Completely Integrable Equation for the Quantum Correlation Function of Nonlinear Schr\"odinger Eqaution

Correlation functions of exactly solvable models can be described by differential equation [Barough, McCoy, Wu]. In this paper we show that for non free fermionic case differential equations should be replaced by integro-differential equations. We derive an integro-differential equation, which describes time and temperature dependent correlation function $<\psi(0,0)\psi^\dagger(x,t)>_T$ of penetrable Bose gas. The integro-differential equation turns out be the continuum generalization of classical nonlinear Schr\"odinger equation.