| Abstract: |
| We investigate the transfer of energy to high frequencies in a quasi-linear Schr\{o}dinger equation with sublinear dispersion relation on the one-dimensional torus. Specifically, we construct initial data that undergo finite but arbitrarily large Sobolev norm explosions: starting with arbitrarily small norms in Sobolev spaces of high regularity, these norms become arbitrarily large at later times.
Our analysis introduces a novel instability mechanism. By applying para-differential normal forms, we derive an effective equation that governs the dynamics, whose leading term is a non-trivial transport operator with non-constant coefficients. Using a positive commutator method, inspired by Mourre`s commutator theory, we demonstrate that this operator drives the energy cascade, leading to the observed instability. |
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