Long time existence for fully nonlinear NLS with small Cauchy data on the circle

By R. Feola, Felice Iandoli. 2018

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In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrodinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and of size $\varepsilon$ sufficiently small, the lifespan of the solution is of order $\varepsilon^{-N}$ for any $N\in\mathbb{N}$ if some non resonance conditions are fulfilled. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e9f2d190fe29bddda7a8c069369a85eb44b4b440