Stabilization of small solutions of discrete NLS with potential having two eigenvalues

References

• Eisenberg HS, Morandotti R, Silberberg Y, et al. Optical discrete solitons in waveguide arrays. I. Soliton formation. J Opt Soc Am B. 2002;19(12):2938–2944.
   Web of Science ®Google Scholar
• Peschel U, Morandotti R, Arnold JM, et al. Optical discrete solitons in waveguide arrays. 2. Dynamic properties. J Opt Soc Am B. 2002;19(11):2637–2644.
   Web of Science ®Google Scholar
• Efremidis NK, Sears S, Christodoulides DN, et al. Discrete solitons in photorefractive optically induced photonic lattices. Phys Rev E. 2002;66:046602.
   Web of Science ®Google Scholar
• Sukhorukov AA, Kivshar YS, Eisenberg HS, et al. Spatial optical solitons in waveguide arrays. IEEE J Quantum Electron. 2003;39(1):31–50.
   Web of Science ®Google Scholar
• Cataliotti FS, Burger S, Fort C, et al. Josephson junction arrays with bose-einstein condensates. Science. 2001;293(5531):843–846.
   PubMed Web of Science ®Google Scholar
• Hadad Y, Khanikaev AB, Alù A. Self-induced topological transitions and edge states supported by nonlinear staggered potentials. Phys. Rev. B. 2016;93:155112.
   Web of Science ®Google Scholar
• Flach S, Gorbach AV. Discrete breathers – advances in theory and applications. Phys Rep. 2008;467(1):1–116.
   Google Scholar
• Flach S, Willis CR. Discrete breathers. Phys Rep. 1998;295(5):181–264.
   Google Scholar
• Bambusi D, Penati T. Continuous approximation of breathers in one- and two-dimensional DNLS lattices. Nonlinearity. 2010;23(1):143–157.
   Web of Science ®Google Scholar
• Stefanov A, Kevrekidis PG. Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations. Nonlinearity. 2005;18(4):1841–1857.
   Web of Science ®Google Scholar
• Pelinovsky DE, Stefanov A. On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension. J Math Phys. 2008;49(11):113501, 17.
   Web of Science ®Google Scholar
• Komech AI, Kopylova EA, Kunze M. Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations. Appl Anal. 2006;85(12):1487–1508.
   Google Scholar
• Mielke A, Patz C. Dispersive stability of infinite-dimensional Hamiltonian systems on lattices. Appl Anal. 2010;89(9):1493–1512.
   Web of Science ®Google Scholar
• Cuccagna S, Tarulli M. On asymptotic stability of standing waves of discrete Schrödinger equation in Z. SIAM J Math Anal. 2009;41(3):861–885.
   Web of Science ®Google Scholar
• Kevrekidis PG, Pelinovsky DE, Stefanov A. Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation. SIAM J Math Anal. 2009;41(5):2010–2030.
   Web of Science ®Google Scholar
• Mizumachi T, Pelinovsky D. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete Contin Dyn Syst Ser S. 2012;5(5):971–987.
   Web of Science ®Google Scholar
• Gustafson S, Nakanishi K, Tsai T-P. Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves. Int Math Res Not. 2004;2004(66):3559–3584.
   Web of Science ®Google Scholar
• Mizumachi T. Asymptotic stability of small solitons for 2D nonlinear Schrödinger equations with potential. J Math Kyoto Univ. 2007;47(3):599–620.
   Google Scholar
• Mizumachi T. Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential. J Math Kyoto Univ. 2008;48(3):471–497.
   Google Scholar
• Pillet C-A, Wayne CE. Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J Differ Equ. 1997;141(2):310–326.
   Web of Science ®Google Scholar
• Soffer A, Weinstein MI. Multichannel nonlinear scattering for nonintegrable equations. Comm Math Phys. 1990;133(1):119–146.
   Web of Science ®Google Scholar
• Maeda M. Existence and asymptotic stability of quasi-periodic solution of discrete NLS with potential. SIAM J Math Anal. 2017;49(5):3396–3426.
   Web of Science ®Google Scholar
• Soffer A, Weinstein MI. Selection of the ground state for nonlinear Schrödinger equations. Rev Math Phys. 2004;16(8):977–1071.
   Web of Science ®Google Scholar
• Tsai T-P, Yau H-T. Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data. Adv Theor Math Phys. 2002;6(1):107–139.
