arXiv subjects

D. Bambusi

Publications and source records attributed to D. Bambusi.

9 recordsLinked to original sources

arXiv2021

On the stable eigenvalues of perturbed anharmonic oscillators in dimension two

We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr-Sommerfeld quantization rule applied to the normal form Hamiltonian and also admit an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in [BLM20b] (see also [PS10]) for the particular case of $T^d$.

math-ph↗

arXiv2017

Asymptotic stability of synchronous orbits for a gravitating elastic sphere

We study the dynamics of an elastic body whose shape and position evolve due to the gravitational forces exerted by a pointlike planet. We work in the quadrupole approximation. We consider the solution in which the center of mass of the body moves on a circular orbit, and the body rotates in a synchronous way about its axis, so that it always shows the same face to the planet as the Moon does with the Earth. We prove that if any internal deformation of the body dissipates some energy, then such an orbit is locally asymptotically stable. The proof is based on the construction of a suitable system of coordinates and on the use of LaSalle's principle. A large part of the paper is devoted to the analysis of the kinematics of an elastic body interacting with a gravitational field. We think this could have some interest in itself.

math-ph↗

arXiv2013

Normal Forms, symmetry, and linearization of dynamical systems

We discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system - or a parameter dependent family - via the Poincaré Normal Form approach. We discuss this at first formally, and later pay attention to the convergence of the linearizing procedure. We also discuss some generalizations of our main result

math-ph↗

arXiv2012

On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential

In 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.

math.AP↗

arXiv2009

Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein--Gordon lattices

We construct small amplitude breathers in 1D and 2D Klein--Gordon infinite lattices. We also show that the breathers are well approximated by the ground state of the nonlinear Schroedinger equation. The result is obtained by exploiting the relation between the Klein Gordon lattice and the discrete Non Linear Schroedinger lattice. The proof is based on a Lyapunov-Schmidt decomposition and continuum approximation techniques introduced in [7], actually using its main result as an important lemma.

math.DS↗

arXiv2009

Continuous approximation of breathers in one and two dimensional DNLS lattices

In this paper we construct and approximate breathers in the DNLS model starting from the continuous limit: such periodic solutions are obtained as perturbations of the ground state of the NLS model in $H^1(\RR^n)$, with $n=1,2$. In both the dimensions we recover the Sievers-Takeno (ST) and the Page (P) modes; furthermore, in $\RR^2$ also the two hybrid (H) modes are constructed. The proof is based on the interpolation of the lattice using the Finite Element Method (FEM).

math.DS↗

arXiv2008

Boundary effects on the dynamics of chains of coupled oscillators

We study the dynamics of a chain of coupled particles subjected to a restoring force (Klein-Gordon lattice) in the cases of either periodic or Dirichlet boundary conditions. Precisely, we prove that, when the initial data are of small amplitude and have long wavelength, the main part of the solution is interpolated by a solution of the nonlinear Schrödinger equation, which in turn has the property that its Fourier coefficients decay exponentially. The first order correction to the solution has Fourier coefficients that decay exponentially in the periodic case, but only as a power in the Dirichlet case. In particular our result allows one to explain the numerical computations of the paper \cite{BMP07}.

math.DS↗

arXiv2008

Invariant tori for commuting Hamiltonian PDEs

We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\leq n$. The result we get ensures the persistence of an $r$-parameter family of $r$-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunof-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2 dimensional tori, while in the second case we construct 3 dimensional tori.

math.FA↗

arXiv2004

Birfhoff Normal Form for PDEs with Tame Modulus

We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that we call of Tame Modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions and we use it to study some concrete equations (NLW,NLS) with different boundary conditions. An application to a nonlinear Schrödinger equation on the $d$-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular we get lower bounds on the existence time of solutions.

math-ph↗