Discrete structures equivalent to nonlinear Dirichlet and wave equations

Lovison, Alberto; Cardin, Franco; Bobbo, Alessia
June 2009
Continuum Mechanics & Thermodynamics;Jun2009, Vol. 21 Issue 1, p27
Academic Journal
The Amann–Conley–Zehnder (ACZ) reduction is a global Lyapunov–Schmidt reduction for PDEs based on spectral decomposition. ACZ has been applied in conjunction to diverse topological methods, to derive existence and multiplicity results for Hamiltonian systems, for elliptic boundary value problems, and for nonlinear wave equations. Recently, the ACZ reduction has been translated numerically for semilinear Dirichlet problems and for modeling molecular dynamics, showing competitive performances with standard techniques. In this paper, we apply ACZ to a class of nonlinear wave equations in $${\mathbb{T}^n}$$, attaining to the definition of a finite lattice of harmonic oscillators weakly nonlinearly coupled exactly equivalent to the continuum model. This result can be thought as a thermodynamic limit arrested at a small but finite scale without residuals. Reduced dimensional models reveal the macroscopic scaled features of the continuum, which could be interpreted as collective variables.


Related Articles

  • Classification of the simplest diffeomorphisms of the sphere S2 with one stability modulus. Mitryakova, T. N.; Pochinka, O. V. // Journal of Mathematical Sciences;Apr2009, Vol. 158 Issue 2, p261 

    In this work, the authors obtain a topological classification of the simplest diffeomorphisms of the sphere S2 having exactly one orbit of one-sided heteroclinic tangency.

  • On Weierstrass' Monsters and lineability. Jiménez-Rodríguez, P.; Muñoz-Fernández, G. A.; Seoane-Sepúlveda, J. B. // Bulletin of the Belgian Mathematical Society - Simon Stevin;2013, Vol. 20 Issue 4, p577 

    Let E be a topological vector space and let us consider a property P. We say that the subset M of E formed by the vectors in E which satisfy P is µ-lineable (for certain cardinal µ, finite or infinite) if M∪ {0} contains an infinite dimensional linear space of dimension µ. In 1966...

  • Spin-polarization spectra in a gapped graphene superlattice. Korol', A.; Isai, V.; Medvid', N. // Physics of the Solid State;Feb2015, Vol. 57 Issue 2, p419 

    A one-dimensional superlattice constructed based on a gapped single-layer graphene has been considered. The spin-dependent transport characteristics of this structure have been calculated in the continuum model with the use of the transfer-matrix method. It has been shown that the spin...

  • The influence of surface on regionally aggregated evaporation and energy partitioning. Bunzli, Daniel; Schmid, Hans Peter // Journal of the Atmospheric Sciences;3/15/98, Vol. 55 Issue 6, p961 

    Presents a detail examination on average surface fluxes of sensible heat and latent heat which was evaluated over periodically varying terrain, using a two-dimensional E-... model with high spatial resolution and a parameterization of the local surface energy balance according to Penman and...

  • On One-Sided, D-Chaotic Cellular Automaton, Having Continuum of Fixed Points and Topological Entropy log(3). FORYŚ, WIT; MATYJA, JANUSZ // Journal of Cellular Automata;2013, Vol. 8 Issue 3/4, p131 

    In this paper, generalizing our previous result, we present a one-sided cellular automaton F with radius r = 1 defined over six-element alphabet. We prove that the automaton F is chaotic in the sense of Devaney, non-injective, has continuum of fixed points and topological entropy equal to log(3).

  • Transverse instability of nonlinear longitudinal waves in hexagonal lattices. Porubov, Alexey; Andrianov, Igor; Markert, Berndt // Proceedings of the Estonian Academy of Sciences;2015 Supplement, Vol. 64, p349 

    Various continuum limits of the original discrete hexagonal lattice model are used to obtain transverse weakly nonlinear equations for longitudinal waves. It is shown, that the long wavelength continuum limit gives rise to the Kadomtsev-Petviashvili equation, while another continuum limit...

  • On the relation between topological entropy and entropy dimension. Saltykov, P. S. // Mathematical Notes;Jun2009, Vol. 86 Issue 1/2, p255 

    For the Lipschitz mapping of a metric compact set into itself, there is a classical upper bound on topological entropy, namely, the product of the entropy dimension of the compact set by the logarithm of the Lipschitz constant. The Ghys conjecture is that, by varying the metric, one can...

  • Generating ensembles and measuring mixing in a model granular system. Puckett, James G.; Lechenault, Frédéric; Daniels, Karen E. // AIP Conference Proceedings;7/1/2009, Vol. 1145 Issue 1, p675 

    A major open question in the field of granular materials is the identification of relevant state variables which can predict macroscopic behavior. We experimentally investigate the mixing properties of an idealized granular liquid in the vicinity of its jamming transition, through the generation...

  • Thermodynamics of G·A mispairs in DNA: Continuum electrostatic model. Berashevich, Julia; Chakraborty, Tapash // Journal of Chemical Physics;1/7/2009, Vol. 130 Issue 1, p015101 

    An analysis of the stability of a duplex containing G·A mispairs or G·A/A·G tandem during the DNA melting has shown that the duplex stability depends on both DNA sequences and the conformations of the G·A mispairs. The thermodynamics of single pair opening for the G(anti)·A(syn)...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics