Exact Global Control of Small Divisors in Rational Normal Form

Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schr\"{o}dinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.

[1]  Lufang Mi,et al.  Exponential stability estimate for the derivative nonlinear Schrödinger equation , 2022, Nonlinearity.

[2]  Qiaoling Chen,et al.  Long time stability result for 1-dimensional nonlinear Schrödinger equation , 2022, Journal of Differential Equations.

[3]  J. Bernier,et al.  Long Time Dynamics for Generalized Korteweg–de Vries and Benjamin–Ono Equations , 2020, Archive for Rational Mechanics and Analysis.

[4]  Jessica Elisa Massetti,et al.  An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS , 2020 .

[5]  Lufang Mi,et al.  A Nekhoroshev type theorem for the derivative nonlinear Schrödinger equation , 2020 .

[6]  Jing Zhang.  Almost Global Solutions to Hamiltonian Derivative Nonlinear Schrödinger Equations on the Circle , 2019, Journal of Dynamics and Differential Equations.

[7]  E. Faou,et al.  LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE $d$-DIMENSIONAL TORUS , 2019, Forum of Mathematics, Sigma.

[8]  Chunyong Liu,et al.  A Nekhoroshev type theorem for the nonlinear wave equation , 2020 .

[9]  E. Faou,et al.  Rational Normal Forms and Stability of Small Solutions to Nonlinear Schrödinger Equations , 2018, Annals of PDE.

[10]  M. Berti,et al.  Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle , 2018 .

[11]  D. Bambusi,et al.  A large probability averaging theorem for the defocusing NLS , 2018, Nonlinearity.

[12]  Yunfeng Shi,et al.  The Stability of Full Dimensional KAM tori for Nonlinear Schr\"odinger equation , 2017, 1705.01658.

[13]  Jean-Marc Delort,et al.  Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres , 2015 .

[14]  Jing Zhang,et al.  Long Time Stability of Hamiltonian Partial Differential Equations , 2014, SIAM J. Math. Anal..

[15]  L. Jianjun,et al.  Stability of KAM tori for nonlinear Schrodinger equation , 2014, 1404.7557.

[16]  Erwan Faou,et al.  A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus , 2013 .

[17]  Qidi Zhang.  Long-Time Existence for Semi-Linear Klein–Gordon Equations with Quadratic Potential , 2008, 0811.1152.

[18]  Benoît Grébert,et al.  Normal Forms for Semilinear Quantum Harmonic Oscillators , 2008, 0808.0995.

[19]  Dario Bambusi,et al.  Birkhoff normal form for partial differential equations with tame modulus , 2006 .

[20]  B. Grébert,et al.  Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds , 2005, math/0510292.

[21]  Dario Bambusi,et al.  Birkhoff Normal Form for Some Nonlinear PDEs , 2003 .

[22]  Jean Bourgain,et al.  On diffusion in high-dimensional Hamiltonian systems and PDE , 2000 .

[23]  Dario Bambusi,et al.  Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations , 1999 .

[24]  Jean Bourgain,et al.  Construction of approximative and almost periodic solutions of perturbed linear schrödinger and wave equations , 1996 .

[25]  Jean-Marc Delort,et al.  A quasi-linear Birkhoff normal forms method : application to the quasi-linear Klein-Gordon equations on S[1] , 2012 .

[26]  Dario Bambusi,et al.  A Birkhoff normal form theorem for some semilinear PDEs , 2008 .

[27]  J. Pöschel,et al.  Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation , 1996 .

[28]  Yu. L. Dalecky,et al.  Gaussian measures in Hilbert space , 1991 .