Statics and Dynamics of an Inhomogeneously-Nonlinear Lattice

We introduce an inhomogeneously-nonlinear Schr{\"o}dinger lattice, featuring a defocusing segment, a focusing segment and a transitional interface between the two. We illustrate that such inhomogeneous settings present vastly different dynamical behavior than the one expected in their homogeneous counterparts in the vicinity of the interface. We analyze the relevant stationary states, as well as their stability by means of perturbation theory and linear stability analysis. We find good agreement with the numerical findings in the vicinity of the anti-continuum limit. For larger values of the coupling, we follow the relevant branches numerically and show that they terminate at values of the coupling strength which are larger for more extended solutions. The dynamical development of relevant instabilities is also monitored in the case of unstable solutions.Comment: 14 pages, 4 figure

Asymptotic behavior of small solutions for the discrete nonlinear Schr\"odinger and Klein-Gordon equations

We show decay estimates for the propagator of the discrete Schr\"odinger and Klein-Gordon equations in the form \norm{U(t)f}{l^\infty}\leq C (1+|t|)^{-d/3}\norm{f}{l^1}. This implies a corresponding (restricted) set of Strichartz estimates. Applications of the latter include the existence of excitation thresholds for certain regimes of the parameters and the decay of small initial data for relevant $l^p$ norms. The analytical decay estimates are corroborated with numerical results.Comment: 13 pages, 4 figure

Nonlinear Lattices Generated from Harmonic Lattices with Geometric Constraints

Geometrical constraints imposed on higher dimensional harmonic lattices generally lead to nonlinear dynamical lattice models. Helical lattices obtained by such a procedure are shown to be described by sine- plus linear-lattice equations. The interplay between sinusoidal and quadratic potential terms in such models is shown to yield localized nonlinear modes identified as intrinsic resonant modes

Homoclinic standing waves in focussing DNLS equations --Variational approach via constrained optimization

We study focussing discrete nonlinear Schr\"{o}dinger equations and present a new variational existence proof for homoclinic standing waves (bright solitons). Our approach relies on the constrained maximization of an energy functional and provides the existence of two one-parameter families of waves with unimodal and even profile function for a wide class of nonlinearities. Finally, we illustrate our results by numerical simulations.Comment: new version with revised introduction and improved condition (A3); 16 pages, several figure

Perturbation-induced radiation by the Ablowitz-Ladik soliton

An efficient formalism is elaborated to analytically describe dynamics of the Ablowitz-Ladik soliton in the presence of perturbations. This formalism is based on using the Riemann-Hilbert problem and provides the means of calculating evolution of the discrete soliton parameters, as well as shape distortion and perturbation-induced radiation effects. As an example, soliton characteristics are calculated for linear damping and quintic perturbations.Comment: 13 pages, 4 figures, Phys. Rev. E (in press

Framework for a Protein Ontology

Biomedical ontologies are emerging as critical tools in genomic and proteomic research, where complex data in disparate resources need to be integrated. A number of ontologies describe properties that can be attributed to proteins. For example, protein functions are described by the Gene Ontology (GO) and human diseases by SNOMED CT or ICD10. There is, however, a gap in the current set of ontologies – one that describes the protein entities themselves and their relationships. We have designed the PRotein Ontology (PRO) to facilitate protein annotation and to guide new experiments. The components of PRO extend from the classification of proteins on the basis of evolutionary relationships to the representation of the multiple protein forms of a gene (products generated by genetic variation, alternative splicing, proteolytic cleavage, and other post-translational modifications). PRO will allow the specification of relationships between PRO, GO and other ontologies in the OBO Foundry. Here we describe the initial development of PRO, illustrated using human and mouse proteins involved in the transforming growth factor-beta and bone morphogenetic protein signaling pathways