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

  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},
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
On the existence time for the Kirchhoff equation with periodic boundary conditions
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
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
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
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
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
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
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


Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation
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
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
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
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
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
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
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