zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. (English) Zbl 1200.39001
The authors consider the nonlinear discrete periodic system $$a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n-\omega u_n=\sigma f_n(u_n),\quad n\in\mathbb{Z},$$ where $f_n(u)$ is continuous in $u$ and with saturable nonlinearity for each $n\in\mathbb{Z}$, $f_{n+T}(u)=f_n(u)$, $\{a_n\},\{b_n\}$ are real valued $T$-periodic sequences. They are interested in the existence of nontrivial homoclinic solutions for this equation; this problem appears when one looks for the discrete solitons of the periodic discrete nonlinear Schrödinger equations. A new sufficient condition guaranteeing the existence of homoclinic solutions is obtained by using critical point theory. It is proved that it is also necessary in some special cases. Moreover, the rate of decay is established.

39A12Discrete version of topics in analysis
39A70Difference operators
39A23Periodic solutions (difference equations)
37C29Homoclinic and heteroclinic orbits
Full Text: DOI
[1] Arioli, G.; Gazzola, F.: Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear anal. 26, 1103-1114 (1996) · Zbl 0867.70004 · doi:10.1016/0362-546X(94)00269-N
[2] Aubry, S.: Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D 103, 201-250 (1997) · 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, Physica D 216, 1-30 (2006) · 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 1, 271-298 (2001) · Zbl 1092.37523 · doi:10.3934/dcdsb.2001.1.271
[5] Christodoulides, D. N.; Lederer, F.; Silberberg, Y.: Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature 424, 817-823 (2003)
[6] Cuevas, J.; Kevrekidis, P. G.; Frantzeskakis, D. J.; Malomed, B. A.: Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D 238, 67-76 (2009) · Zbl 1153.82321 · doi:10.1016/j.physd.2008.08.013
[7] Efremidis, N. K.; Sears, S.; Christodoulides, D. N.; Fleischer, J. W.; Segev, M.: Discrete solitons in photorefractive optically induced photonic lattices, Phys. rev. E 66, 046602 (2002)
[8] Flach, S.; Gorbach, A. V.: Discrete breathers --- advance in theory and applications, Phys. rep. 467, 1-116 (2008) · Zbl 1218.37107
[9] Flach, S.; Willis, C. R.: Discrete breathers, Phys. rep. 295, 181-264 (1998)
[10] Fleischer, J. W.; Carmon, T.; Segev, M.; Efremidis, N. K.; Christodoulides, D. N.: Observation of discrete solitons in optically induced real time waveguide arrays, Phys. rev. Lett. 90, 023902 (2003)
[11] Fleischer, J. W.; Segev, M.; Efremidis, N. K.; Christodoulides, D. N.: Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature 422, 147-150 (2003)
[12] Gatz, S.; Herrmann, J.: Soliton propagation in materials with saturable nonlinearity, J. opt. Soc. amer. B 8, 2296-2302 (1991)
[13] Gatz, S.; Herrmann, J.: Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change, Opt. lett. 17, 484-486 (1992)
[14] Gorbach, A. V.; Johansson, M.: Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. phys. J. D 29, 77-93 (2004)
[15] James, G.: Centre manifold reduction for quasilinear discrete systems, J. nonlinear sci. 13, 27-63 (2003) · Zbl 1185.37158 · doi:10.1007/s00332-002-0525-x
[16] Livi, R.; Franzosi, R.; Oppo, G. -L.: Self-localization of Bose -- Einstein condensates in optical lattices via boundary dissipation, Phys. rev. Lett. 97, 060401 (2006)
[17] Mackay, R. S.; Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7, 1623-1643 (1994) · Zbl 0811.70017 · doi:10.1088/0951-7715/7/6/006
[18] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems, (1989) · Zbl 0676.58017
[19] Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity 19, 27-40 (2006) · Zbl 1220.35163 · doi:10.1088/0951-7715/19/1/002
[20] Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach, Discrete contin. Dyn. syst. 19, 419-430 (2007) · Zbl 1220.35164 · doi:10.3934/dcds.2007.19.419
[21] Pankov, A.; Rothos, V.: Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A 464, 3219-3236 (2008) · Zbl 1186.35206 · doi:10.1098/rspa.2008.0255
[22] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS reg. Conf. ser. Math. 65 (1986) · Zbl 0609.58002
[23] Shi, H.; Zhang, H.: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations, J. math. Anal. appl. 361, 411-419 (2010) · Zbl 1178.35351 · doi:10.1016/j.jmaa.2009.07.026
[24] Sukhorukov, A. A.; Kivshar, Y. S.: Generation and stability of discrete gap solitons, Opt. lett. 28, 2345-2347 (2003)
[25] Teschl, G.: Jacobi operators and completely integrable nonlinear lattices, Math. surveys monogr. 72 (2000) · Zbl 1056.39029
[26] Yan, Z.: Envelope solution profiles of the discrete nonlinear Schrödinger equation with a saturable nonlinearity, Appl. math. Lett. 22, 448-452 (2009) · Zbl 1170.35540 · doi:10.1016/j.aml.2008.06.015
[27] Yu, J.; Guo, Z.: On boundary value problems for a discrete generalized Emden -- Fowler equation, J. differential equations 231, 18-31 (2006) · Zbl 1112.39011 · doi:10.1016/j.jde.2006.08.011
[28] Zhou, Z.; Yu, J.; Guo, Z.: Periodic solutions of higher-dimensional discrete systems, Proc. roy. Soc. Edinburgh sect. A 134, 1013-1022 (2004) · Zbl 1073.39010 · doi:10.1017/S0308210500003607