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A Hopf bifurcation of interfaces in a Dirichlet problem.

Ham, YoonMee, Lee, Sang-Gu (1999)

Southwest Journal of Pure and Applied Mathematics [electronic only]

Bifurcation of free vibrations for completely resonant wave equations

Massimiliano Berti, Philippe Bolle (2004)

Bollettino dell'Unione Matematica Italiana

We prove existence of small amplitude, 2p/v-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency $ω$ belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.

Nonlinear oscillations of completely resonant wave equations

Berti, Massimiliano (2007)

Proceedings of Equadiff 11

Nonlinear vibrations of completely resonant wave equations

Massimiliano Berti (2007)

Banach Center Publications

We present recent existence results of small amplitude periodic and quasi-periodic solutions of completely resonant nonlinear wave equations. Both infinite-dimensional bifurcation phenomena and small divisors difficulties occur. The proofs rely on bifurcation theory, Nash-Moser implicit function theorems, dynamical systems techniques and variational methods.

Quasi-periodic solutions of Hamiltonian PDEs

Massimiliano Berti (2011)

Journées Équations aux dérivées partielles

We overview recent existence results and techniques about KAM theory for PDEs.

Quasi-periodic solutions with Sobolev regularity of NLS on $𝕋^{d}$ with a multiplicative potential

Massimiliano Berti, Philippe Bolle (2013)

Journal of the European Mathematical Society

We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on $𝕋^{d}, d \geq 1$ , finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are $C^{\infty}$ then the solutions are $C^{\infty}$ . The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators...