Noticias en español
Uniqueness and stability of semi-wavefronts for KPP-Fisher equation with delay
Abstract: In this talk I will preset some recent results on the stability and uniqueness of semi-wavefronts of the equation u_t(t,x)=u_{xx}(t,x)+u(t,x)(1-u(t-h,x)), t >0, x in \R; where the parameter h>0 is a delay. The uniqueness (up to translations) of semi-wavefronts (i.e., solutions in the form u(t,x)=\phi_c(x+ct) satisfying $\phi_c(-\infty)=0$ and $\liminf_{z\to +\infty}\phi_c(z)>0$) is `largely open’ problem. By a simple approach we have obtained the uniqueness (up to translations) of semi-wavefronts for all speed, i.e., c >= 2, and the stability on each...
Read MoreBreathers and the dynamics of solutions to the KdV type equations
Abstract: Our first aim is to identify a large class of non-linear functions f(⋅) for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or “small” breathers solutions. Also we prove that all small, uniformly in time L^1 ∩ H^1 bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order t^1/2 around any compact set in space. This set is included in the linearly dominated dispersive region x≪t. Moreover, we prove this result independently of the...
Read MoreSEMINAR CAPDE de EDPs
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...
Read MoreDynamics of strongly interacting 2-solitons for dispersive equations
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...
Read MoreOn breather solutions of some hierarchies of nonlinear dispersive equations
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.
Read MoreSemiclassical Trace Formula and Spectral Shift Function for Schrödinger Operators with Matrix-Valued Potentials.
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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