Infinite dimensional symplectic capacity and nonsqueezing property for the Zakharov system on the $1$-dimensional torus

TL;DR

This paper establishes the invariance of symplectic capacity for the Zakharov system on a 1D torus, leading to a nonsqueezing property by analyzing a modified infinite dimensional Hamiltonian system.

Contribution

It introduces a new approach to prove symplectic capacity invariance for the Zakharov system, relaxing conditions of previous Hamiltonian systems and connecting it to nonsqueezing.

Findings

Proves symplectic capacity invariance for the Zakharov system.

Shows the Zakharov solution map acts as a symplectomorphism.

Establishes the nonsqueezing property for the system.

Abstract

We prove the invariant of the symplectic capacity for the Zakharov system on a torus. If the Zakharov solution map is well-defined, then it can be regarded as a symplectomorphism. Thus, we first show the global well-posedness via the local well-posedness and the conservation law. The invariant of the symplectic capacity can be obtained using an approximation method. Many authors use an approximation method to obtain the nonsqueezing theorem, instead of an invariant of the symplectic capacity. However, the conditions of the Hamiltonian system introduced by Kuksin can be relaxed by a new modified infinite dimensional Hamiltonian system. Thus we can back to the symplectic capacity which contains the nonsqueezing property. Heuristically, we obtain the invariant by using the Hamiltonian system which has linear flow at high frequencies and nonlinear flow at low frequencies.