In this paper, we consider the following periodic discrete nonlinear Schrödinger equation
Lun−ωun=gn(un),n=(n1,n2,...,nm)∈Zm,
where ω∉σ(L)(the spectrum of L) and gn(s) is super or asymptotically linear as |s|→∞. Under weaker conditions on gn, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.
Citation: Xionghui Xu, Jijiang Sun. Ground state solutions for periodic Discrete nonlinear Schrödinger equations[J]. AIMS Mathematics, 2021, 6(12): 13057-13071. doi: 10.3934/math.2021755
[1] | Kun-Peng Jin, Can Liu . RETRACTED ARTICLE: Decay estimates for the wave equation with partial boundary memory damping. Networks and Heterogeneous Media, 2024, 19(3): 1402-1423. doi: 10.3934/nhm.2024060 |
[2] | Yaru Xie, Genqi Xu . The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks and Heterogeneous Media, 2016, 11(3): 527-543. doi: 10.3934/nhm.2016008 |
[3] | Ye Sun, Daniel B. Work . Error bounds for Kalman filters on traffic networks. Networks and Heterogeneous Media, 2018, 13(2): 261-295. doi: 10.3934/nhm.2018012 |
[4] | Zhong-Jie Han, Enrique Zuazua . Decay rates for 1−d heat-wave planar networks. Networks and Heterogeneous Media, 2016, 11(4): 655-692. doi: 10.3934/nhm.2016013 |
[5] | Abdelaziz Soufyane, Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Imad Kissami, Mostafa Zahri . Stability results of a swelling porous-elastic system with two nonlinear variable exponent damping. Networks and Heterogeneous Media, 2024, 19(1): 430-455. doi: 10.3934/nhm.2024019 |
[6] | Clinton Innes, Razvan C. Fetecau, Ralf W. Wittenberg . Modelling heterogeneity and an open-mindedness social norm in opinion dynamics. Networks and Heterogeneous Media, 2017, 12(1): 59-92. doi: 10.3934/nhm.2017003 |
[7] | Serge Nicaise, Julie Valein . Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks and Heterogeneous Media, 2007, 2(3): 425-479. doi: 10.3934/nhm.2007.2.425 |
[8] | Gildas Besançon, Didier Georges, Zohra Benayache . Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks and Heterogeneous Media, 2009, 4(2): 211-221. doi: 10.3934/nhm.2009.4.211 |
[9] |
Linglong Du .
Long time behavior for the visco-elastic damped wave equation in |
[10] | Yacine Chitour, Guilherme Mazanti, Mario Sigalotti . Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Networks and Heterogeneous Media, 2016, 11(4): 563-601. doi: 10.3934/nhm.2016010 |
In this paper, we consider the following periodic discrete nonlinear Schrödinger equation
Lun−ωun=gn(un),n=(n1,n2,...,nm)∈Zm,
where ω∉σ(L)(the spectrum of L) and gn(s) is super or asymptotically linear as |s|→∞. Under weaker conditions on gn, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.
The journal retracts the paper entitled "Decay estimates for the wave equation with partial boundary memory damping" [1].
After the article was published, the authors decided to withdraw the article.
This retraction was approved by the Editor in Chief of the journal Networks and Heterogeneous Media.
[1] |
S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Phys. D, 103 (1997), 201–250. doi: 10.1016/S0167-2789(96)00261-8
![]() |
[2] | S. Aubry, G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc., 3 (1980), 133–164. |
[3] |
S. Aubry, G. Kopidakis, V. Kadelburg, Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems, Discrete Contin. Dyn. Syst. B, 1 (2001), 271–298. doi: 10.3934/dcdsb.2001.1.271
![]() |
[4] | M.Ya. Azbel, Energy spectrum of a conduction electron in a magnetic field, Sov. Phys. JETP, 19 (1964), 634–645. |
[5] | G. W. Chen, S. W. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218 (2012), 5496–5507. |
[6] |
G. W. Chen, S. W. Ma, Ground state and geometrically distinct solitons of discrete nonlinear Schrödinger equations with saturable nonlinearities, Stud. Appl. Math., 131 (2013), 389–413. doi: 10.1111/sapm.12016
![]() |
[7] |
G. W. Chen, S. W. Ma, Z. Q. Wang, Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities, J. Differ. Equ., 261 (2016), 3493–3518. doi: 10.1016/j.jde.2016.05.030
![]() |
[8] |
D. N. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 424 (2003), 817–823. doi: 10.1038/nature01936
![]() |
[9] |
J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis, B. A. Malomed, Discrete solitons in nonlinear Schrodinger lattices with a power-law nonlinearity, Phys. D, 238 (2009), 67–76. doi: 10.1016/j.physd.2008.08.013
![]() |
[10] |
S. Flach, C. R. Willis, Discrete breathers, Phys Rep., 295 (1998), 181–264. doi: 10.1016/S0370-1573(97)00068-9
