A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle

Abstract

In this paper we consider an abstract class of quasi-linear para-differential equations on the circle. For each equation in the class we prove the existence of a change of coordinates which conjugates the equation to a diagonal and constant coefficient para-differential equation. In the case of Hamiltonian equations we also put the system in Poincaré-Birkhoff normal forms. We apply this transformation to quasi-linear perturbations of the Schrödinger and Beam equations, obtaining a long time existence result without requiring any symmetry on the initial data. We also provide the local in time well-posedness for quasi-linear perturbations of the Benjamin-Ono equation.

arXiv commentSome typos have been corrected, the presentation improved

References in corpus (3)

• Time quasi-periodic gravity water waves in finite depth
• Quasi-periodic solutions for fully nonlinear forced reversible Schroedinger equations
• Reducibility of non-resonant transport equation on $T^d$ with unbounded perturbations

Cited by in corpus (5)

• Long-time existence for semi-linear beam equations on irrational tori
• Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori
• On the stability of periodic multi-solitons of the KdV equation
• About linearization of infinite-dimensional Hamiltonian systems
• Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori