Noticias en español
Strongly interacting kink-antikink pairs for scalar fields on a line
Abstract: I will present a recent joint work with Michał Kowalczyk and Andrew Lawrie. A nonlinear wave equation with a double-well potential in 1+1 dimension admits stationary solutions called kinks and antikinks, which are minimal energy solutions connecting the two minima of the potential. We study solutions whose energy is equal to twice the energy of a kink, which is the threshold energy for a formation of a kink-antikink pair. We prove that, up to translations in space and time, there is exactly one kink-antikink pair having this threshold energy. I will explain the main...
Read MoreVortices induced by topological forcing in nematic liquid crystal layes
Abstract: Liquid crystals with negative anisotropic dielectric constant and homeotropic anchoring are a natural physical context where dissipative vortices are observed. Dissipative vortices are known in this context as \emph{umbilical defects}.Major problems arise when practical implementations are aimed at, because soft-matter defects are dissipative structures that obey a complex Ginzburg-Landau equation (CGLE) and undergo a coarsenig dynamics ruled by their mutual interaction and annhilation. Therefore, they are unstable, usually limited to a single defect pair per sample or a...
Read MoreDecay in the one dimensional generalized improved Boussinesq equation.
Abstract: The purpose of this talk is to explain the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity $p$ is sufficiently large. In this talk we remove that condition on the power $p$ and prove decay to zero in terms of the energy space norm $L^2\times H^1$, for any $p>1$, in two...
Read MoreDispersive blow-up and persistence properties for the Schrödinger-Korteweg-de Vries system.
Abstract: In this talk, we shall prove the existence of dispersive blow-up for the Schrödinger-Korteweg-de Vries system. Roughly, dispersive blow-up has being called to the development of point singularities due to the focussing of short or long waves. In mathematical terms, we show that the existence of this kind of singularities is provided by the linear dispersive solution by proving that the Duhamel term is smoother. This result is the first one regarding systems of nonlinear dispersive equations. To obtain our results we use, in addition to smoothing properties, persistence properties...
Read MoreConnecting orbits in Hilbert spaces and applications to P.D.E.
Abstract: Functional Analysis methods are often useful to solve efficiently P.D.E. problems. The idea is to view a solution of a P.D.E. as a map with values in a space of functions, and reduce the initial P.D.E. to an O.D.E. problem. I will establish a general theorem on the existence of heteroclinic orbits in Hilbert spaces. As a first application, I will present a new construction, in a more general setting, of the heteroclinic double layers (initially constructed by Schatzman). As a second application, I will show the existence of the heteroclinic double layers for a fourth order...
Read MoreProfile of touch-down solution to a non-local MEMS model
Abstract: In this talk, I am at showing the construction of solutions with extinction (touch-down) for the equation of MEMS devices (Micro-electromechanical systems). In addition to that, we also describe the asymptotic behavior of the solution at the extinction domain.
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