scholarly journals On the analytic Birkhoff normal form of the Benjamin–Ono equation and applications

2022 ◽  
Vol 216 ◽  
pp. 112687
Author(s):  
Patrick Gérard ◽  
Thomas Kappeler ◽  
Petar Topalov
Keyword(s):  
Normal Form ◽  
Birkhoff Normal Form

Quasi-Periodic Solutions of Nonlinear Wave Equation with x-Dependent Coefficients

2015 ◽  
Vol 25 (03) ◽  
pp. 1550043 ◽  
Author(s):  
Yixian Gao ◽  
Weipeng Zhang ◽  
Shuguan Ji
Keyword(s):  
Wave Equation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Seismic Waves ◽  
Birkhoff Normal Form ◽  
General Boundary Conditions ◽  
Kam Theorem ◽  
Infinite Dimensional

This paper is devoted to the study of quasi-periodic solutions of a nonlinear wave equation with x-dependent coefficients. Such a model arises from the forced vibration of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Based on the partial Birkhoff normal form and an infinite-dimensional KAM theorem, we can obtain the existence of quasi-periodic solutions for this model under the general boundary conditions.

An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

2019 ◽  
Vol 375 (3) ◽  
pp. 2089-2153 ◽  
Author(s):  
Luca Biasco ◽  
Jessica Elisa Massetti ◽  
Michela Procesi
Keyword(s):  
Normal Form ◽  
Exponential Type ◽  
Birkhoff Normal Form ◽  
Form Theorem ◽  
Normal Form Theorem

scholarly journals LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS

2020 ◽  
Vol 8 ◽  
Author(s):  
JOACKIM BERNIER ◽  
ERWAN FAOU ◽  
BENOÎT GRÉBERT
Keyword(s):  
Normal Form ◽  
Weak Interactions ◽  
Birkhoff Normal Form ◽  
Global Dynamics ◽  
General Strategy ◽  
Time Behavior ◽  
Dimensional Torus ◽  
Nonresonance Condition ◽  
Long Time ◽  
Form Transformation

We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

scholarly journals Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part II. Abstract splitting

2009 ◽  
Vol 114 (3) ◽  
pp. 459-490 ◽  
Author(s):  
Erwan Faou ◽  
Benoît Grébert ◽  
Eric Paturel
Keyword(s):  
Normal Form ◽  
Splitting Methods ◽  
Birkhoff Normal Form ◽  
Hamiltonian Pdes

scholarly journals On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form

1996 ◽  
Vol 140 (2) ◽  
pp. 335-358 ◽  
Author(s):  
D Bättig ◽  
T Kappeler ◽  
B Mityagin
Keyword(s):  
Normal Form ◽  
Vries Equation ◽  
Birkhoff Normal Form ◽  
De Vries

Birkhoff Normal Form for Some Nonlinear PDEs

2003 ◽  
Vol 234 (2) ◽  
pp. 253-285 ◽  
Author(s):  
Dario Bambusi
Keyword(s):  
Normal Form ◽  
Nonlinear Pdes ◽  
Birkhoff Normal Form

scholarly journals Chaotic-Like Transfers of Energy in Hamiltonian PDEs

2021 ◽  
Author(s):  
Filippo Giuliani ◽  
Marcel Guardia ◽  
Pau Martin ◽  
Stefano Pasquali
Keyword(s):  
Normal Form ◽  
Time Scales ◽  
Symbolic Dynamics ◽  
Exchange Energy ◽  
Birkhoff Normal Form ◽  
Smale Horseshoe ◽  
Beam Equations ◽  
Different Types ◽  
Hamiltonian Pdes

AbstractWe consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $${\mathbb {T}}^2$$ T 2 and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered “chaotic-like” since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.

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