   Google Scholar
• Cuccagna S, Maeda M. On small energy stabilization in the NLS with a trapping potential. Anal PDE. 2015;8(6):1289–1349.
   Web of Science ®Google Scholar
• Cuccagna S, Maeda M, Phan TV. On small energy stabilization in the NLKG with a trapping potential. Nonlinear Anal. 2016;146:32–58.
   Web of Science ®Google Scholar
• Comech A, Van Phan T, Stefanov A. Asymptotic stability of solitary waves in generalized Gross-Neveu model. Ann Inst H Poincaré Anal Non Linéaire. 2017;34(1):157–196.
   Web of Science ®Google Scholar
• Cuccagna S, Tarulli M. On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential. J Math Anal Appl. 2016;436(2):1332–1368.
   Web of Science ®Google Scholar
• Pelinovsky DE, Stefanov A. Asymptotic stability of small gap solitons in nonlinear dirac equations. J Math Phys. 2012;53(7):073705.
   Web of Science ®Google Scholar
• Mandelik D, Lahini Y, Silberberg Y. Nonlinearly induced relaxation to the ground state in a two-level system. Phys Rev Lett. 2005;95:073902.
   PubMed Web of Science ®Google Scholar
• Cuccagna S. Lp continuity of wave operators in Z. J Math Anal Appl. 2009;354(2):594–605.
   Web of Science ®Google Scholar
• Gang Z, Weinstein MI. Equipartition of mass in nonlinear Schrödinger/Gross-Pitaevskii equations. Appl Math Res Express. 2011;AMRX(2):123-–181.
   Google Scholar
• Rose HA, Weinstein MI. On the bound states of the nonlinear Schrödinger equation with a linear potential. Phys D. 1988;30(1–2):207–218.
   Google Scholar
• Fukuizumi R, Ohta M. Stability of standing waves for nonlinear Schrödinger equations with potentials. Differ Integral Equ. 2003;16(1):111–128.
   Google Scholar
• Cuccagna S. On instability of excited states of the nonlinear Schrödinger equation. Phys D. 2009;238(1):38–54.
   Web of Science ®Google Scholar
• Cuccagna S, Maeda M. On orbital instability of spectrally stable vortices of the NLS in the plane. J Nonlinear Sci. 2016;26(6):1851–1894.
   Web of Science ®Google Scholar
• Kevrekidis PG, Pelinovsky DE, Saxena A. When linear stability does not exclude nonlinear instability. Phys Rev Lett. 2015;114:214101.
   PubMed Web of Science ®Google Scholar
• Mizumachi T. Instability of vortex solitons for 2D focusing NLS. Adv Differ Equ. 2007;12(3):241–264.
   Google Scholar
• Maeda M, Masaki S. An example of stable excited state on nonlinear Schrödinger equation with nonlocal nonlinearity. Differ Integral Equ. 2013;26(7–8):731–756.
   Web of Science ®Google Scholar
• Bambusi D. Asymptotic stability of ground states in some Hamiltonian PDEs with symmetry. Comm Math Phys. 2013;320(2):499–542.
   Web of Science ®Google Scholar
• Bambusi D, Cuccagna S. On dispersion of small energy solutions to the nonlinear Klein Gordon equation with a potential. Amer J Math. 2011;133(5):1421–1468.
   Web of Science ®Google Scholar
• Cuccagna S. The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states. Comm Math Phys. 2011;305(2):279–331.
   Web of Science ®Google Scholar
• Cuccagna S. On the Darboux and Birkhoff steps in the asymptotic stability of solitons. Rend Istit Mat Univ Trieste. 2012;44:197–257.
   Google Scholar
• Cuccagna S. On asymptotic stability of moving ground states of the nonlinear Schrödinger equation. Trans Amer Math Soc. 2014;366(6):2827–2888.
   Web of Science ®Google Scholar
• Weinstein MI. Localized states and dynamics in the nonlinear schroedinger / Gross-Pitaevskii equations. In: Dynamics of partial differential equations. Cham: Springer; 2015. p. 41–79. (Frontiers in applied dynamics: reviews and tutorials; vol. 3).
   Google Scholar
• Egorova I, Kopylova E, Teschl G. Dispersion estimates for one-dimensional discrete Schrödinger and wave equations. J Spectr Theory. 2015;5(4):663–696.
   Web of Science ®Google Scholar