# On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances

@article{Feola2019OnTI,
title={On the integrability of Degasperis–Procesi equation: Control of the Sobolev norms and Birkhoff resonances},
author={Roberto Feola and Filippo Giuliani and Stefano Pasquali},
journal={Journal of Differential Equations},
year={2019}
}
• Published 31 January 2018
• Mathematics, Physics
• Journal of Differential Equations
We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$ with $\mathtt{c}\neq 0$. In \cite{Deg} the authors proved that this equation possesses infinitely many conserved quantities. We prove that, in a neighborhood of the origin, there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of some $H^s$ Sobolev space, both on $\mathbb… Expand 9 Citations On the existence time for the Kirchhoff equation with periodic boundary conditions • Mathematics, Physics • 2018 We consider the Cauchy problem for the Kirchhoff equation on$\mathbb{T}^d$with initial data of small amplitude$\varepsilon$in Sobolev class. We prove a lower bound$\varepsilon^{-4}$for theExpand New blow-up criterion for the Degasperis–Procesi equation with weak dissipation • Xijun Deng • Physics • 2021 In this paper, we investigate the Cauchy problem of the Degasperis–Procesi equation with weak dissipation. We establish a new local-in-space blow-up criterion of the dissipative Degasperis–ProcesiExpand Reducible KAM Tori for the Degasperis–Procesi Equation • Mathematics, Medicine • Communications in mathematical physics • 2020 The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Expand A Nekhoroshev type theorem for the nonlinear wave equation • Mathematics • 2020 Abstract We prove a Nekhoroshev type theorem for the nonlinear wave equation (NLW) (0.1) u t t = u x x − m u − f ( u ) under Dirichlet boundary conditions where f is an analytic function. MoreExpand A Nekhoroshev type theorem for the derivative nonlinear Schrödinger equation • Mathematics • 2020 Abstract It is proved a Nekhoroshev type theorem for the derivative nonlinear Schrodinger equation in a Gevrey space. More precisely, we prove that if the norm of initial datum is equal to e / 2 ,Expand About linearization of infinite-dimensional Hamiltonian systems • Mathematics • 2021 This article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of itsExpand Blow-up criteria for the generalized Degasperis–Procesi equation In this paper, we investigate the initial value problem of the generalized Degasperis–Procesi equation. We establish some new local-in-space blow-up criterion. Our results extend the corresponding ... Time quasi-periodic traveling gravity water waves in infinite depth • Mathematics, Physics • 2020 We present the recent result [8] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence resultExpand Quasi-periodic traveling waves on an infinitely deep fluid under gravity • Mathematics, Physics • 2020 We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic inExpand #### References SHOWING 1-10 OF 57 REFERENCES Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation • Mathematics, Physics • 2015 We study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formalExpand Quasi-Linear Perturbations of Hamiltonian Klein-Gordon Equations on Spheres The Hamiltonian$\int_X(\abs{\partial_t u}^2 + \abs{\nabla u}^2 + \m^2\abs{u}^2)\,dx$, defined on functions on$\R\times X$, where$X$is a compact manifold, has critical points which are solutionsExpand Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs • Physics, Computer Science • SIAM J. Appl. Dyn. Syst. • 2003 The effects of the balance parameter b and the kernel g(x) on the solitary wave structures are studied and their interactions analytically for$\nu=0$and numerically for small or zero viscosity are investigated. Expand On well-posedness of the Degasperis-Procesi equation • Mathematics • 2011 It is shown in both the periodic and the non-periodic cases that the data-to-solution map for the Degasperis-Procesi (DP) equation is not a uniformly continuous map on bounded subsets of SobolevExpand On the well-posedness of the Degasperis-Procesi equation • Mathematics • 2006 We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis–Procesi equation (DP)∂tu-∂txx3u+4u∂xu=3∂xu∂xx2u+u∂xxx3u. This equationExpand Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions • Mathematics • 2017 The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data ofExpand A Riemann-Hilbert approach for the Degasperis-Procesi equation • Mathematics, Physics • 2011 We present an inverse scattering transform approach to the Cauchy problem on the line for the Degasperis--Procesi equation$u_t-u_{txx}+3\omega u_x+4uu_x=3u_xu_{xx}+uu_{xxx}\$ in the form of anExpand
Birkhoff normal form for partial differential equations with tame modulus
• Mathematics
• 2006
We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that we call ofExpand
Birkhoff Normal Form for Some Nonlinear PDEs
Abstract: We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear waveExpand
A quasi-linear Birkhoff normal forms method : application to the quasi-linear Klein-Gordon equations on S[1]
Consider a nonlinear Klein-Gordon equation on the unit circle, with small smooth data . A solution u which, for any interger N, may be extended as a smooth solution on a time-interval of lengthExpand