Event Date: Sep 03, 2018 in CAPDE, Seminars

SEMINAR CAPDE de EDPs   Primera Sesión 16:00 hs. Expositor Panayotis Smyrnelis DIM-CMM Universidad de Chile   Title Minimal heteroclinics for second and fourth order O.D.E systems   Segunda Sesión 17:00 hrs.   Expositor Chulkwang Kwak (PUC)   Title Well-posedness issues of some dispersive equations under the periodic boundary condition.   Abstract: In this talk, we are going to discuss about the well-posedness theory of dispersive equations (KdV- and NLS-type equations) posed on T, via analytic methods. I am going to briefly explain some notions and methodologies required to study the...

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Dynamics of strongly interacting 2-solitons for dispersive equations

Event Date: Aug 20, 2018 in CAPDE, Seminars

Abstract:   The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once there are nonlinear effects, many nonlinear dispersive equations (for example: NLS, gKdV, coupled NLS,…) admit special “compact” solutions, called solitary wave or solitons, whose shape does not change in time. A multi-soliton is a solution which is close to a superposition of several solitons. The problem we address is the one of the dynamics of relative distance for...

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On breather solutions of some hierarchies of nonlinear dispersive equations

Event Date: Aug 13, 2018 in CAPDE, Seminars

Abstract:   In this talk I will briefly introduce hierarchies of some nonlinear dispersive equations, namely KdV, mKdV and Gardner hierarchies. We will see that some of  these hierarchies have soliton and breather solutions, suited to the level of the hierarchy. I will show that these soliton and breather solutions satisfy the same nonlinear ODE characterizing them for any member of the hierarchy and I will present a stability result for breather solution of some higher order mKdV equations.

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Semiclassical Trace Formula and Spectral Shift Function for Schrödinger Operators with Matrix-Valued Potentials.

Event Date: Aug 06, 2018 in CAPDE, Seminars

Abstract: In this talk, I will present some recent results on the spectral properties of semiclassical systems of pseudodifferential operators. We developed a stationary approach for the study of the Spectral Shift Function for a pair of self-adjoint Schrödinger operators with matrix-valued potentials. A Weyl-type semiclassical asymptotics with sharp remainder estimate for the SSF is obtained, and under the existence condition of a scalar escape function, a full asymptotic expansion for its derivatives is proved. This last result is a generalization of the result of Robert-Tamura (1984)...

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Operator algebras, compact operators, and the essential spectrum of the N-body problems

Event Date: Jul 23, 2018 in CAPDE, Seminars

Abstract:  I will begin by reviewing a general method to determine the essential spectrum of Schrodinger-type operators. The method is based first on the fact that an operator is Fredholm if, and only if, it is inversible modulo the compacts (Atkinson’s theorem). This reduces the study of certain quotients by the compact operators. To study the invertibility in these quotients, one uses, following Georgescu, Mantoiu, and others, a determination of the spectrum of a suitable operator algebra and of the action of the translation group on its spectrum.   I will give an example of how...

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The singular Yamabe problem and a fully nonlinear generalization & Vortex desingularization for the 2D Euler equations

Event Date: Jul 17, 2018 in CAPDE, Seminars

(16:00 hrs.) Title: The singular Yamabe problem and a fully nonlinear generalization  Abstract: I will begin with an overview of the work of Loewner-Nirenberg on constructing complete conformal metrics of constant negative scalar curvature on domains in Euclidean space, and its extension to Riemannian manifolds with boundary.  I will then describe some fully nonlinear generalizations.  Finally, I will discuss a certain geometric invariant of solutions, called the renormalized volume, and some recent work with Robin Graham on computing closed formulas for these invariants in dimension four....

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