On the integrability of DegasperisProcesi equation: Control of the Sobolev norms and Birkhoff resonances
Abstract
We consider the dispersive DegasperisProcesi equation u_{t} u_{xxt}  cu_{xxx} + 4 cu_{x}  uu_{xxx}  3u_{x}u_{xx} + 4 uu_{x} = 0 with c ∈ R ∖ { 0 }. In [15] the authors proved that this equation possesses infinitely many conserved quantities. We prove that 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 the Sobolev space H^{s} with s ≥ 2, both on R and T. By the analysis of these conserved quantities we deduce a result of global wellposedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the DegasperisProcesi at any order is actionpreserving.
 Publication:

Journal of Differential Equations
 Pub Date:
 March 2019
 DOI:
 10.1016/j.jde.2018.09.003
 arXiv:
 arXiv:1802.00035
 Bibcode:
 2019JDE...266.3390F
 Keywords:

 Mathematics  Analysis of PDEs;
 37K55;
 35L02