Orateur
Description
In this talk, we will discuss finite-time blow-up solutions for the focusing Calogero–Sutherland derivative nonlinear Schrödinger equation on the torus. While global well-posedness below the critical mass threshold was previously known, we construct a family of smooth finite-time blow-up solutions with supercritical mass.
The construction is based on an analysis of the explicit formula. We consider a class of finite-gap potentials and identify a precise resonant condition under which the corresponding solution blows up in finite time. This resonance is detected through the appearance of a unimodular eigenvalue of the operator arising in the explicit formula. We then use the explicit formula to obtain a complete description of the blow-up dynamics, including the blow-up rate of all Sobolev norms.
We will also explain the complementary non-resonant case, where the finite-gap family gives global solutions with uniform-in-time Sobolev bounds. This yields a sharp resonance/non-resonance dichotomy and shows instability of the constructed blow-up solutions. This is joint work with Enno Lenzmann.