August  2017, 37(8): 4309-4328. doi: 10.3934/dcds.2017184

Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one

1. 

West Building, Office No. 5W443, Department of Mathematics Education, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon, 402-751, South Korea

2. 

Dipartimento di Matematica, Largo Bruno Pontecorvo n. 5, 56127, Pisa (PI), Italy

3. 

Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

× Corresponding author: Daniele Garrisi

Received  November 2016 Revised  May 2017 Published  April 2017

Fund Project: The first author was supported by INHA UNIVERSITY Research Grant through the project number 51747-01 titled "Stability in non-linear evolution equations". The second author was supported by University of Pisa, project no. PRA-2016-41 "Fenomeni singolari in problemi deterministici e stocastici ed applicazioni"; by INDAM, GNAMPA -Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni and by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University.

We prove that standing-waves which are solutions to the non-linear Schrödinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.

Citation: Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184
References:
[1]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993, Corrected reprint of the 1993 original.  Google Scholar

[2]

J. BellazziniV. BenciM. Ghimenti and A. M. Micheletti, On the existence of the fundamental eigenvalue of an elliptic problem in $\mathbb{R}^N$, Adv. Nonlinear Stud., 7 (2007), 439-458.  doi: 10.1515/ans-2007-0306.  Google Scholar

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J. Bellazzini and G. Siciliano, Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.  doi: 10.1016/j.jfa.2011.06.014.  Google Scholar

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V. Benci and D. Fortunato, Hylomorphic solitons and charged Q-balls: Existence and stability, Chaos Solitons Fractals, 58 (2014), 1-15.  doi: 10.1016/j.chaos.2013.10.005.  Google Scholar

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H. Berestycki and P. -L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

T. Cazenave and P. -L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. , 85 (1982), 549-561, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1103921547.MR0677997 doi: 10.1007/BF01403504.  Google Scholar

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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

[9]

J. DávilaM. del Pino and I. Guerra, Non-uniqueness of positive ground states of non-linear Schrödinger equations, Proc. Lond. Math. Soc. (3), 106 (2013), 318-344.  doi: 10.1112/plms/pds038.  Google Scholar

[10]

D. Garrisi, On the orbital stability of standing-waves solutions to a coupled non-linear KleinGordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.  Google Scholar

[11]

M. GrillakisJ. Shatah and W. 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

[12]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[13]

P. -L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145, URL http://www.numdam.org/item?id=AIHPC_1984__1_2_109_0. doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[14]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. , 100 (1985), 173-190, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1103943442. doi: 10.1007/BF01212446.  Google Scholar

[15]

M. Shibata, Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term, Manuscripta Math., 143 (2014), 221-237.  doi: 10.1007/s00229-013-0627-9.  Google Scholar

[16]

X. Song, Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities, J. Math. Anal. Appl., 366 (2010), 345-359.  doi: 10.1016/j.jmaa.2009.12.011.  Google Scholar

[17]

T. TaoM. Visan and X. Zhang, The nonlinear Schrödinger equation with combined powertype nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[18]

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

[19]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993, Corrected reprint of the 1993 original.  Google Scholar

[2]

J. BellazziniV. BenciM. Ghimenti and A. M. Micheletti, On the existence of the fundamental eigenvalue of an elliptic problem in $\mathbb{R}^N$, Adv. Nonlinear Stud., 7 (2007), 439-458.  doi: 10.1515/ans-2007-0306.  Google Scholar

[3]

J. Bellazzini and G. Siciliano, Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.  doi: 10.1016/j.jfa.2011.06.014.  Google Scholar

[4]

V. Benci and D. Fortunato, Hylomorphic solitons and charged Q-balls: Existence and stability, Chaos Solitons Fractals, 58 (2014), 1-15.  doi: 10.1016/j.chaos.2013.10.005.  Google Scholar

[5]

H. Berestycki and P. -L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

T. Cazenave and P. -L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. , 85 (1982), 549-561, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1103921547.MR0677997 doi: 10.1007/BF01403504.  Google Scholar

[8]

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

[9]

J. DávilaM. del Pino and I. Guerra, Non-uniqueness of positive ground states of non-linear Schrödinger equations, Proc. Lond. Math. Soc. (3), 106 (2013), 318-344.  doi: 10.1112/plms/pds038.  Google Scholar

[10]

D. Garrisi, On the orbital stability of standing-waves solutions to a coupled non-linear KleinGordon equation, Adv. Nonlinear Stud., 12 (2012), 639-658.  doi: 10.1515/ans-2012-0311.  Google Scholar

[11]

M. GrillakisJ. Shatah and W. 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

[12]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[13]

P. -L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145, URL http://www.numdam.org/item?id=AIHPC_1984__1_2_109_0. doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[14]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. , 100 (1985), 173-190, URL http://projecteuclid.org/getRecord?id=euclid.cmp/1103943442. doi: 10.1007/BF01212446.  Google Scholar

[15]

M. Shibata, Stable standing waves of nonlinear Schrödinger equations with a general nonlinear term, Manuscripta Math., 143 (2014), 221-237.  doi: 10.1007/s00229-013-0627-9.  Google Scholar

[16]

X. Song, Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities, J. Math. Anal. Appl., 366 (2010), 345-359.  doi: 10.1016/j.jmaa.2009.12.011.  Google Scholar

[17]

T. TaoM. Visan and X. Zhang, The nonlinear Schrödinger equation with combined powertype nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[18]

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

[19]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

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