Pankov, Alexander; Rothos, Vassilis Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity. (English) Zbl 1186.35206 Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 464, No. 2100, 3219-3236 (2008). Summary: We demonstrate the existence of solutions in the discrete nonlinear Schrödinger equation (DNLS) with saturable nonlinearity. We consider two types of solutions to DNLS periodic and vanishing at infinity. Calculus of variations and the Nehari manifolds are employed to establish the existence of these solutions. We present some extensions of our results, combining the Nehari manifold approach and the Mountain Pass argument. Cited in 49 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B10 Periodic solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs Keywords:periodic and decaying solutions; discrete nonlinear Schrödinger; the Nehari manifolds PDFBibTeX XMLCite \textit{A. Pankov} and \textit{V. Rothos}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 464, No. 2100, 3219--3236 (2008; Zbl 1186.35206) Full Text: DOI References: [1] Christodoulides, Nature; Physical Science (London) 424 (6950) pp 817– (2003) [2] Canadian Journal of Physics = Journal Canadien de Physique 64 pp 311– (1986) [3] Physical Review Letters 81 pp 3383– (1998) [4] PHYS REV E 56 pp 7267– (1997) [5] J PHYS A 38 pp 807– (2005) · Zbl 1069.81016 [6] Krolikowski, Optics letters 22 (6) pp 369– (1997) [7] IEEE J QUANT ELECTRON 39 pp 3– (2003) [8] TRANS AM MATH SOC 95 pp 101– (1960) [9] MILAN J MATH 73 pp 259– (2005) · Zbl 1225.35222 [10] 19 pp 27– (2006) · Zbl 1220.35163 [11] Snyder, Optics letters 18 (7) pp 482– (1993) [12] ARCH RAT MECH ANAL 125 pp 145– (1993) · Zbl 0801.35136 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.