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)
Bifurcations of travelling wave solutions in the discrete NLS equations. (English) Zbl 1086.34059
The paper is concerned with changes in behaviour of solutions to nonlinear discrete Schrödinger equations. The paper is motivated particularly by the need to clarify some contradictions in the existing literature which are identified by the authors in the introduction. The initial analysis is based on a reduction of the discrete NLS equation to a linear mixed advance-delay differential equation. One can derive an equation for the resonances of the travelling wave solutions and the authors explain why one should investigate repeated roots of this equation. The later analysis is based on a discrete dynamical systems approach and the derivation of a centre manifold and normal form reduction. In conclusion, the authors have shown that they have cleared up some of the earlier contradictory results, and have identified two open problems.

34K18Bifurcation theory of functional differential equations
34K06Linear functional-differential equations
37D10Invariant manifold theory
Full Text: DOI
[1] Ablowitz, M. J.; Ladik, J. F.: Nonlinear differential -- difference equations and Fourier analysis. J. math. Phys. 17, 1011-1018 (1976) · Zbl 0322.42014
[2] Ablowitz, M. J.; Musslimani, Z. H.; Biondini, G.: Methods for discrete solitons in nonlinear lattices. Phys. rev. E 65, 026602 (2002)
[3] Ablowitz, M. J.; Musslimani, Z. H.: Discrete spatial solitons in a diffraction-managed nonlinear waveguide array: a unified approach. Physica D 184, 276-303 (2003) · Zbl 1030.78007
[4] Aigner, A. A.; Champneys, A. R.; Rothos, V. M.: A new barrier to the existence of moving kinks in Frenkel -- Kontorova lattices. Physica D 186, No. 3 -- 4, 148-170 (2003) · Zbl 1037.74026
[5] Carr, J.: Application of centre manifold theory. (1981) · Zbl 0464.58001
[6] Calvo, D. C.; Akylas, T. R.: On the formation of bound states by interacting nonlocal solitary waves. Physica D 101, 270-288 (1997) · Zbl 0899.76090
[7] Champneys, A. R.; Harterich, J.: Cascades of homoclinic orbits to a saddle-focus for reversible and perturbed Hamiltonian systems. J. dyn. Stabil. syst. 15, 231-252 (2000) · Zbl 1003.37033
[8] Champneys, A.; Kivshar, Yu.S.: Origin of multi-kinks in nonlinear dispersive systems. Phys. rev. E 61, 2551-2554 (2000)
[9] Claude, C.; Kivshar, Y. S.; Kluth, O.; Spatschek, K. H.: Moving localized modes in nonlinear lattices. Phys. rev. B 47, 14228-14232 (1993) · Zbl 0900.35382
[10] Duncan, D. B.; Eilbeck, J. C.; Feddersen, H.; Wattis, J. A. D.: Solitons on lattices. Physica D 68, 1-11 (1993) · Zbl 0785.35087
[11] Eilbeck, J. C.; Johansson, M.: The discrete nonlinear Schrödinger equation --- 20 years on. Proceedings of the third conference localization and energy transfer in nonlinear systems, 44-67 (2003) · Zbl 1032.81007
[12] Elbert, A. E.: Solutions of the extended nonlinear Schrödinger equation that oscillate at infinity. Proceedings of the Steklov institute of mathematics, suppl. 1, S54-S67 (2003)
[13] Feddersen, H.: Solitary wave solutions to the discrete nonlinear Schrödinger equation. Nonlinear coherent structures in physics and biology (Dijon, 1991), lecture notes in physics, vol. 393 393, 159-167 (1991)
[14] Flach, S.; Kladko, K.: Moving discrete breathers?. Physica D 127, 61-72 (1999) · Zbl 0947.70016
[15] Flach, S.; Zolotaryuk, Y.; Kladko, K.: Moving lattice kinds and pulses: an inverse method. Phys. rev. E 59, 6105-6115 (1999)
[16] Grimshaw, R.: Weakly nonlocal solitary waves in a singularly perturbed nonlinear Schrödinger equation. Stud. appl. Math. 94, 257-270 (1995) · Zbl 0826.35117
[17] Iooss, G.: Travelling waves in the Fermi -- pasta -- Ulam lattice. Nonlinearity 13, 849-866 (2000) · Zbl 0960.37038
[18] Iooss, G.; Adelmeyer, M.: Topics in bifurcation theory and applications. (1998) · Zbl 0968.34027
[19] Iooss, G.; Kirchgassner, K.: Travelling waves in a chain of coupled nonlinear oscillators. Comm. math. Phys. 211, 439-464 (2000) · Zbl 0956.37055
[20] Hirota, R.: Exact envelope -- soliton solutions of a nonlinear wave equation. J. math. Phys. 14, 805-809 (1973) · Zbl 0257.35052
[21] Hennig, D.; Tsironis, G.: Wave transmission in nonlinear lattices. Phys. rep. 307, 333-432 (1999)
[22] Kevrekidis, P. G.: On a class of discretizations of Hamiltonian nonlinear partial differential equations. Physica D 183, 68-86 (2003) · Zbl 1031.70001
[23] Kevrekidis, P. G.; Rasmussen, K. O.; Bishop, A. R.: The discrete nonlinear Schrödinger equation: a survey of recent results. Int. J. Mod. phys. B 15, 2833-2900 (2001)
[24] Klauder, M.; Laedke, E. W.; Spatschek, K. H.; Turitsyn, S. K.: Pulse propagation in optical fibers near the zero dispersion point. Phys. rev. E 47, R3844-R3847 (1993)
[25] Kolossovski, K.; Champneys, A. R.; Buryak, A. V.; Sammut, R. A.: Multi-pulse embedded solitons as bound states of quasi-solitons. Physica D 171, 153-177 (2002) · Zbl 1010.35087
[26] 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
[27] Mallet-Paret, J.: The global structure of traveling waves in spatially discrete dynamical systems. J. dyn. Diff. eqs. 11, 49-127 (1999) · Zbl 0921.34046
[28] Mielke, A.; Holmes, P.; O’reilly, O.: Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center. J. dyn. Diff. eqs. 4, 95-126 (1992) · Zbl 0749.58022
[29] Yang, J.: Stable embedded solitons. Phys. rev. Lett. 91, 143903 (2003)
[30] Yang, J.; Akylas, T. R.: Continuous families of embedded solitons in the third-order nonlinear Schrödinger equation. Stud. appl. Math. 111, 359-375 (2003) · Zbl 1141.35460