130,482 research outputs found
De Toda à KdV
Get PDFRésumé:
On considère la limite à grand nombre de particules d'un système hamiltonien de type « Toda périodique » pour une famille de conditions initiales proches de la solution d'équilibre. On montre que, dans la formulation de paire de Lax, les deux bords des spectres des matrices de Jacobi des conditions initiales sont déterminés, à une erreur près, par ceux de deux opérateurs de Hill, associés à la famille de conditions initiales considérées. On en déduit que les spectres des matrices de Jacobi, lors de l'évolution limite donnée par KdV, restent constants à une erreur près que nous estimons. Enfin on montre que les actions du système Toda, convenablement renormalisées, tendent vers celles des deux équations de KdV. Pour citer cet article : D. Bambusi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).
Abstract:
We consider the large number of particles limit of a periodic Toda lattice for a family of initial data close to the equilibrium state. We show that each of the two edges of the spectra of the corresponding Jacobi matrices is up to an error, determined by the spectra of two Hill operators, associated to this family. We then show that the spectra of the Jacobi matrices remain almost constant when the matrices evolve along the two limiting KdV equations. Finally we prove that the Toda actions, when appropriately renormalized, converge to the ones of KdV. To cite this article: D. Bambusi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009)
On the Stable Eigenvalues of Perturbed Anharmonic Oscillators in Dimension Two
Get PDFWe 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 admits an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in Bambusi et al. (Commun Part Differ Equ 45:1-18, 2020) (see also Parnovski and Sobolev in Invent Math 181(3):467-540, 2010) for the particular case of T-d
Small amplitude breathers in 1D and 2D Klein-Gordon lattices
No full textWe construct small amplitude breathers in one-dimensional (1D) and two-dimensional (2D) Klein-Gordon (KG) infinite lattices. We also show that the breathers are well-approximated by the ground state of the nonlinear Schrdinger equation. The result is obtained by exploiting the relation between the KG lattice and the discrete nonlinear Schrdinger model. The proof is based on a Lyapunov-Schmidt decomposition and continuum approximation techniques introduced in [Bambusi and Penati, Continuous approximation of ground states in DNLS lattices, Nonlinearity 23 (2010), pp. 143-157], actually using its main result as an important lemma
Boundary effects on the dynamics of chains of coupled oscillators
No full textWe 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 a 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 (Bambusi et al 2007 Phys. Lett. A)
Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow
Get PDFIn this paper we prove an abstract result of almost global existence for small and smooth solutions of some semilinear PDEs on Riemannian manifolds with globally integrable geodesic flow. Some examples of such manifolds are Lie groups (including flat tori), homogeneous spaces and rotational invariant surfaces. As applications of the abstract result we prove almost global existence for a nonlinear Schrödinger equation with a convolution potential and for a nonlinear beam equation. We also prove Hs stability of the ground state in NLS equation. The proof is based on a normal form procedure and the combination of the arguments used in Bambusi and Langella (2022 arXiv:2202.04505) to bound the growth of Sobolev norms in linear systems and a generalization of the arguments in Bambusi et al (2024 Commun. Math. Phys. 405 253-85)
Reducibility of 1-d Schrödinger Equation with Time Quasiperiodic Unbounded Perturbations, II
Get PDFWe study the Schrödinger equation on R with a potential behaving as x2l at infinity, l∈ [ 1 , + ∞) and with a small time quasiperiodic perturbation. We prove that if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible
Birkhoff normal form for partial differential equations with tame modulus
No full textWe prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations (PDEs). The theorem applies to semilinear equations with nonlinearity satisfying a property that we call 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 (nonlinear wave (NLW) equation, nonlinear Schrödinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the -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
Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations
No full textWe prove a Nekhoroshev type result [26,27] for the nonlinear Schrödinger equation iut = -uxx - mu - uφ(\u\2) , (0.1) with vanishing or periodic boundary conditions on [0, π]; here m ∈ R is a parameter and φ : R → R, is a function analytic in a neighborhood of the origin and such that φ(0) = 0, φ′(0) ≠ 0. More precisely, we consider the Cauchy problem for (0.1 ) with initial data which extend to analytic entire functions of finite order, and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum. The proof is obtained by a method which applies to Hamiltonian perturbations of linear PDE's with the following properties: (i) the linear dynamics is periodic (ii) there exists a finite order Birkhoff normal form which is integrable and quasi convex as a function of the action variables. Eq. (0.1) satisfies (i) and (ii) when restricted to a level surface of \\u\\L2, which is an integral of motion. The main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here
Asymptotic Stability of Ground States in Some Hamiltonian PDEs with Symmetry
No full textWe consider a ground state (soliton) of a Hamiltonian PDE. We prove that if the soliton is orbitally stable, then it is also asymptotically stable. The main assumptions are transversal nondegeneracy of the manifold of the ground states, linear dispersion (in the form of Strichartz estimates) and nonlinear Fermi Golden Rule. We allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues. The theory is tailor made for the application to the translational invariant NLS in space dimension 3. The proof is based on the extension of some tools of the theory of Hamiltonian systems (reduction theory, Darboux theorem, normal form) to the case of systems invariant under a symmetry group with unbounded generators
- …
