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

Abstract

We consider the dispersive Degasperis-Procesi equation ut−uxxt−cuxxx+4cux−uuxxx−3uxuxx+4uux = 0 with c ∈ R \ {0}. In [12] 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 well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis-Procesi at any order is action-preserving. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry".

Reducible KAM Tori for the Degasperis–Procesi Equation

“…This means that a resonant monomial contributes to H Birk if and only if it is resonant for all the constants of motion. This was proved in detail in [32] at the level of formal power series. Here we adapt this result to the Eq.…”

“…A way to deal with this problem is to exploit the integrability of the DP equation. In [32] the authors construct an infinite number of conserved quantities K n for the Eq. (1.1) with f = 0 starting from the ones given in [28].…”

“…In order to prove the Proposition 3.2 above we need some preliminary results proved in detail in [32].…”

“…In [32] it has been proved that Ker(H (2) ) − triv {F (3) , H (3) } = 0 (see the notation of Definition 7.8) and hence…”

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

“…Other applications of normal form techniques to get long-time existence for nonlinear PDEs on compact domains can be found in the work of Bambusi, Nekhoroshev, Grébert, Delort and Szeftel, see e.g. [12], [11], [10], and Feola, Giuliani and Pasquali [24]. approach and by PRIN 2015 Variational methods, with applications to problems in mathematical physics and geometry.…”