Quantum nonlinear Schrödinger equation and its classical counterpart (Q1078399)

The author considers the quantum nonlinear Schrödinger equation of the form \[ i\psi_ t=-\psi_{xx}+2c\psi^{\dag}(x)\psi (x)^ 2, \] where \(\psi^{\dag}(x,t)\) and \(\psi\) (y,t) are the time-dependent creation and annihilation operators respectively in the Fock space \({\mathcal H}=\oplus^{\infty}_{N=0}{\mathcal H}_ N\) satisfying the canonical commutation relations \[ [\psi (x,t), \psi (y,t)]=0,\quad [\psi (x,t), \psi^{\dag}(y,t)]=\delta (x-y), \] and such that \(\psi\) (x,t) annihilate the vacuum vector \(\Omega\in {\mathcal H}_ 0\). This problem and its classical counterpart are discussed.