Symplectic dynamics for infinite dimensional Hamiltonian equations

Abstract

We consider an invariance of the symplectic capacity and the nonsqueezing theorem for infinite dimensional Hamiltonian systems. The symplectic capacity for infinite dimensional Hamiltonian systems was first introduced by Kuksin[34]. This result also contained the invariance of the symplectic capacity for Hamiltonian systems in the specific conditions. Many authors have studied the nonsqueezing theorem for the Hamiltonian systems which do not satisfy Kuksin’s condition. They only proved the nonsqueezing theorem by estimating between original and frequency truncated solution flow, without considering the symplectic capacity. For example, Bourgain [7] proved the nonsqueezing property of the 1D cubic nonlinear $Schr \ddot{o} dinger$ equation. However, we focus back to the symplectic capacity. Applying the idea of Bourgain [7] to the method of Kuksin [34], we relax the conditions of the Hamiltonian system, which was used by Kuksin [34]. Heuristically, we prove the invariance of the symplectic capacity by approximating the solutions to the original infinite dimensional Hamiltonian system by a modified Hamiltonian system which has linear flow on high frequencies and nonlinear flow on low frequencies. We also consider concrete examples such as the higher-order KdV equation and the Zakharov system. The nonsqueezing property of the higher-order Korteweg-de Vries flow was proved by Hong and Kwak [25], but we can extend this result to the symplectic capacity. Furthermore, we also prove the invariance of the symplectic capacity by the Zakharov flow on $L^2_x(\BbbT) \times H^{-1/2}_x,(\BbbT) \times H^{-3/2}_x (\BbbT)$, the sharp space has that the local well-posedness which can be obtained by $X^{s,b}$ space.