## Well-posedness and long time behavior for the Schrödinger-Korteweg-de Vries interactions on the half-Line

Abstract: In this talk we discuss about the initial-boundary value problem for the Schrödinger-Korteweg-de Vries system on the left and right half-line for a wide class of initial-boundary data, including the energy regularity H1(R±) × H1(R±) for initial data. Assuming homogeneous boundary conditions it is shown for positive coupling interactions that local solutions can be extended globally in time for initial data in the energy space; furthermore, for negative coupling interactions it was proved, for a certain class of regular initial data, the following result: if the respective...

Read More## Resonances in deformed tubes: twisting and bending & Gross-Pitaevskii equation and integrable systems

16:00 hrs. Expositor Pablo Miranda, (USach) Título Resonances in deformed tubes: twisting and bending Abstract: In this talk we will consider an infinite straight tube and we will deform it by a periodic twisting and a local bending. On the deformed tube we will define the Laplacian and will study the existence of scattering resonances created by the deformations. We will show the existence of exactly one resonance or one eigenvalue near the bottom of the essential spectrum, depending on the strength of the twisting and the bending. We will also obtain the asymptotic behavior of the...

Read More## 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 More## Breathers 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 More## SEMINAR 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 More## Dynamics 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...

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