Local well-posedness for quasi-linear NLS with large Cauchy data on the circle

TL;DR

This paper establishes local well-posedness for a broad class of quasilinear Schrödinger equations on the circle by transforming the system into a form with constant, purely imaginary symbols, enabling energy estimates.

Contribution

It introduces a novel transformation technique for quasilinear NLS on the circle, facilitating well-posedness proofs for large initial data.

Findings

Proves local well-posedness for large Cauchy data.

Transforms quasilinear NLS into a paradifferential form with constant, imaginary symbols.

Provides a priori energy estimates for solutions.

Abstract

We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schr\"odinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.