Branching of Cantor manifolds of elliptic tori and applications to PDEs. (English) Zbl 1230.37092
The authors consider a Hamiltonian system with infinitely many degrees of freedom and investigate the dynamics in the vicinity of an elliptic invariant torus. They construct Cantor manifolds of tori of arbitrarily large, finite dimension, which accumulate on a given, lower dimensional, KAM elliptic torus.
The proof is based on an averaging procedure and on an improved KAM theorem, whose main advantage consists in an explicit description of the Cantor set of “good” parameters in terms of the final frequency. This considerably simplifies the measure estimates.
Moreover, a new branching phenomenon of Cantor manifolds of elliptic tori of increasing dimension is detected, namely that in the neighborhood of an elliptic equilibrium, there are finitely many branches of tori bifurcating from Cantor manifolds of finite dimension. In this setting, passing from finite to infinite seems a very deep and interesting problem.
As an application, the authors construct new solutions of the nonlinear wave equation that accumulate on a torus.
Finally, a positive answer is given to a conjecture of Bourgain, showing the existence of elliptic tori whose tangential frequency is constrained to a fixed Diophantine direction.
The proof is based on an averaging procedure and on an improved KAM theorem, whose main advantage consists in an explicit description of the Cantor set of “good” parameters in terms of the final frequency. This considerably simplifies the measure estimates.
Moreover, a new branching phenomenon of Cantor manifolds of elliptic tori of increasing dimension is detected, namely that in the neighborhood of an elliptic equilibrium, there are finitely many branches of tori bifurcating from Cantor manifolds of finite dimension. In this setting, passing from finite to infinite seems a very deep and interesting problem.
As an application, the authors construct new solutions of the nonlinear wave equation that accumulate on a torus.
Finally, a positive answer is given to a conjecture of Bourgain, showing the existence of elliptic tori whose tangential frequency is constrained to a fixed Diophantine direction.
Reviewer: Enrico Valdinoci (Roma)
MSC:
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35L71 | Second-order semilinear hyperbolic equations |
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