zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Homoclinic solutions in periodic difference equations with saturable nonlinearity. (English) Zbl 1239.39010

This paper is mainly concerned with the homoclinic solutions of the following difference equation

Lu n -ωu n =σχ n u n 3 1+c n u n 2 ,

where

Lu n =a n u n+1 +a n-1 u n-1 +b n u n ,

and {a n }, {b n }, {c n } and {χ n } are positive real valued T-periodic sequences. By using the linking theorem in combination with periodic approximations, the authors establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions for the above difference equation. Their results solve an open problem proposed by A. Pankov [Nonlinearity 19, No. 1, 27–40 (2006; Zbl 1220.35163)]. This paper is interesting and a good contribution in this area.

MSC:
39A23Periodic solutions (difference equations)
39A20Generalized difference equations
References:
[1]Arioli G, Gazzola F. Periodic motions of an infinite lattice of particles with nearest neighbor interaction. Nonlinear Anal, 1996, 26: 1103–1114 · Zbl 0867.70004 · doi:10.1016/0362-546X(94)00269-N
[2]Aubry S. Breathers in nonlinear lattices: existence, linear stability and quantization. Phys D, 1997, 103: 201–250 · Zbl 1194.34059 · doi:10.1016/S0167-2789(96)00261-8
[3]Aubry S. Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems. Phys D, 2006, 216: 1–30 · Zbl 1159.82312 · doi:10.1016/j.physd.2005.12.020
[4]Aubry S, Kopidakis G, Kadelburg V. Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems. Discrete Contin Dyn Syst Ser B, 2001, 1: 271–298 · Zbl 1092.37523 · doi:10.3934/dcdsb.2001.1.271
[5]Bruno G, Pankov A, Tverdokhleb Yu. On almost-periodic operators in the spaces of sequences. Acta Appl Math, 2001, 65: 153–167 · Zbl 0993.39013 · doi:10.1023/A:1010695824612
[6]Chang K C. Critical Point Theory and Its Applications (in Chinese). Shanghai: Shanghai Kexue Jishu Chubanshe, 1986
[7]Christodoulides D N, Lederer F, Silberberg Y. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature, 2003, 424: 817–823 · doi:10.1038/nature01936
[8]Cuevas J, Kevrekidis P G, Frantzeskakis D J, et al. Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity. Phys D, 2009, 238: 67–76 · Zbl 1153.82321 · doi:10.1016/j.physd.2008.08.013
[9]Flach S, Gorbach A V. Discrete breathers advance in theory and applications. Phys Rep, 2008, 467: 1–116 · doi:10.1016/j.physrep.2008.05.002
[10]Flach S, Willis C R. Discrete breathers. Phys Rep, 1998, 295: 181–264 · doi:10.1016/S0370-1573(97)00068-9
[11]Fleischer J W, Carmon T, Segev M, et al. Observation of discrete solitons in optically induced real time waveguide arrays. Phys Rev Lett, 2003, 90: 023902 · doi:10.1103/PhysRevLett.90.023902
[12]Fleischer J W, Segev M, Efremidis N K, et al. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature, 2003, 422: 147–150 · doi:10.1038/nature01452
[13]Gatz S, Herrmann J. Soliton propagation in materials with saturable nonlinearity. J Opt Soc Amer B, 1991, 8: 2296–2302 · doi:10.1364/JOSAB.8.002296
[14]Gatz S, Herrmann J. Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change. Opt Lett, 1992, 17: 484–486 · doi:10.1364/OL.17.000484
[15]Gorbach A V, Johansson M. 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
[16]James G. Centre manifold reduction for quasilinear discrete systems. J Nonlinear Sci, 2003, 13: 27–63 · Zbl 1185.37158 · doi:10.1007/s00332-002-0525-x
[17]Kopidakis G, Aubry S, Tsironis G P. Targeted energy transfer through discrete breathers in nonlinear systems. Phys Rev Lett, 2001, 87: 165501 · doi:10.1103/PhysRevLett.87.165501
[18]Livi R, Franzosi R, Oppo G L. Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation. Phys Rev Lett, 2006, 97: 060401 · doi:10.1103/PhysRevLett.97.060401
[19]MacKay R S, Aubry S. Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity, 1994, 7: 1623–1643 · Zbl 0811.70017 · doi:10.1088/0951-7715/7/6/006
[20]Pankov A. Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity, 2006, 19: 27–40 · Zbl 1220.35163 · doi:10.1088/0951-7715/19/1/002
[21]Pankov A. Gap solitons in periodic discrete nonlinear Schrödinger equations. II. A generalized Nehari manifold approach. Discrete Contin Dyn Syst, 2007, 19: 419–430 · Zbl 1220.35164 · doi:10.3934/dcds.2007.19.419
[22]Pankov A, Rothos V. Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity. Proc R Soc A, 2008, 464: 3219–3236 · Zbl 1186.35206 · doi:10.1098/rspa.2008.0255
[23]Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Regional Conference Series in Mathematics, No. 65. Providence, RI: American Mathematical Society, 1986
[24]Shi H, Zhang H. Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J Math Anal Appl, 2010, 361: 411–419 · Zbl 1178.35351 · doi:10.1016/j.jmaa.2009.07.026
[25]Sukhorukov A A, Kivshar Y S. Generation and stability of discrete gap solitons. Opt Lett, 2003, 28: 2345–2347 · doi:10.1364/OL.28.002345
[26]Teschl G. Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, No. 72. Providence, RI: American Mathematical Society, 2000.
[27]Yan Z. Envelope solution profiles of the discrete nonlinear Schrödinger equation with a saturable nonlinearity. Appl Math Lett, 2009, 22: 448–452 · Zbl 1170.35540 · doi:10.1016/j.aml.2008.06.015
[28]Zhou Z, Yu J. On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. J Differential Equations, 2010, 249: 1199–1212 · Zbl 1200.39001 · doi:10.1016/j.jde.2010.03.010