arXiv subjects

Andrea Pezzi

Publications and source records attributed to Andrea Pezzi.

3 recordsLinked to original sources

arXiv2026

Recursive behavior in a diatomic FPUT lattice

We study the diatomic FPUT lattice with cubic anharmonic potential, and analyze the recurrent behaviour of its solutions. We find that two distinct types of recurrence occur. One type is the classic FPUT recurrence; for such recurrence, we find that the relation between recurrence period and nonlinear strength is similar to that in the monatomic case. The other type, which cannot exist in the monatomic lattice, is the recurrence due to the interactions between modes in the two branches of the dispersion relation. Indeed, we prove the existence of the optical-acoustical-acoustical resonant interaction between three Fourier modes for which a recurrent behavior in the distribution of the energy is observed. In addition, we develop a reduced Fourier-space dynamical model that reproduces the same recurrent behavior. We assess the robustness of our results through numerical simulations of the diatomic Toda lattice and the diatomic granular chain; in both cases, the same recursive behavior is observed. Finally, in the continuous limit, we derive from the diatomic model a system of three coupled PDEs which are known to be integrable.

nlin.CD↗

arXiv2023

Three- and four-wave resonances in the nonlinear quadratic Kelvin lattice

In this paper we investigate analytically and numerically the nonlinear Kelvin lattice, namely a chain of masses and nonlinear springs, as in the alpha-Fermi-Pasta-Ulam-Tsingou (FPUT) chain, where, in addition, each mass is connected to a nonlinear resonator, i.e., a second mass free to oscillate. Both nonlinearities are quadratic in the equations of motion. This setup represents the simplest prototype of nonlinear wave propagation on a nonlinear metamaterial. In the linear case, we diagonalize the system, and the two branches of the dispersion relation can be found. Using this result, we derive in the nonlinear case the equations of motion for the normal variables in Fourier space, obtaining a system governed by triad interactions among the two branches of the dispersion relation. We find that the transfer of energy between these two branches is ruled by three- and four-wave resonant interactions. We perform numerical simulations of the primitive equations of motion and highlight the role of resonances as an efficient mechanism for transferring energy. Moreover, as predicted by the theory, we provide direct evidence that four-wave resonances appear on a time scale that is longer than the time scale for three-wave resonances. We also assess the recurrence behaviour (usual in the FPUT system) for the nonlinear Kelvin lattice, and we show that, while recurrence is observed if all the energy is placed, at time t=0, in the lowest mode of the acoustical branch, a non-recurrent behaviour is observed if the initial energy is located in the optical branch.

nlin.PS↗

arXiv2017

2nd order PDEs: geometric and functional considerations

INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the applied physicist, but with a weakness for formalization: look inside the black box of the formulas, try to compact them (for example, proceeding from an inverse transformation of coordinates) and make them smart (in the context, reformulating the theory by means of differential operators and related invariants), applying them with awareness and then connecting them to geometry or to spatial categories, which are in mathematics what is closest to the sensible reality. Finally, proposing examples that are exercise and corroborating for theory. TOPICS The geometric meaning of invariant to a differential operator. Operator Principal Part and its factorization: commutativity and product with and without residues(first order terms). Related conditions by operators and invariants derivatives. Coordinate transformation by invariants and expression of the hyperbolic and parabolic operators in the new coordinates. Properties of the Jacobian Matrix and relations between invariants derivatives and inverse coordinates transformation or the initial variables derivatives. Commutativity conditions and product without residues in terms of inverse coordinate transformations that allow to build commutative differential operators or whose product is without residues (or both). Diffeomorphisms and plane transformations: new operators and invariants in the new coordinate space which lead to the chain rule in compact form. Conclusive considerations and examples who compares different methods of solution.

math.GM↗