PERIODIC DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS WITH PERTURBED AND SUB-LINEAR TERMS

Jie Yang; Guanwei Chen

doi:10.11948/20210330

2022 Volume 12 Issue 6

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Article navigation > Journal of Applied Analysis & Computation > 2022 > 12(6): 2220-2229

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Jie Yang, Guanwei Chen. PERIODIC DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS WITH PERTURBED AND SUB-LINEAR TERMS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2220-2229. doi: 10.11948/20210330

Citation:

Jie Yang, Guanwei Chen. PERIODIC DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS WITH PERTURBED AND SUB-LINEAR TERMS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2220-2229. doi: 10.11948/20210330

PERIODIC DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS WITH PERTURBED AND SUB-LINEAR TERMS

Jie Yang¹,
Guanwei Chen^1, ,

1.
School of Mathematical Sciences, University of Jinan, Jinan 250022, Shandong Province, China

Corresponding author: Email address:guanweic@163.com(G. Chen)

Abstract

Abstract

In this paper, we study a class of perturbed discrete nonlinear Schrödinger equations with sub-linear nonlinearities at infinity and obtain the existence of solitons for this class of equations by using a generalized saddle point theorem. To the best of our knowledge, there is no published result focusing on this class of perturbed discrete nonlinear equations by this method.
- Discrete nonlinear Schrödinger equations /
- sub-linear /
- solitons /
- generalized saddle point theorem
MSC: 35Q51, 35Q55, 39A12, 39A70

