Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-11T01:24:52.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Perturbed Schrödinger lattice systems: existence of homoclinic solutions

Part of: Difference and functional equations Equations of mathematical physics and other areas of application Difference equations

Published online by Cambridge University Press:  27 December 2018

Guanwei Chen  and
Shiwang Ma
Guanwei Chen
Affiliation:
School of Mathematical Sciences, University of Jinan, Jinan 250022, Shandong Province, China (guanweic@163.com)
Shiwang Ma
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China (shiwangm@163.net)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a class of Schrödinger lattice systems with sublinear nonlinearities and perturbed terms. We get an interesting result that the systems do not have nontrivial homoclinic solutions if the perturbed terms are removed, but the systems have ground state homoclinic solutions if the perturbed terms are added. Besides, we also study the continuity of the homoclinic solutions in the perturbation terms at zero. To the best of our knowledge, there is no published result focusing on the perturbed Schrödinger lattice systems.

Keywords

Schrödinger lattice systemsperturbed termssublinearhomoclinic solutions

MSC classification

Primary: 35Q51: Soliton-like equations
Secondary: 35Q55: NLS-like equations (nonlinear Schrödinger) 39A12: Discrete version of topics in analysis 39A70: Difference operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 1083 - 1096
Copyright
Copyright © Royal Society of Edinburgh 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chen, G. and Ma, S.. Discrete nonlinear Schrödinger equations with superlinear nonlinearities. Appl. Math. Comput. 218 (2012), 5496–5507.Google Scholar
2Chen, G. and Ma, S.. Ground state and geometrically distinct solitons of discrete nonlinear Schrödinger equations with saturable nonlinearities. Stud. Appl. Math. 131 (2013), 389–413.Google Scholar
3Chen, G. and Ma, S.. Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms. Appl. Math. Comput. 232 (2014), 787–798.Google Scholar
4Chen, G. and Schechter, M.. Non-periodic discrete Schrödinger equations. Ground state solutions. Z. Angew. Math. Phys. 67 (2016a), 1–15.Google Scholar
5Chen, G., Ma, S. and Wang, Z-Q.. Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities. J. Differ. Equ. 261 (2016b), 3493–3518.Google Scholar
6Christodoulides, D. N., Lederer, F. and Silberberg, Y.. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424 (2003), 817–823.Google Scholar
7Ekeland, I.. Non-convex minimization problems. Bull. Amer. Math. Soc. 1 (1979), 443–474.Google Scholar
8Jia, L. and Chen, G.. Discrete Schrödinger equations with sign-changing nonlinearities. Infinitely many homoclinic solutions. J. Math. Anal. Appl. 452 (2017), 568–577.Google Scholar
9Jia, L., Chen, J. and Chen, G.. Discrete Schrödinger equations in the non-periodic and superlinear cases: Homoclinic solutions. Adv. Differ. Equ. 2017 (2017), 1–15.Google Scholar
10Kopidakis, G., Aubry, S. and Tsironis, G. P.. Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett. 87 (2001), 165501.Google Scholar
11Livi, R., Franzosi, R. and Oppo, G-L.. Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation. Phys. Rev. Lett. 97 (2006), 060401.Google Scholar
12Pankov, A.. Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity 19 (2006), 27–40.Google Scholar
13Pankov, A.. Gap solitons in periodic discrete nonlinear Schrödinger equations II. A generalized Nehari manifold approach. Discrete Contin. Dyn. Syst. 19 (2007), 419–430.Google Scholar
14Pankov, A.. Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearity. J. Math. Anal. Appl. 371 (2010), 254–265.Google Scholar
15Pankov, A. and Rothos, V.. Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity. Proc. R. Soc. A 464 (2008), 3219–3236.Google Scholar
16Pankov, A. and Zhang, G.. Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity. J. Math. Sci. 177 (2011), 71–82.Google Scholar
17Shi, H.. Gap solitons in periodic discrete Schrödinger equations with nonlinearity. Acta Appl. Math. 109 (2010a), 1065–1075.Google Scholar
18Shi, H. and Zhang, H.. Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl. 361 (2010b), 411–19.Google Scholar
19Teschl, G.. Jacobi operators and completely integrable nonlinear lattices (Mathematical Surveys and Monographs, vol 72) (Providence, RI: American Mathematical Society, 2000).Google Scholar
20Yang, M., Chen, W. and Ding, Y.. Solutions for discrete periodic Schrödinger equations with spectrum 0. Acta. Appl. Math. 110 (2010), 1475–1488.Google Scholar
21Zhang, G. and Pankov, A.. Standing waves of the discrete nonlinear Schrödinger equations with growing potentials. Commun. Math. Anal. 5 (2008), 38–49.Google Scholar
22Zhang, G. and Pankov, A.. Standing wave solutions of the discrete non-linear Schrödinger equations with unbounded potentials, II. Appl. Anal. 89 (2010), 1541–1557.Google Scholar
23Zhou, Z. and Ma, D.. Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials. Sci. China Math. 58 (2015), 781–790.Google Scholar
24Zhou, Z. and Yu, J.. On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. J. Differ. Equ. 249 (2010a), 1199–1212.Google Scholar
25Zhou, Z., Yu, J. and Chen, Y.. On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity 23 (2010b), 1727–1740.Google Scholar