1,720,973 research outputs found
Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations
Full text linkIn this paper we consider a class of fully nonlinear forced and reversible Schrödinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such as diffeomorphisms of the torus and pseudo-differential operators. This procedure automatically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability
Finite dimensional invariant KAM tori for tame vector fields
Full text linkWe discuss a Nash-Moser/KAM algorithm for the construction
of invariant tori for tame vector fields. Similar algorithms have been studied
widely both in finite and infinite dimensional contexts: we are particularly
interested in the second case where tameness properties of the vector fields
become very important. We focus on the formal aspects of the algorithm and
particularly on the minimal hypotheses needed for convergence. We discuss
various applications where we show how our algorithm allows one to reduce to
solving only linear forced equations. We remark that our algorithm works at
the same time in analytic and Sobolev classes
Birkhoff Normal form for Gravity Water Waves
Full text linkWe consider the gravity water waves system with a one-dimensional periodic interface in infinite depth, and present the proof of the rigorous reduction of these equations to their cubic Birkhoff normal form (Berti et al. in Birkhoff normal form and long-time existence for periodic gravity Water Waves. arXiv:1810.11549, 2018). This confirms a conjecture of Zakharov–Dyachenko (Phys Lett A 190:144–148, 1994) based on the formal Birkhoff integrability of the water waves Hamiltonian truncated at degree four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order ε−3
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
Birkhoff normal form and long time existence for periodic gravity water waves
Full text linkWe consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This proves a conjecture of Zakharov-Dyachenko [55] suggested by the formal Birkhoff integrability of the water waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order ε−3. This time scale is expected to be optimal. Main difficulties in the proof are the presence of non-trivial resonant four-waves interactions, the so-called Benjamin-Feir resonances, the small divisors arising from near-resonances and the quasilinear nature of the equations. Some of the main ingredients that we use are: (1) a reduction procedure to constant coefficient operators up to smoothing remainders, that, together with the verification of key algebraic cancellations of the system, implies the integrability of the equations at non-negative orders; (2) a Poincaré-Birkhoff normal form of the smoothing remainders that deals with near-resonances; (3) an a priori algebraic identification argument of the above Poincaré- Birkhoff normal form equations with the formal Hamiltonian computations of [55, 19, 27, 17], that allows us to handle the Benjamin-Feir resonances
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
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
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