Localized solutions of a nonlinear diatomic lattice
Authors:
V. V. Konotop and G. Perla Menzala
Journal:
Quart. Appl. Math. 63 (2005), 201-223
MSC (2000):
Primary 39A10, 39B72
DOI:
https://doi.org/10.1090/S0033-569X-05-00952-6
Published electronically:
February 18, 2005
MathSciNet review:
2150770
Full-text PDF Free Access
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Abstract: We consider a coupled system of differential-difference nonlinear equations. We study the dynamics of such a diatomic lattice showing global existence and uniqueness in an appropriate function space. Our approach based on energy estimates allows us to prove the result only in the case where nonlinear force constants are positive and equal. All other situations remain at this point as open problems.
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[1]1 Abdullaev, F. Kh., Darmanyan, S.A. and Garnier J., Modulational instability of electromagnetic waves in inhomogeneous and in discrete media. In Progress in Optics, Edit. E. Wolf, 44, 303-366 (2002).
[2]2 Bona, J.L. and Saut, J.L., “Dispersive blowup of solutions of generalized Korteweg-de Vries equations”. J. Diff. Equations 103, 1993, 3-57.
[3]3 Konotop, V.V., “Small-amplitude envelope solitons in nonlinear lattices”. Phys. Rev. E 53, 2843 (1996).
[4]4 Perla Menzala, G. and Konotop, V.V., “On global existence of localized solutions of some nonlinear lattices”, Applicable Analysis, 75, 157-173 (2000).
[5]5 Perla Menzala, G. and Konotop, V.V., “Uniform decay rates of solutions of some nonlinear lattices”, Nonlinear Analysis, 54, 2003, 261-278.
[6]6 Scott, A.C., Nonlinear Science, Oxford University Press (1999).
[7]7 Sulem, C. and Sulem, P., Nonlinear Schrödinger equation: Self-focusing and wave collapse, Springer-Verlag, Berlin (1999).
[8]8 Ziman, J.M., Principles of the Theory of Solids (Cambridge University Press, 1964).
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Additional Information
V. V. Konotop
Affiliation:
Centro de Física Teórica e Computacional, Universidade de Lisboa, Complexo Interdisciplinar, Av. Professor Gama Pinto 2, 1649-003 Lisbon, Portugal
Email:
konotop@cii.fc.ul.pt
G. Perla Menzala
Affiliation:
National Laboratory of Scientific Computation, LNCC/MCT, Rua Getulio Vargas 333, Petropolis, RJ, CEP 25651-070, Brasil and Federal University of Rio de Janeiro, Institute of Mathematics, P.O. Box 68530, 21945-970, Rio de Janeiro, RJ, Brasil
Email:
perla@lncc.br
Keywords:
Nonlinear lattices,
high interaction,
diatomic case
Received by editor(s):
March 20, 2003
Published electronically:
February 18, 2005
Article copyright:
© Copyright 2005
Brown University
