1,720,966 research outputs found
Sobolev instability in the cubic NLS equation with convolution potentials on irrational tori
No full textIn this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to the case of square (and rational) tori. We weaken the regularity assumptions on the convolution potentials, required in a previous work by Guardia (2014) [11] for the square case, to obtain the Hs-instability (s>1) of the elliptic equilibrium u=0. We also provide the existence of solutions u(t) with arbitrarily small L2 norm which achieve a prescribed growth within a time T satisfying polynomial estimates
KAM for quasi-linear PDE's
Full text linkIn this Thesis we present two new results of existence and stability of Cantor families of small amplitude quasi-periodic in time solutions for quasi-linear Hamiltonian PDE's arising as models for shallow water phenomena.\\
The considered problems present serious small divisors difficulties and the results are achieved by implementing Nash-Moser algorithms and by exploiting pseudo differential calculus techniques.
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The first result concerns a generalized quasi-linear KdV equation
u_t+u_{xxx}+\mathcal{N}_2(x, u, u_x, u_{xx}, u_{xxx})=0, \quad x\in \T,
where is a nonlinearity originating from a cubic Hamiltonian.\\
The nonlinear part depends upon some parameters and it is intriguing to study how the choice of these parameters affects the bifurcation analysis.\\
The linearized equation at the origin is resonant, namely the linear solutions are all periodic, hence the existence of the expected quasi-periodic solutions is due only to the presence of the nonlinearities.\\
The nonlinear terms of these equations are quadratic and contains derivatives of the same order of the linear part, thus they produce strong perturbative effect near the origin.
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The second result is the first KAM result for quasi-linear PDE's with asymptotically linear dispersion law and it implies the first existence result for quasi-periodic solutions of the Degasperis-Procesi equation.\\
We consider Hamiltonian perturbations of the Degasperis-Procesi equation
u_t-u_{x x t}+u_{xxx}-4 u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x+\mathcal{N}_6( u, u_x, u_{xx}, u_{xxx})=0, \quad x\in \T,
where is a nonlinearity originating from a Hamiltonian density with a zero of order seven at the origin.\\
We exploit the integrable structure of the unperturbed equation to overcome some small divisors problems.\\
The complicated symplectic structure and the asymptotically linear dispersion law make harder the analysis of the linearized operator in a neighborhood of the origin, which is required by the Nash-Moser scheme, and the measure estimates for the frequencies of the expected quasi-periodic solutions
Reducible KAM Tori for the Degasperis–Procesi Equation
Full text linkWe develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa–Holm equation
Reducibility for a class of weakly dispersive linear operators arising from the Degasperis–Procesi equation
Full text linkWe prove reducibility of a class of quasi-periodically forced linear equations of the form
∂tu−∂x∘(1+a(ωt,x))u+(ωt)u=0x∈:=R/2πZ,
where u=u(t,x),a is a small, C∞ function, is a pseudo-differential operator of order −1, provided that ω∈Rν satisfies appropriate non-resonance conditions. Such PDEs arise by linearizing the Degasperis–Procesi (DP) equation at a small amplitude quasi-periodic function. Our work provides a first fundamental step in developing a KAM theory for perturbations of the DP equation on the circle. Following [3], our approach is based on two main points: first a reduction in orders based on an Egorov type theorem then a KAM diagonalization scheme. In both steps the key difficulties arise from the asymptotically linear dispersion law. In view of the application to the nonlinear context we prove sharp tame bounds on the diagonalizing change of variables. We remark that the strategy and the techniques proposed are applicable for proving reducibility of more general classes of linear pseudo differential first order operators
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Effective chaos for the Kirchhoff equation on tori
No full textWe consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillate in a chaotic way on certain long timescales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, which we name effective chaos (it occurs over a long, but finite, timescale), is a consequence of the existence of symbolic dynamics for an effective system. Since the first-order resonant dynamics has been proved to be essentially stable, we need to perform a second-order analysis to find an effective model displaying chaotic dynamics. More precisely, after some nontrivial reductions, this model behaves as two weakly coupled pendulums
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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