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We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint -pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the...
For a riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the riemannian exponential...
For a Riemannian structure on a
semidirect product of Lie groups, the variational problems can be
reduced using the group symmetry.
Choosing the Levi-Civita connection of a positive definite
metric tensor,
instead of any of the canonical connections for the Lie group,
simplifies the reduction of the variations but complicates the
expression for the Lie algebra valued covariant derivatives.
The origin of the discrepancy is in the semidirect product
structure, which implies that the Riemannian
exponential...
In this paper, the classical Lie theory is applied to study the Benjamin-Bona-Mahony (BBM) and modified Benjamin-Bona-Mahony equations (MBBM) to obtain their symmetries, invariant solutions, symmetry reductions and differential invariants. By observation of the the adjoint representation of Mentioned symmetry groups on their Lie algebras, we find the primary classification (optimal system) of their group-invariant solutions which provides new exact solutions to BBM and MBBM equations. Finally, conservation...
We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Mañé conjecture.
We study relations between functions on the cotangent bundle of a spacetime which are constants of motion for geodesics and functions on the odd-dimensional phase space conserved by the Reeb vector fields of geometrical structures generated by the metric and an electromagnetic field.
We study the resurgent structure associated with a Hamilton-Jacobi equation. This
equation is obtained as the inner equation when studying the separatrix splitting problem
for a perturbed pendulum via complex matching. We derive the Bridge equation, which
encompasses infinitely many resurgent relations satisfied by the formal solution and the
other components of the formal integral.
A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce open sets () of symplectic diffeomorphisms and Hamiltonian systems, exhibitinglargerobustly transitive sets. We show that the closure of such open sets contains a variety of systems, including so-calleda priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of...
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