![]() |
[11] | S. Flach, K. Kladko, Moving discrete breathers Phys. D, 127 (1999), 61–72. |
[12] |
S. Flach, A. V. Gorbach, Discrete breathers-advances in theory and applications, Phys Rep., 467 (2008), 1–116. doi: 10.1016/j.physrep.2008.05.002
![]() |
[13] |
J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, D. N. Christodoulides, Observation of discrete solitons in optically induced real time waveguide arrays, Phys. Rev. Lett., 90 (2003), 023902. doi: 10.1103/PhysRevLett.90.023902
![]() |
[14] |
J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 422 (2003), 147–150. doi: 10.1038/nature01452
![]() |
[15] |
P. G. Harper, Single Band Motion of Conduction Electrons in a Uniform Magnetic Field, Proc. Phys. Soc. Sect. A, 68 (1955), 874–878. doi: 10.1088/0370-1298/68/10/304
![]() |
[16] |
D. Hennig, G. P. Tsironis, Wave transmission in nonlinear lattices, Physics Reports, 307 (1999), 333–432. doi: 10.1016/S0370-1573(98)00025-8
![]() |
[17] |
S. Iubini, A. Politi, Chaos and localization in the discrete nonlinear Schrödinger equation, Chaos, Solitons and Fractals, 147 (2021), 110954. doi: 10.1016/j.chaos.2021.110954
![]() |
[18] |
L. Jeanjean, K. Tanaka, A positive solution for an asymptotically linear elliptic problem on RN autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597–614. doi: 10.1051/cocv:2002068
![]() |
[19] | P. G. Kevrekidis, K. Rasmussen, A. R. Bishop, The discrete nonlinear Schrödinger equation: a survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2883–2900. |
[20] |
G. Kopidakis, S. Aubry, G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 165501. doi: 10.1103/PhysRevLett.87.165501
![]() |
[21] |
G. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763–776. doi: 10.1142/S0219199702000853
![]() |
[22] |
S. Liu, On superlinear Schrdinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1–9. doi: 10.1007/s00526-011-0447-2
![]() |
[23] |
R. Livi, R. Franzosi, G. L. Oppo, Self-localization of Bose Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 060401. doi: 10.1103/PhysRevLett.97.060401
![]() |
[24] | D. Ma, Z. Zhou, Existence and multiplicity results of homoclinic solutions for the DNLS equations with unbounded potentials, Abstr. Appl. Anal., 2012 (2012), 703596. |
[25] | A. Mai, Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities, Abstr. Appl. Anal., 2013 (2013), 317139. |
[26] | A. Mai, Z. Zhou, Discrete solitons for periodic discrete nonlinear Schrödinger equations, Appl. Math. Comput., 222 (2013), 34–41. |
[27] |
D. V. Makarov, M. Yu. Uleysky, Chaos-assisted formation of immiscible matter-wave solitons and self-stabilization in the binary discrete nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 227–238. doi: 10.1016/j.cnsns.2016.07.006
![]() |
[28] |
A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27–40. doi: 10.1088/0951-7715/19/1/002
![]() |
[29] |
A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations. II. A generalized Nehari manifold approach, Discrete Contin. Dyn. Syst. Ser. A, 19 (2007), 419–430. doi: 10.3934/dcds.2007.19.419
![]() |
[30] |
H. Shi, Gap solitons in periodic discrete nonlinear Schrödinger equations with nonlinearity, Acta Appl. Math., 109 (2010), 1065–1075. doi: 10.1007/s10440-008-9360-x
![]() |
[31] | J. J. Sun, S. W. Ma, Multiple solutions for discrete periodic nonlinear Schrödinger equations, J. Math. Phys., 56 (2015), 1413–1442. |
[32] |
A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Func. Anal., 257 (2009), 3802–3822. doi: 10.1016/j.jfa.2009.09.013
![]() |
[33] |
X. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud., 14 (2014), 361–373. doi: 10.1515/ans-2014-0208
![]() |
[34] | X. Tang, Non-nehari-manifold method for asymptotically linear schrodinger equation, J. Math. Phys., 56 (2015), 1413–1442. |
[35] | A. Trombettoni, A. Smerzi, Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates, Phys. Rev. Lett., 16 (2001), 2353–2356. |
[36] |
Z. Yang, W. Chen, Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum 0, Acta Appl. Math., 110 (2010), 1475–1488. doi: 10.1007/s10440-009-9521-6
![]() |
[37] |
L. Zhang, S. Ma, Ground state solutions for periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, Adv. Difference Equ., 2018 (2018), 1–13. doi: 10.1186/s13662-017-1452-3
![]() |
[38] |
Z. Zhou, J. Yu, Y. Chen, On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity, Nonlinearity, 23 (2010), 1727–1740. doi: 10.1088/0951-7715/23/7/011
![]() |