scholar

[21]

S. Cuccagna and M. Maeda, On orbital instability of spectrally stable vortices of the NLS in the plane, J. Nonlinear Sci., 26 (2016), 1851-1894.  doi: 10.1007/s00332-016-9322-9.  Google Scholar

[22]

S. De Bièvre and S. Rota Nodari, Orbital stability via the energy-momentum method: The case of higher dimensional symmetry groups, Arch. Ration. Mech. Anal., 231 (2019), 233-284.  doi: 10.1007/s00205-018-1278-5.  Google Scholar

[23]

M. P. Do Carmo, Riemannian Geometry, Birkhäuser Boston, Inc., Boston, MA, 1992. Second edn.  Google Scholar

[24]

A. Galindo, A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento (2), 20 (1977), 210-212.  doi: 10.1007/BF02785129.  Google Scholar

[25]

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

[26]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.  doi: 10.1215/S0012-7094-79-04631-3.  Google Scholar

[27]

K. Nakanishi and W. Schlag, Global dynamics above the ground state for the nonlinear Klein–Gordon equation without a radial assumption, Arch. Ration. Mech. Anal., 203 (2012), 809-851.  doi: 10.1007/s00205-011-0462-7.  Google Scholar

[28]

P. J. Olver, Applications of Lie groups to differential equations, vol. 107 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[29]

D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys., 53 (2012), 073705, 27 pp. doi: 10.1063/1.4731477.  Google Scholar

[30]

G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 1051-1095.  doi: 10.1081/PDE-200033754.  Google Scholar

[31]

C. Radford, Dirac equation revisited: Charge quantization as a consequence of the Dirac equation, Phys. Rev. D, 27 (1983), 1970-1971.  doi: 10.1103/PhysRevD.27.1970.  Google Scholar

[32] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.   Google Scholar
[33] W. Rossmann, Lie Groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, 5. Oxford University Press, Oxford, 2002.   Google Scholar
[34]

I. M. Sigal, Nonlinear wave and Schrödinger equations. Ⅰ. Instability of periodic and quasiperiodic solutions, Comm. Math. Phys., 153 (1993), 297-320.  doi: 10.1007/BF02096645.  Google Scholar

[35]

A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math., 136 (1999), 9-74.  doi: 10.1007/s002220050303.  Google Scholar

[36]

W. A. Strauss, Nonlinear Wave Equations, vol. 73 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989. doi: 10.1090/cbms/073.  Google Scholar

[37]

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

[38]

M. I. Weinstein, Localized states and dynamics in the nonlinear Schrödinger / Gross–Pitaevskii equation, Dynamics of Partial Differential Equations. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 3. Springer, Cham. 41–79. doi: 10.1007/978-3-319-19935-1_2.  Google Scholar

[39]

J. A. Wolf, Harmonic Analysis on Commutative Spaces, vol. 142 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/142.  Google Scholar

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