Initial value problem of the discrete nonlinear Schrödinger equation with complex potential

References

• Hennig D, Tsironis GP. Wave transmission in nonlinear lattices. Phys Repts. 1999;309:333–432.
   Web of Science ®Google Scholar
• Kevrekidis PG. The discrete nonlinear Schrödinger equation: mathematical analysis, numerical computations and physical perspectives. Berlin: Springer; 2009.
   Google Scholar
• Kevrekidis PG, Ø. Rasmussen K, Bishop AR. The discrete nonlinear Schrödinger equation: a survey of recent results. Intern J Modern Phys B. 2001;15:2833–2900.
   Web of Science ®Google Scholar
• Flach S, Gorbach AV. Discrete breathers – advances in theory and applications. Phys Repts. 2008;467:1–116.
   Web of Science ®Google Scholar
• Flach S, Willis CR. Discrete breathers. Phys Repts. 1998;295:181–264.
   Web of Science ®Google Scholar
• Cheng M, Pankov A. Gap solitons in periodic nonlinear Schrödinger equations with nonlinear hopping. Electr J Differ Equ. 2016;2016(287):1–14.
   Google Scholar
• Pankov A. Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity. 2006;19:27–40.
   Web of Science ®Google Scholar
• Pankov A. Gap solitons in periodic discrete nonlinear Schrödinger equations, II: a generalized Nehari manifold approach. Discr Cont Dyn Syst A. 2007;19:419–430.
   Web of Science ®Google Scholar
• Pankov A. Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities. J Math Anal Appl. 2010;371:254–265.
   Web of Science ®Google Scholar
• Pankov A, Rothos V. Periodic and decaying solutions in discrete nonliinear Schrödinger equations with saturable nonlinearity. Proc Roy Soc A. 2008;464:3219–3236.
   Google Scholar
• Pankov A, Zhang G. Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearities. J Math Sci. 2011;177:71–82.
   Google Scholar
• Zhang G. Breather solutions of the discrete nonlinear Schrödinger equation with unbounded potential. J Math Phys. 2009;50:013505.
   Web of Science ®Google Scholar
• Zhang G. Breather solutions of the discrete nonlinear Schrödinger equation with sign changing nonlinearity. J Math Phys. 2011;52:043516.
   Web of Science ®Google Scholar
• Zhang G, Liu F. Existence of breather solutions of the DNLS equation with unbounded potential. Nonlin Anal. 2009;71:e786–e792.
   Web of Science ®Google Scholar
• Zhang G, Pankov A. Standing waves of the discrete nonlinear Schrödinger equations with growing potentials. Commun Math Anal. 2008;5(2):38–49.
   Google Scholar
• Zhang G, Pankov A. Standing wave solutions for the discrete nonlinear Schrödinger equations with unbounded potentials, II. Applicable Anal. 2011;89:1541–1557.
   Web of Science ®Google Scholar
• Weinstein MI. Excitation threshold for nonlinear localized modes on lattices. Nonlinearity. 1999;19:673–691.
   Web of Science ®Google Scholar
• Karachalios NI, Yannacopoulos AN. Global existence and compact attractors for the discrete nonlinear Schrödinger equations. J Differ Equ. 2005;217:88–123.
   Web of Science ®Google Scholar
• Karachalios NI, Yannacopoulos AN. The existence of global attractor for the discrete nonlinear Schrödinger equation.II. Compactness without tail estimates in ZN, N≥1, lattices. Proc Roy Soc Edinburgh. 2007;137A:63–76.
   Google Scholar
• N'Guérékata G, Pankov A. Global well-posedness for discrete nonlinear Schrödinger equation. Appl Anal. 2010;89:1513–1521.
   Web of Science ®Google Scholar
• Pacciani P, Konotop VV, Perla Menzala G. On localized solutions of discrete nonlinear Schrödinger equation: an exact result. Phys D. 2005;204:122–133.
   Web of Science ®Google Scholar
• Bates PW, Liu K, Wang B. Attractors for lattice dynamical systems. J Bifur Chaos Appl Sci Eng. 2001;11:143–153.
   Web of Science ®Google Scholar
• Zhou S. Attractors for second order lattice dynamical systems. J Differ Equ. 2002;179:605–624.
   Web of Science ®Google Scholar
• Zhou S. Attractors for first order dissipative lattices. Phys D. 2003;178:51–61.
   Web of Science ®Google Scholar
• Zhou S. Attractors and approximations for lattice dynamical systems. J Differ Equ. 2004;200:342–368.
   Web of Science ®Google Scholar
• Zhou S, Shi W. Attractors and dimension for dissipative lattice systems. J Differ Equ. 2006;224:172–204.
   Web of Science ®Google Scholar
• Engel K-J, Nagel R. A short course on operator semigroups. New York: Springer; 2006.
   Google Scholar
• Pazy A. Semigroups of linear operators and applications. New York: Springer; 1983.
   Google Scholar
• Cazenave T, Haraux A. An introduction to semilinear evolution equations. Oxford University Press; Oxford, 1998. Translation.
   Google Scholar
• Temam R. Infinite-Dimensional dynamical systems in mathematics and physics. New York: Springer; 1997.
   Google Scholar
• Chepyzhov VV, Vishik MI. Attractors for equations of mathematical physics. Providence (RI): Colloquium Publication; 2002. (American Math Soc.; 49).
   Google Scholar
• Karachalios NI, Sánchez-Rey B, Kevrekidis PG, et al. Breathers for the discrete nonlinear Schrödinger equation with nonlinear hopping. J Nonlin Sci. 2013;23:205–239.
   Web of Science ®Google Scholar