Bifurcating electric wires

The stationary states of a string through which an electric current is sent and which is placed in an axial magnetic field, are investigated. Using methods of constrained variational principles, it is shown that, in case the string is inextensible, only those stationary states which have least total potential energy are stable

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We show that, given an approximate solution of the invariance equation which satisfies some non-degeneracy conditions, there is a true solution nearby. We call this an {\sl a posteriori} approach. The proof of the main theorems is based on an iterative method to solve the functional equation. The theorems do not assume that the system is close to integrable nor that it is written in action-angle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant. The a posteriori formulation allows us to justify approximate solutions produced by many non-rigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on succesive corrections. This makes it possible to adapt the method almost verbatim to several infinite-dimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.Comment: To appear in JD

Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

We construct quasi-periodic and almost periodic solutions for coupled Hamiltonian systems on an infinite lattice which is translation invariant. The couplings can be long range, provided that they decay moderately fast with respect to the distance. For the solutions we construct, most of the sites are moving in a neighborhood of a hyperbolic fixed point, but there are oscillating sites clustered around a sequence of nodes. The amplitude of these oscillations does not need to tend to zero. In particular, the almost periodic solutions do not decay at infinity. We formulate an invariance equation. Solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation. We show that, if there is an approximate solution of the invariance equation that satisfies some non-degeneracy conditions, there is a true solution close by. The proof of this \emph{a-posteriori} theorem is based on a Nash-Moser iteration, which does not use transformation theory. Simpler versions of the scheme were developed in E. Fontich, R. de la Llave,Y. Sire \emph{J. Differential. Equations.} {\bf 246}, 3136 (2009). One technical tool, important for our purposes, is the use of weighted spaces that capture the idea that the maps under consideration are local interactions. Using these weighted spaces, the estimates of iterative steps are similar to those in finite dimensional spaces. In particular, the estimates are independent of the number of nodes that get excited. Using these techniques, given two breathers, we can place them apart and obtain an approximate solution, which leads to a true solution nearby. By repeating the process infinitely often, we can get solutions with infinitely many frequencies which do not tend to zero at infinity.Comment: This is a revised version of the paper located at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=12-2

Non-radial sign-changing solutions for the Schroedinger-Poisson problem in the semiclassical limit

We study the existence of nonradial sign-changing solutions to the Schroedinger-Poisson system in dimension N>=3. We construct nonradial sign-changing multi-peak solutions whose peaks are displaced in suitable symmetric configurations and collapse to the same point. The proof is based on the Lyapunov-Schmidt reduction