One dimensional energy cascades in a fractional quasilinear NLS

TL;DR

This paper demonstrates how energy can transfer to high frequencies in a one-dimensional fractional quasilinear Schrödinger equation, leading to large Sobolev norm growth from small initial data, revealing a new instability mechanism.

Contribution

It introduces a novel instability mechanism based on paradifferential normal forms and Mourre's commutator theory for energy cascades in a fractional quasilinear NLS.

Findings

Finite but arbitrary large Sobolev norm explosion from small initial data

Identification of a transport operator responsible for energy cascades

Development of a new analytical framework for instability analysis

Abstract

We consider the problem of transfer of energy to high frequencies in a quasilinear Schr\"odinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, which is based on extracting, via paradifferential normal forms, an effective equation driving the dynamics whose leading term is a non-trivial transport operator with non-constant coefficients. We prove that such operator is responsible for energy cascades via a positive commutator estimate inspired by Mourre's commutator theory.