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This paper is devoted to the construction of periodic solutions of nonlinear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash–Moser methods. Our approach avoids this, exploiting the possibility of reducing, through paradifferential conjugation, the equation under study to an equivalent form for which periodic solutions may be constructed by a classical iteration scheme.
For all dimensions we prove there exist solutions to the focusing cubic nonlinear Schrödinger equations that blow up on a set of codimension two. The blowup set is identified both as the site of concentration and by a bounded supercritical norm outside any neighborhood of the set. In all cases, the global norm grows at the log-log rate.
We consider the resonances of a quantum graph that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of in a disc of a large radius. We call a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of . We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.
We prove some improved estimates for the Ginzburg–Landau energy (with or without a magnetic field) in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localization of the “ball construction method” combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy “displaced” from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite order “renormalized energy” of vortex interaction, up to the best possible precision, i.e., with only a error per vortex, and is complemented by local compactness results on the vortices. Besides being used crucially in a forthcoming paper, our result can serve to provide lower bounds for weighted Ginzburg–Landau energies.