Completely integrable equation for the quantum correlation function of nonlinear Schrödinger equation

Correlation functions of exactly solvable models can be described by differential equations [1]. In this paper we show that for the non-free fermionic case, differential equations should be replaced by integro-differential equations. We derive an integro-differential equation, which describes a time and temperature dependent correlation function 〈ψ(0,0)ψ^†(x,t)〉_T of the penetrable Bose gas. The integro-differential equation turns out to be the continuum generalization of the classical nonlinear Schrodinger equation.