1,720,993 research outputs found
Linear behavior in Covid19 epidemic as an effect of lockdown
Get PDFWe propose a mechanism explaining the approximately linear growth of Covid19 world total cases as well as the slow linear decrease of the daily new cases (and daily deaths) observed (in average) in USA and Italy. In our explanation, we regard a given population (the whole world or a single nation) as composed by many sub-clusters which, after lockdown, evolve essentially independently. The interaction is modeled by the fact that the outbreak time of the epidemic in a sub-cluster is a random variable with probability density slowly varying in time. The explanation is independent of the law according to which the epidemic evolves in the single sub cluster
Energy cascade in Fermi-Pasta-Ulam models
No full textWe show that, for long-wavelength initial conditions, the FPU dynamics is described, up to a certain time, by two KdV-like equations, which represent the resonant Hamiltonian normal form of the system. The energy cascade taking place in the system is then qualitatively characterized by arguments
of dimensional analysis based on such equations
The nonlinear Schrodinger equation as a resonant normal form
No full textAveraging theory is used to study the dynamics of dispersive equations taking the nonlinear Klein Gordon equation on the line as a model problem: For approximatively monochromatic initial data of amplitude E, we show that the corresponding solution consists of two non interacting wave packets, each one being described by a nonlinear Schrodinger equation. Such solutions are also proved to be stable over times of order 1/epsilon(2). We think that this approach puts into a new light the problem of obtaining modulations equations for general dispersive equations. The proof of our results requires a new use of normal forms as a tool for constructing approximate solutions
Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam
No full textWe address the problem of equipartition in a long Fermi-Pasta-Ulam (FPU) chain. After giving a precise relation between FPU and Korteweg-de Vries we use the latter equation to show that, corresponding to initial data a la Fermi, the time average of the energy on the kth mode decreases exponentially with k/N. The result persists in the thermodynamic limit
A Birkhoof-Lewis type theorem for some Hamiltonian PDE's
No full textIn this paper we give an extension of the Birkhoff–Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schroedinger equation with
smoothing nonlinearity
Freezing of Energy of a Soliton in an External Potential
No full textIn this paper we study the dynamics of a soliton in the generalized NLS with a small external potential ϵV of Schwartz class. We prove that there exists an effective mechanical system describing the dynamics of the soliton and that, for any positive integer r, the energy of such a mechanical system is almost conserved up to times of order ϵ−r. In the rotational invariant case we deduce that the true orbit of the soliton remains close to the mechanical one up to times of order ϵ−r
On metastability in FPU
No full textWe present an analytical study of the Fermi-Pasta-Ulam (FPU) alpha-model with periodic boundary conditions. We analyze the dynamics corresponding to initial data with one low frequency Fourier mode excited. We show that, correspondingly, a pair of KdV equations constitute the resonant normal form of the system. We also use such a normal form in order to prove the existence of a metastability phenomenon. More precisely, we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes. Subsequently, the distribution remains unchanged up to the time scales of validity of our approximation. The phenomenon is controlled by the specific energy
On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential
No full textIn this paper we study small amplitude solutions of nonlinear Klein Gordon equations with a potential. Under smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity, we prove that all small amplitude initial data with finite energy give rise to solutions asymptotically free. In the case where the linear system has at most one bound state the result was already proved by Soffer and Weinstein: we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states. The proof is based on a combination of Birkhoff normal form techniques and dispersive estimates
Birkhoff coordinates for the Toda lattice in the limit of infinitely many particles with an application to FPU
Get PDFIn this paper we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number N of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius R/Nα (in discrete Sobolev-analytic norms) into a ball of radius R′/Nα (with R,R′>0 independent of N) if and only if α≥2. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size R/N2, 0<R≪1, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself
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