# American Institute of Mathematical Sciences

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. [2] J. Bellazzini, V. Benci, M. 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. [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. [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. [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. [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. [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. [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. [9] J. Dávila, M. 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. [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. [11] M. Grillakis, J. 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. [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. [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. [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. [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. [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. [17] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined powertype nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. [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. [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.

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##### 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. [2] J. Bellazzini, V. Benci, M. 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. [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. [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. [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. [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. [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. [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. [9] J. Dávila, M. 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. [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. [11] M. Grillakis, J. 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. [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. [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. [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. [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. [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. [17] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined powertype nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. [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. [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.
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