First Hamiltonian structure of the KdV equation ($J=-2{\partial }_x$)

One normalization of the first Hamiltonian structure uses the constant differential Poisson operator $J=-2{\partial }_x$. For functionals with variational derivatives $f,g$, its Poisson bracket is $\int fJgdx=-2\int f{g}^{\prime }dx$ under periodic or vanishing-boundary conventions. Thus the field bracket is $\{u(x),u(y)\}=-2{\partial }_x\delta (x-y)$. The factor and sign change with normalization; the constant first-order operator is the invariant structural feature.