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

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