Related papers: A Birkhoff--Lewis Type Theorem for Some Hamiltonia…
Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for…
Determination of periodic orbits for a Hamiltonian system together with their semi-classical quantization has been a long standing problem. We consider here resonances for a $h$-Pseudo-Differential Operator $H(y,hD_y;h)$ induced by a…
This paper is devoted to the study of periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients. Such mathematical model may be described the infinitesimal, free, undamped in-plane bending vibrations of a…
Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is…
We consider the non linear wave equation (NLW) on the d-dimensional torus with a smooth nonlinearity of order at least two at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up…
Periodic orbits for the classical $\phi^4$ theory on the one dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, fixed and free boundary conditions. Through the process, we…
In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0,i_\infty$ at the…
Quantum manifestations of the dynamics around resonant tori in perturbed Hamiltonian systems, dictated by the Poincar\'e--Birkhoff theorem, are shown to exist. They are embedded in the interactions involving states which differ in a number…
The goal of this work is to study the existence of quasi-periodic solutions in time to nonlinear beam equations with a multiplicative potential. The nonlinearities are required to only finitely differentiable and the frequency is along a…
We develop an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. The Hamiltonian, averaged over one of the planetary mean longitude, is expanded in power series of eccentricities and…
We prove the existence of time-periodic solutions to non-linear massive Klein-Gordon equations in Anti-de Sitter as well as their orbital stability over exponentially long times for certain values of the mass corresponding to completely…
Bifurcations of periodic orbits as an external parameter is varied are a characteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of phase space…
We consider an undamped nonlinear hinged-hinged beam with stretching nonlinearity as an infinite dimensional hamiltonian system. We obtain analytically a quantitative Birkhoff Normal Form, via a nonlinear coordinate transformation that…
We prove the existence of infinitely many nontrivial weak periodic solutions for a class of fractional Kirchhoff problems driven by a relativistic Schr\"odinger operator with periodic boundary conditions and involving different types of…
We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be catalogued according to the minimal period and the number of…
One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the…
We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the…
We consider Hamiltonian diffeomorphisms of the Euclidean space, generated by compactly supported time-dependent perturbations of hyperbolic quadratic forms. We prove that, under some natural assumptions, such a diffeomorphism must have…
This paper is devoted to the study of periodic (in time) solutions to an one-dimensional semilinear wave equation with $x$-dependent coefficients under various homogeneous boundary conditions. Such a model arises from the forced vibrations…
In this paper we study second order non-linear periodic systems driven by the ordinary vector $p$-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical…