The nonlinear Schrodinger equation is $$ ih\frac{\partial \psi}{\partial t} = -\frac{h^2}{2}\Delta \psi + V\psi-|\psi|^{p-1}\psi $$ From Wiki, I know the Hamiltonian system is
And from Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, the author say the nonlinear Schrodinger equation (NLS) is Hamiltonian system. But for the given $E(\phi)$ , what is $p$ and $q$ as in definition of Hamiltonian system?
Define the Hamiltonian density functional $\mathcal{H}$ as $$ \mathcal{H}=\frac{\hbar^2}{2}\nabla\Psi\cdot\nabla\Psi^*+(V-|\Psi|^p)\Psi^*\Psi. $$ The evolution equation is given by $$ i\hbar\frac{\partial\Psi}{\partial t}=\frac{\delta\mathcal{H}}{\delta\Psi^*}=\{\Psi,\mathcal{H}\}, $$ where the commutator between two functionals $\mathcal{F}$ and $\mathcal{G}$ is given by $$ \{\mathcal{F},\mathcal{G}\}=\frac{\delta\mathcal{F}}{\delta\Psi}\frac{\delta\mathcal{G}}{\delta\Psi^*}-\frac{\delta\mathcal{F}}{\delta\Psi^*}\frac{\delta\mathcal{G}}{\delta\Psi}, $$ with $\frac{\delta\mathcal{F}}{\delta\Psi}$ being the functional derivative. The canonical pair analogous to $(q,p)$ is given here by $(\Psi,\Psi^*)$.