Rational normal forms and stability of small solutions to nonlinear Schrödinger equations

Abstract

We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\epsilon$, for generic initial data of size $\epsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order ${\epsilon }^{M+1}$. This implies that for such initial data $u(0)$ we control the Sobolev norm of the solution $u(t)$ for time of order ${\epsilon }^{-M}$. Furthermore this property is locally stable: if $v(0)$ is sufficiently close to $u(0)$ (of order ${\epsilon }^{3/2}$) then the solution $v(t)$ is also controled for time of order ${\epsilon }^{-M}$.

References in corpus (5)

• Birkhoff Normal Form and Hamiltonian PDEs
• Resonant dynamics for the quintic non linear Schrödinger equation
• Energy cascades for NLS on the torus
• KAM for beating solutions of the quintic NLS
• Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions

Cited by in corpus (5)

• Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations
• On the stability of periodic multi-solitons of the KdV equation
• Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori
• Exponential stability of solutions to the Schr{ö}dinger-Poisson equation
• Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$