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References

[1]	S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 1997, 103, 201-250. doi: 10.1016/S0167-2789(96)00261-8 CrossRef Google Scholar
[2]	D. N. Christodoulides, F. Lederer and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 2003, 424, 817-823. doi: 10.1038/nature01936 CrossRef Google Scholar
[3]	J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 2009, 238, 67-76. doi: 10.1016/j.physd.2008.08.013 CrossRef Google Scholar
[4]	G. Chen and S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 2012, 218, 5496-5507. Google Scholar
[5]	G. Chen and S. Ma, Ground State and Geometrically Distinct Solitons of Discrete Nonlinear Schrödinger Equations with Saturable Nonlinearities, Stud. Appl. Math., 2013, 131, 389-413. doi: 10.1111/sapm.12016 CrossRef Google Scholar
[6]	G. Chen, S. Ma and Z. Wang, Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities, J. Differential Equations, 2016, 261, 3493-3518. doi: 10.1016/j.jde.2016.05.030 CrossRef Google Scholar
[7]	Y. Ding, Variational Methods for Strongly Indefinite Problems, Interdisciplinary Mathematical Sciences, World Scientific, Hackensack, 2007, 7. Google Scholar
[8]	L. Erbe, B. Jia and Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, J. Appl. Anal. Comput., 2019, 9, 271-294. Google Scholar
[9]	S. Flach and C. R. Willis, Discrete breathers, Phys. Rep., 1998, 295, 181-264. doi: 10.1016/S0370-1573(97)00068-9 CrossRef Google Scholar
[10]	S. Flach and A. V. Gorbach, Discrete breathers-advance in theory and applications, Phys. Rep., 2008, 467, 1-116. doi: 10.1016/j.physrep.2008.05.002 CrossRef Google Scholar
[11]	J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis and D. N. Christodoulides, Observation of discrete solitons in optically induced real time waveguide arrays, Phys. Rev. Lett., 2003, 90, 023902. doi: 10.1103/PhysRevLett.90.023902 CrossRef Google Scholar
[12]	J. W. Fleischer, M. Segev, N. K. Efremidis and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 2003, 422, 147-150. doi: 10.1038/nature01452 CrossRef Google Scholar
[13]	A. V. Gorbach and M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. Phys. J. D, 2004, 29, 77-93. doi: 10.1140/epjd/e2004-00017-3 CrossRef Google Scholar
[14]	Y. Hanif and U. Saleem, Degenerate and non-degenerate solutions of PT-symmetric nonlocal integrable discrete nonlinear Schrödinger equation, Phys. Lett. A, 2020, 384(32), 126834. doi: 10.1016/j.physleta.2020.126834 CrossRef Google Scholar
[15]	G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 2003, 13, 27-63. doi: 10.1007/s00332-002-0525-x CrossRef Google Scholar
[16]	G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 2001, 87, 165501. doi: 10.1103/PhysRevLett.87.165501 CrossRef Google Scholar
[17]	G. Lin and Z. Zhou, Homoclinic solutions of discrete ϕ-Laplacian equations with mixed nonlinearities, Comm. Pure Appl. Anal., 2018, 17, 1723-1747. doi: 10.3934/cpaa.2018082 CrossRef Google Scholar
[18]	G. Lin, J. Yu and Z. Zhou, Homoclinic solutions of discrete nonlinear Schrödinger equations with partially sublinear nonlinearities, Electron. J. Differ. Equ., 2019, 96, 1-14. Google Scholar
[19]	G. Lin, Z. Zhou and J. Yu, Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials, J. Dynam. Differential Equations, 2020, 32, 527-555. doi: 10.1007/s10884-019-09743-4 CrossRef Google Scholar
[20]	S. Liu and Z. Shen, Generalized saddle point theorem and asymptotically linear problems with periodic potential, Nonlinear Anal., 2013, 86, 52-57. doi: 10.1016/j.na.2013.03.005 CrossRef Google Scholar
[21]	R. Livi, R. Franzosi and G. L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 2006, 97, 060401. doi: 10.1103/PhysRevLett.97.060401 CrossRef Google Scholar
[22]	M. I. Molina, The two-dimensional fractional discrete nonlinear Schrödinger equation, Phys. Lett. A, 2020, 384(33), 126835. doi: 10.1016/j.physleta.2020.126835 CrossRef Google Scholar
[23]	A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 2006, 19, 27-40. doi: 10.1088/0951-7715/19/1/002 CrossRef Google Scholar
[24]	A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations Ⅱ: A generalized Nehari manifold approach, Discrete Contin. Dyn. Syst., 2007, 19, 419-430. doi: 10.3934/dcds.2007.19.419 CrossRef Google Scholar
[25]	A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A, 2008, 464, 3219-3236. doi: 10.1098/rspa.2008.0255 CrossRef Google Scholar
[26]	A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearity, J. Math. Anal. Appl., 2010, 371, 254-265. doi: 10.1016/j.jmaa.2010.05.041 CrossRef Google Scholar
[27]	M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 2003, 9, 601-619. doi: 10.1051/cocv:2003029 CrossRef Google Scholar
[28]	H. Shi and H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 2010, 361, 411-19. doi: 10.1016/j.jmaa.2009.07.026 CrossRef Google Scholar
[29]	H. Shi, Gap solitons in periodic discrete Schrödinger equations with nonlinearity, Acta Appl. Math., 2010, 109, 1065-1075. doi: 10.1007/s10440-008-9360-x CrossRef Google Scholar
[30]	A. A. Sukhorukov and Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett., 2003, 28, 2345-2347. doi: 10.1364/OL.28.002345 CrossRef Google Scholar
[31]	A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 2009, 12, 3802-3822. Google Scholar
[32]	G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices (Mathematical Surveys and Monographs vol 72) (Providence, RI: American Mathematical Society), 2000. Google Scholar
[33]	M. Yang, W. Chen and Y. Ding, Solutions for Discrete Periodic Schrödinger Equations with Spectrum 0, Acta Appl. Math., 2010, 110, 1475-1488. doi: 10.1007/s10440-009-9521-6 CrossRef Google Scholar
[34]	Z. Zhou and J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 2010, 249, 1199-1212. doi: 10.1016/j.jde.2010.03.010 CrossRef Google Scholar
[35]	Z. Zhou, J. Yu and Y. Chen, On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity, Nonlinearity, 2010, 23, 1727-1740. doi: 10.1088/0951-7715/23/7/011 CrossRef Google Scholar