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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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

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