Nekhoroshev type theorem of KdV type equation

Abstract

We prove the exponential stability (namely, Nekhoroshev type theorem) of Korteweg-de-Vries (KdV) type equation with potential term, $$u_t = \partial_x \left(- \partial _{xx} u + V * u + g\left(u\right)\right), \qquad \left(t,x\right) \in \mathbb{R} \times S^1,$$ where $V$ is a smooth convolution potential and $g\left(u\right)$ is certain polynomial of $u$. We can show the periodic KdV equation as an infinite dimensional nearly integrable Hamiltonian. Hence, this result is obtained by the Birkhoff normal form in infinite dimension.