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Hamiltonian form of the cubic nonlinear Schrodinger equations

The focusing and defocusing equations

i ψ_{t} + ψ_{xx} \pm 2∣ ψ ∣^{2} ψ = 0

have Hamiltonians

H_{\pm} = \int_{R} (∣ ψ_{x} ∣^{2} \mp ∣ ψ ∣^{4}) d x

under the convention

i ψ_{t} = δ H_{\pm} / δ \overline{ψ}

.

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