Long time stability of Hamiltonian Korteweg–de Vries equation

We investigate the long time stability of the solutions to the Korteweg–de Vries equation under periodic boundary condition. It is proved that if the initial datum is ɛ-small in the modified Sobolev space, then their evolution remains in a ball of radius 2ɛ for time intervals of order ε−94⋅e1ε. Our research enhances the previously established stability time of ɛ−1 from the local existence theory to an exponential scale of e1ε. This result can be realized by applying Birkhoff normal form method as well as the properties of topological spaces and the structure of normal forms.