2 4 M ay 2 00 5 Solution of polynomial systems derived from differential equations

Tau Numerical Solution of Volterra Integro-Differential Equations With Arbitrary Polynomial Bases

5 M ay 2 00 8 Local time steps for a finite volume scheme

Contents 1 Introduction 3 2 Description of the local time stepping strategy 4 2.

M ay 2 00 2 Center conditions : Rigidity of logarithmic differential equations

In this paper we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko’s result on Hamiltonian differential equations. The main tools are PicardLefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A’Campo’s theorem o...

1 M ay 2 00 5 What does integrability of finite - gap / soliton potentials mean ?

In the example of the Schrödinger/KdV equation we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville’s integrability of finite-dimensional Hamiltonian systems (stationary KdV–equations). Three key objects in this field: the explicit Ψ-function, trace formula and...

ar X iv : 0 80 5 . 37 89 v 1 [ m at h . A P ] 2 4 M ay 2 00 8 The balance between diffusion and absorption in semilinear parabolic equations ∗

Let h : [0,∞) 7→ [0,∞) be continuous and nondecreasing, h(t) > 0 if t > 0, and m,q be positive real numbers. We investigate the behavior when k → ∞ of the fundamental solutions u = uk of ∂tu − ∆u m + h(t)u = 0 in Ω × (0, T ) satisfying uk(x, 0) = kδ0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0, 0), or a solution of the asso...